MatheMatical ForMulas and scientiFic data. A Quick Reference Guide 9781683928652, 2022952304

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Mathematical Formulas and

Scientific Data

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license, disclaimer of liability, and limited warranty By purchasing or using this book and disc (the “Work”), you agree that this license grants permission to use the contents contained herein, including the disc, but does not give you the right of ownership to any of the textual content in the book / disc or ownership to any of the information or products contained in it. This license does not permit uploading of the Work onto the Internet or on a network (of any kind) without the written consent of the Publisher. Duplication or dissemination of any text, code, simulations, images, etc. contained herein is limited to and subject to licensing terms for the respective products, and permission must be obtained from the Publisher or the owner of the content, etc., in order to reproduce or network any portion of the textual material (in any media) that is contained in the Work. Mercury Learning and Information (“MLI” or “the Publisher”) and anyone involved in the creation, writing, or production of the companion disc, accompanying algorithms, code, or computer programs (“the software”), and any accompanying Web site or software of the Work, cannot and do not warrant the performance or results that might be obtained by using the contents of the Work. The author, developers, and the Publisher have used their best efforts to insure the accuracy and functionality of the textual material and/or programs contained in this package; we, however, make no warranty of any kind, express or implied, regarding the performance of these contents or programs. The Work is sold “as is” without warranty (except for defective materials used in manufacturing the book or due to faulty workmanship). The author, developers, and the publisher of any accompanying content, and anyone involved in the composition, production, and manufacturing of this work will not be liable for damages of any kind arising out of the use of (or the inability to use) the algorithms, source code, computer programs, or textual material contained in this publication. This includes, but is not limited to, loss of revenue or profit, or other incidental, physical, or consequential damages arising out of the use of this Work. The sole remedy in the event of a claim of any kind is expressly limited to replacement of the book and/or disc, and only at the discretion of the Publisher. The use of “implied warranty” and certain “exclusions” vary from state to state and might not apply to the purchaser of this product.

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Mathematical Formulas and

Scientific Data A Quick Reference Guide

C. P. Kothandaraman, PhD

MERCURY LEARNING AND INFORMATION Dulles, Virginia Boston, Massachusetts New Delhi

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Reprint & Revision Copyright © 2023 by Mercury Learning and Information LLC. All rights reserved. Original Copyright © 2021 by NEW AGE International Publishers. This publication, portions of it, or any accompanying software may not be reproduced in any way, stored in a retrieval system of any type, or transmitted by any means, media, electronic display or mechanical display, including, but not limited to, photocopy, recording, Internet postings, or scanning, without prior permission in writing from the publisher. Publisher: David Pallai Mercury Learning and Information 22841 Quicksilver Drive Dulles, VA 20166 [email protected] www.merclearning.com (800) 232-0223 C. P. Kothandaraman. Mathematical Formulas and Scientific Data. ISBN: 978-1-68392-865-2 The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarks or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc. is not an attempt to infringe on the property of others. Library of Congress Control Number: 2022952304 232425321  Printed on acid-free paper in the United States of America. Our titles are available for adoption, license, or bulk purchase by institutions, corporations, etc. For additional information, please contact the Customer Service Dept. at (800) 232-0223(toll free). Digital versions of our titles are available at numerous electronic vendors. The sole obligation of Mercury Learning and Information to the purchaser is to replace the book and/or disc, based on defective materials or faulty workmanship, but not based on the operation or functionality of the product.

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Contents Preface vii PART I: MATHEMATICAL FORMULAS Chapter 1  Algebra Chapter 2   Elementary Geometry Chapter 3  Trigonometry Chapter 4   Analytic Geometry Chapter 5   Differential Calculus Chapter 6   Integral Calculus—I Chapter 7   Integral Calculus—II Chapter 8  Vectors

PART II: SCIENTIFIC DATA FOR ENGINEERS Chapter 1   SI Units Chapter 2   The Fundamental Mechanical Units Chapter 3   The Fundamental Electrical and Magnetic Units Chapter 4   Relations Between the Systems of Electrical Units Chapter 5   Rationalization of MKS Units Chapter 6  Definitions of Electric and Magnetic Quantities in SI Chapter 7  Definitions in the CGS Electromagnetic System of Units Chapter 8   Heat Units and Definitions Chapter 9   Photometric and Optical Units and Definitions Chapter 10  Acoustical Units and Definitions Chapter 11  Properties of the Elements Chapter 12  Properties of Metallic Solids (at 293 K) Chapter 13  Properties of Non-Metallic Solids (at 293 K)

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1 3 19 27 43 61 75 87 133

137 139 143 145 147 149 151 155 159 163 165 167 173 179

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vi • Contents

Chapter 14  Properties of Liquids (at 293 K) Chapter 15  Properties of Gases at STP Chapter 16  Mechanical Data Chapter 17 Relative Humidities From Wet-and Dry-Bulb Thermometers (Exposed in Standard Screen) Chapter 18  Acoustic Data Chapter 19  Astronomical Data Chapter 20  Terrestrial and Geodetic Data Chapter 21 Radioactivity Chapter 22  Properties of Inorganic Compounds Chapter 23  Properties of Organic Compounds (at 293 K) Chapter 24  The Fundamental Constants

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185 189 193 199 203 207 211 217 221 229 233

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Preface This will serve as an excellent reference for in-service professionals in the areas of mathematics, science, and engineering. It will also help students as a quick review for examinations. Formulas from algebra to vectors are discussed in Part I of the book. Technical terms, theorems, and applicable laws are defined, and their uses are explained. Answers to standard equations in these areas are provided. Equations from straight lines to conics are derived and applications are provided. In the sections on differential and integral calculus, solutions are included for a large number of situations. Part II is unique in providing numerical data in astronomical, acoustical, optical, and geophysical areas. Properties such as density, specific heat, thermal conductivity, and expansion coefficients are tabulated for a large number of metallic, nonmetallic, organic, and inorganic materials. This information and data will be particularly useful for practicing engineers. C.P. Kothandaraman

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PART I

Mathematical Formulas

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CHAPTER

1

Algebra 1

FUNDAMENTAL LAWS (a) Commutative law: a + b = b + a, ab = ba. (b) Associative law: a + (b + c) = (a + b) + c, a(bc) = (ab)c. (c) Distributive law: a(b + c) = ab + ac.

2

LAWS OF EXPONENTS (a) ax · ay = ax+y, (ab)x = ax . bx, (ax)y = axy. (b) a0 = 1 if a ≠ 0, a−x = x

(c) a y =

3

y

1

1 ax , y = a x−y . x a a

y

ax , a y = a .

OPERATIONS WITH ZERO  a − a = 0, a · 0 = 0 · a = 0. 0 If a ≠ 0, = 0, 0a = 0, a0 = 1  (division by zero undefined). a

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4 • Mathematical Formulas and Scientific Data

4

COMPLEX NUMBERS A number of the form a + ib, where a and b are real. i = −1 =i2 = −l, i3 = − i, i4 = l, i5 = i, etc. a + ib = c + id, if and only if a = c, b = d. (a + ib) + (c + id) = (a + c) + id (b + d). (a + ib) (c + id) = (ac − bd) + i (ad + bc).



a + ib (a + ib)(c − id) ac + bd bc − ad = = + i. (c + id)(c − id) c2 + d 2 c2 + d 2 c + id



5

LAWS OF LOGARITHMS If M, N, and b are positive numbers and b ≠ l then (a) logbM N = logbM + logbN, (c) logbMp = p . logbM, 1  (e) log b   = − log b M , M

M = logbM − logbN, N 1 q (d) log b M = ⋅ log b M q (b) log b

(f) log b b = 1,  logb1 = 0

Change of base of Logarithms (c ≠ l): log b M = log c M ⋅ log b c =

6

log c M . log c b

BINOMIAL THEOREM (n a positive integer). n(n − 1) n− 2 2 a b 2! n(n − 1)(n − 2) n− 3 3 + a b + …+ nabn−1 + bn 3!

(a + b)n =a n + na n−1 b +

where

n! = n = l · 2 · 3 (n − l) ………n.

Example 5! = 5 × 4 × 3 × 2 × l

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Algebra • 5

7

EXPANSIONS AND FACTORS (i) (a ± b)2 = a2 ± 2ab + b2. (ii) (a ± b)3 = a3 ± 3a2 b + 3ab2 ± b3. (iii) (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc. (iv) a2 − b2 = (a − b) (a + b). (v)

a3 − b3 = (a − b) (a2 + ab + b2).

(vi) a3 + b3 = (a + b) (a2 − ab + b2). (vii) an − bn = (a − b) (an–1 + an–2 b + ... + bn–1). (viii) an − bn = (a + b) (an–1 − an–2 b + ... − bn–1), for n an even integer. (ix) an + bn = (a + b) (an–1 − an–2 b + ... + bn–1), for n an odd integer. (x)

8

a4 + a2 b2 + b4 = (a2 + ab + b2) (a2 − ab + b2).

RATIO AND PROPORTION If a : b = c : d or

a c a c = , then ad = bc, = . b d b d

a c e If = = = = k, then b d f a + c + e +  pa + qc + re +  = k = b + d + f +  pb + qd + rf + 

9 CONSTANT FACTOR OF PROPORTIONALITY (OR VARIATION), k (i) If y varies directly as x, or y is proportional to x, y = kx. (ii) If y varies inversely as x, or y is inversely proportional to x, k y= , x

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6 • Mathematical Formulas and Scientific Data

(iii) If y varies jointly as x and z, y = kxz. (viii) If y varies directly as x and inversely as z, y=

kx . z

10 ARITHMETIC PROGRESSION a, a + d, a + 2d, a + 3d, .... . If a is the first term, d is the common difference, n is the number of terms, l is the last term, and S is the sum of n terms. l = a + (n − l) d, the sum of the series upto l S=

n n n (a + l)= [a + a + (n − 1)d]= [2 a + (n − 1)d] 2 2 2

The arithmetic mean of a and b is (a + b)/2.

11 GEOMETRIC PROGRESSION a, ar, ar2, ar3, ... . If a is the first term, r is the common ratio, n is the number of terms, l is the last term, and Sn is the sum of n terms,  r n − 1  rl − a l = arn–1,    sum up to n = terms Sn a= .    r −1  r −1 If r2 < l, Sn approaches the limit S∞ as n increases without limit, S∞ =

a . 1−r

The geometric mean of a and b is

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ab .

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Algebra • 7

12 HARMONIC PROGRESSION A sequence of numbers whose reciprocals form an arithmetic progression is called a harmonic progression. Thus 1 1 1 , , , a a + d a + 2d The harmonic mean of a and b is 2ab/(a + b).

13 PERMUTATIONS Each different arrangement of all or a part of a set of things is called a permutation. The number of permutations of n different things taken r at a time is P(n, r) = nPr = n(n − l) (n − 2) ... (n − r + l) =

n! , (n − r )!

where n! = n(n − l) (n − 2) ...(l).

14 COMBINATIONS Each of the groups or relations which can be made by taking part or all of a set of things, without regard to the arrangement of the things in a group, is called a combination. The number of combinations of n different things taken r at a time is C(n,= r)

n

 n = Cr =  r 

Pr n(n − 1)…(n − r + 1) n! . = = r! r (r − 1)…(1) r !(n − r )!

n

15 PROBABILITY If an event may occur in p ways and may fail in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure to occur is q/(p + q).

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8 • Mathematical Formulas and Scientific Data

16 REMAINDER THEOREM If the polynomial f(x) is divided by (x − a), the remainder is f (a). Hence, if a is a root of equation f (x) = 0, then f(x) is divisible by (x − a).

17 DETERMINANTS The determinant D of order n, a11 a21 D= : a n1

a12 a22 : an 2

. . : .

. . : .

. a1 n . a2 n : : . ann

is defined to be the sum Σ (±) a1i a2j a3k ... anl of n! terms, the sign in a given term being taken plus or minus according to the number of inversions (of the numbers l, 2, 3, ..., n) in the corresponding sequence i, j, k, ..., l, is even or odd. The cofactor Aij of the element aij is defined to be the product of (− l)i + j by the determinant obtained from D by deleting the ith row and the jth column. The following theorems are true: (a) If the corresponding rows and columns of D be interchanged, D is unchanged. (b) If any two rows (or columns) of D be interchanged, D is changed to − D. (c) If any two rows (or columns) are alike, then D = 0. (d) If each element of a row (or column) of D be multiplied by m, the new determinant is equal to mD. (e) If each element of a row (or column) is added m times the corresponding element in another row (or column), D is unchanged. (f) D = a1j A1j + a2j A2j + ... + anj Anj ,

j = l, 2, … .., n.

(g) 0 = a1k A1j + a2k A2j + ......... + ank Anj, if j ≠ k.

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Algebra • 9

(h) The solution of the system of equations ai1 x1 + ai2 x2 + ... + ain xn = ci, i = l, 2, …, n,

is unique if D ≠ 0. The solution is given by the equations Dx1 = C1, Dx2 = C2, …, Dxn = Cn,

where Ck is what D becomes when the elements of its kth column are replaced by c1, c2, …, cn, respectively. EXAMPLE 1.



a b c p q r l m n

= a

q r p r p q −b +c  m n l n l m

= a(qn − rm) − b (pn − rl) + c (pm − ql). EXAMPLE 2. Find the values of x, y, and z, which satisfy the system 3x + 2y + z = 1000 4x + y + 3z = 1500 x + y + z = 600 Solution: Here,



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1000 2 1 1500 1 3 600 1 1 −500 = x = = 100 3 2 1 −5 4 1 3 1 1 1

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10 • Mathematical Formulas and Scientific Data





= y

3 1000 1 4 1500 3 1 600 1 −1000 = = 200 3 2 1 −5 4 1 3 1 1 1

= z

3 2 1000 4 1 1500 1 1 600 −1500 = = 300 3 2 1 −5 4 1 3 1 1 1

Interest, Annuities, Sinking Funds. In this section, n is the number of years, and r is the rate of interest expressed as a decimal.

18 AMOUNT A principal P placed at a rate of interest r for n years accumulates to an amount A, as follows: At simple interest:

A = P (l + nr).

At interest compounded annually:

A = P(l + r)n. nq

 4 At interest compounded q times a year: A = P  1 +  . q 

19 NOMINAL AND EFFECTIVE RATES The rate of interest quoted in describing a given variety of compound interest is called the nominal rate. The rate per year at which interest is earned during each year is called the effective rate. The effective rate i corresponding to the nominal rate r, compounded q times a year is: q  r i = 1 +  − 1. q 

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Algebra • 11

20 PRESENT OR DISCOUNTED VALUE OF A FUTURE AMOUNT The present quantity P which in n years will accumulate to the amount A at the rate of interest r, is: A At simple interest: P= . 1 + nr At interest compounded annually:

P=

At interest compounded q times a year:

P=

A . (1 + r )n A  1 + 

r  q

nq

.

P is called the present value of A due in n years at rate r.

21 TRUE DISCOUNT The true discount is: D =A − P.

22 ANNUITY A fixed sum of money paid at regular intervals is called an annuity.

23 AMOUNT OF AN ANNUITY If an annuity P is deposited at the end of each successive year (beginning one year hence), and the interest at rate r, compounded annually, is paid on the accumulated deposit at the end of each year, the total amount N accumulated at the end of n years is N = P.

(1 + r )n − 1 r

N is called the amount of an annuity P.

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12 • Mathematical Formulas and Scientific Data

24 PRESENT VALUE OF AN ANNUITY The total present amount P which will supply an annuity N at the end of each year for n years, beginning one year hence (assuming that in successive years the amount not yet paid out earns interest at rate r, compounded annually), is: (1 + r )n − 1 1 − (1 + r )− n P= N⋅ = N ⋅ . r (1 + r )n r P is called the present value of an annuity.

25 AMOUNT OF A SINKING FUND If a fixed investment N is made at the end of each successive year (beginning at the end of the first year), and interest paid at rate r, compounded annually, is paid on the accumulated amount of the investment at the end of each year, the total amount S accumulated at the end of n years is: S= N ⋅

(1 + r )n − 1 . r

S is called the amount of the sinking fund.

26 FIXED INVESTMENT, OR ANNUAL INSTALLMENT The amount N that must be placed at the end of each year (beginning one year hence), with compound interest paid at rate r on the accumulated deposit, in order to accumulate a sinking fund S in n years is: N= S ⋅

r . (1 + r )n − 1

N is called a fixed investment or annual installment.

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Algebra • 13

Algebraic Equations

27 QUADRATIC EQUATIONS If  ax2 + bx + c = 0, a ≠ 0, x=

then,

− b ± b2 − 4 ac . 2a

If a, b, and c are real and if b2 − 4ac > 0, the roots are real and unequal; if b2 − 4ac = 0, the roots are real and equal, and; if b2 − 4ac < 0, the roots are imaginary.

28 CUBIC EQUATIONS The cubic equation y3 + py2 + qy + r = 0, may be reduced by the substitution p  = y x−  3  to the normal form x3 + ax + b = 0 where a=

1 1 3 q − p2 ) , b = 2 p3 − 9 pq + 27 r ) , ( ( 3 27

which has the solutions x1, x2, and x3, 1 i 3 x1 =+ A B, x2 , x3 = − ( A + B) ± ( A − B), 2 2

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14 • Mathematical Formulas and Scientific Data

where b b2 a3 i2 = −1, A = 3 − + + 2 4 27

b b2 a3 B=3 − − + . 2 4 27

If p, q, and r are real (and hence, if a and b are real), and if 

b2 a3 + > 0,   there are one real root and two conjugate imag4 27

inary roots, if 

b2 a3 + = 0,   there are three real roots of which at least two 4 27

are equal, if 

b2 a3 + < 0,   there are three real and unequal roots. 4 27

If 

b2 a3 + < 0.   the above formulas are impractical. The real 4 27

roots are, a φ  xk =2 − cos  + 120°k  ,    k = 0, 1, 2, 3 3  where cos = φ 

b2  a3  ÷  − , 4  27 

and where the upper sign is to be used if b is positive and the lower sign if b is negative. If

b2 a3 + > 0,  and a > 0 4 27

then the real root is, = x 2

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a cot 2 φ, 3

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Algebra • 15

where ϕ and Ψ are to be computed from cot= 2ψ 

b2 a3 ÷ , tan ϕ = 4 27

3

tan ψ

and where the upper sign is to be used if b is positive and the lower sign if b is negative. b2 a3 + = 0 4 27

If the roots are,

a a a = x 2 − , ± − ,± − , 3 3 3 where the upper sign is to be used if b is positive, and the lower sign if b is negative.

29 BIQUADRATIC (QUARTIC) EQUATION The quartic equation y4 + py3 + qy2 + ry + s = 0 may be reduced to the form x4 + ax2 + bx + c = 0 by the substitution p  = y  x − . 4  Let l, m, and n denote the roots of the resolvent cubic.  a2 − 4 c  b2 a − = t3 +   t2 +  t 0.  2  16  64 The required roots of the reduced quartic are,

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x1

= + l + m + n,

x2

= + l − m − n,

x3

= − l + m − n,

x4

= − l − m + n,

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16 • Mathematical Formulas and Scientific Data

where the selection of the square root to be attached to each of the b quantities l , m , n must give l m n = − . 8

30 GENERAL EQUATIONS OF THE nth DEGREE P ≡ a0xn + a1xn–1 + a2xn–2 + ... + an–1 x + an = 0. If n > 4, there is no formula that gives the roots of this general equation. The following methods may be used to advantage: (a) Roots by Factors: By trial, find a number r such that x = l satisfies the equation, that is, such that a0rn + a1rn–1 + a2rn–2 + ... + an–1r + an = 0, a0 ≠ 0. Then (x − r) is a factor of the left-hand member P of the equation. Divide P by (x − r) leaving an equation of degree one less than that of the original equation. Next, proceed in the same manner with the reduced equation. (All integer roots of P = 0 are divisors of an/a0.) (b) Roots by Approximation: Suppose the coefficients ai in P are real. Let a and b be real numbers. If for x = a and x = b, the left member P of the equation has opposite signs, then a root lies between a and b. By repeated application of this principle, real roots to any desired degree of accuracy may be obtained. (c) Roots by Graphing: If a graph of P is plotted as a function of x, the real roots are the values of x where the graph crosses the x-axis. By increasing the scale of the portion of the graph near an estimated root, the root may be obtained to any desired degree of accuracy. (d) Descartes’ Rule: The number of positive real roots of an equation with real coefficients is either equal to the number of its variations of sign or is less than that number by a positive even integer. A root of multiplicity m is here counted as m roots.

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Algebra • 17

31(a) THE EQUATION xn = a. The n roots of this equation are: 2 kπ 2 kπ   + −1 sin a  cos  , if a > 0, n n   (2 k + 1)π (2 k + 1)π   + −1 sin x = n − a  cos  , if a < 0, n n  

= x

n

where k takes successively the values 0, l, 2, 3, ..., n − l.

31(b) STIRLING’S FORMULA For large values of n, 2 nπ (n / e)n < n! < 2 nπ (n / e)n [1 + 1 / 12 n − 1], where π = 3.l4l59..., e = 2.7l828... 1  log n! ≅  n +  log n − n log e + log 2 π . 2 

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CHAPTER

2

Elementary Geometry Let a, b, c, d, and s denote lengths, A denote areas, and V denote volumes.

32 TRIANGLE A = bh/2, where b denotes the base and h denotes the altitude.

33 RECTANGLE A = ab, where a and b denote the lengths of the sides.

34 PARALLELOGRAM (OPPOSITE SIDES PARALLEL) A = ah = ab sin θ, where a and b denote the sides, h denotes the altitude, and θ denotes the angle between the sides.

35 TRAPEZOID (FOUR SIDES, TWO PARALLEL) A=

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1 h (a + b), where a and b are the sides and h the altitude. 2

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20 • Mathematical Formulas and Scientific Data

36 REGULAR POLYGON OF N SIDES A=

1 2 180° n a co t , 4 n

where a is the length of side.

a 180° R = cosec , 2 n

where R is the radius of circumscribed circle.

a 180° r = cot , 2 n

where r is the radius of inscribed circle.

= α

360° 2 π , = n n

FIGURE 2.1.

radians,

 n−2  n−2 β =  . 180° =  π radians where α and β are the angles  n   n  indicated in Figure 2.l a a = a 2= r tan 2 R sin . 2 2

37 CIRCLE Let

C = circumference,

S = length of arc subtended by θ,

R = radius,

l = chord subtended by arc S,

D = diameter,

h = rise,

`A = area,

θ = central angle in radians.

C = 2πR = πD,

π = 3.l4l59...

S = Rθ =

1 D 2

θ = D cos−1

θ θ d 2 2 R sin = 2 d tan . l = 2 R 2 −= 2 2 θ 1 θ d = 1 4 R 2 −= l 2 R cos = l ctn . 2 2 2 2

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d , R

FIGURE 2.2.

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Elementary Geometry • 21

h = R − d. S 2S d l 1 θ== = 2 cos−1 = 2 tan −1 = 2 sin −1 . R D R 2d D 1 A (circle) = πR2 = πD2. 4 1 1 2 A (sector) = = Rs R θ. 2 2 1 A (segment) = A (sector) − A (triangle) = R 2 (θ − sin θ) 2 ( R − h) = R 2 cos−1 − (R − h) 2 Rh − h2 . R Perimeter of an n-side regular polygon inscribed in a circle. π = 2n R sin n . n 1 2π . Area of inscribed polygon = nR2 sin 2 n Perimeter of an n-side regular polygon circumscribed about a circle π = 2n R tan . n π Area of circumscribed polygon = nR2 tan . n Radius of a circle inscribed in a triangle of sides a, b, and c is r=

(s − a)(s − b)(s − c) 1 , s= (a + b + c). s 2

Radius of a circle circumscribed about a triangle is

R=

abc . 4 s(s − a)(s − b)(s − c)

38 ELLIPSE Perimeter = 4 a. E(k), where k2 = l − (b2/a2) and E(k) is the complete elliptic integral E. A = π ab, where a and b are lengths of semi-major and semi-minor axes, respectively.

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FIGURE 2.3.

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22 • Mathematical Formulas and Scientific Data

39 PARABOLA A = 2ld/3. Height of d1 =

d 2 2 ( l − l1 ) . l2

Width of l1 = l

d − d1 . d

 2  2 d 2 2  2 d  4  l Length of arc = 1 +   −   + ... .   3  l  5  l 

FIGURE 2.4.

40 CATENARY, CYCLOID, ETC. Let P be a point at a distance d from the center of a circle of radius r. The curve traced out P as the circle rolls along a straight line, without slipping, is called a cycloid, as shown in Figure 2.5.

41 AREA BY APPROXIMATION

FIGURE 2.5.

If y0, y1, y2, ..., yn be the lengths of a series of equally spaced parallel chords and if h is their distance apart, the area enclosed by boundary is given approximately by any one of the following formulae:

FIGURE 2.6. 

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Elementary Geometry • 23

1  AT = h  ( y0 + yn ) + y1 + y2 + ... + yn−1  . (Trapezoidal Rule). 2   AD = h 0.4 ( y0 + yn ) + 1.1 ( y1 + yn−1 ) + y2 + y3 + ... + yn− 2  .    (Durand’s Rule). 1 h ( y0 + yn ) + 4 ( y1 + y3 + ... + yn−1 ) + 2 ( y2 + y4 + ... + yn− 2 )  . 3  (n even, Simpson’s Rule). AS =

In general, As gives a more accurate approximation. The greater the value of n, the greater the accuracy of approximation.

42 CUBE V = a3; d = a 3 ; total surface area = 6a2, where a is the length of side and d is the length of diagonal.

43 RECTANGULAR PARALLELOPIPED 2 2 2 V = abc; d = a + b + c ; total surface area = 2(ab + bc + ca), where a, b and c are the lengths of the sides and d is the length of diagonal.

44 PRISM OR CYLINDER V= (area of base) · (altitude). Lateral area= (perimeter of right section). (lateral edge).

45 PYRAMID OR CONE 1 (area of base) ⋅ (altitude). 3 1 Lateral area= of regular pyramid (perimeter of base) ⋅ (slant height). 2 = V

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24 • Mathematical Formulas and Scientific Data

46 (A)  FRUSTUM OF PYRAMID OR CONE

(

)

1 A1 + A2 + A1 ⋅ A2 h, where h is the altitude and A1 and A2 3 are the areas of the bases. 1 Lateral area of a regular figure = (sum of perimeters of base) · 2 (slant height). V=

46 (B) PRISMOID V=

h (A1 + A2 + 4A3), where h = altitude, A1 and A2 are the areas 6

of the bases and A3 is the area of the midsection parallel to the bases.

47 AREA OF SURFACE AND VOLUME OF REGULAR POLYHEDRA OF EDGE L Name

Type of Surface

Area of Surface

Volume

Tetrahedron

4 equilateral triangles

l.73205l2

0.11785l3

Hexahedron (cube) 6 squares

6.00000l2

l.00000l3

Octahedron

8 equilateral triangles

3.464l0l2

0.47l40l3

Dodecahedron

l2 pentagons

20.64578l2 7.663l2l3

Icosahedron

20 equilateral triangles 8.66025l2

2.l8l69l3

48 SPHERE A (sphere) = 4 πR2 = πD2. A (zone) = 2 πRh1 = πDh1. V (sphere) =

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4 3 1 πR = πD3 . 3 6

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Elementary Geometry • 25

FIGURE 2.7.

V (spherical sector) = V (spherical segment of one base) = V (spherical segment of two bases) =

2 2 1 πR h1 = πD2 h1 . 3 6 1 πh3 ( 3 r32 + h32 ) . 6 1 πh2 ( 3 r32 + 3 r22 + h22 ) . 6

A (lune) = 2R2θ, where θ is the angle in radians of lune.

49 SOLID ANGLE (Ψ) The solid angle ψ at any point subtended by a surface S is equal to the area A of the portion of the surface of a sphere of unit radius, center at P, which is cut out by a conical surface, with vertex at P, passing through the perimeter of S. The unit solid angle ψ is called the steradian. The total solid angle about a point is 4π steradians.

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FIGURE 2.8.

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26 • Mathematical Formulas and Scientific Data

50 ELLIPSOID V=

4 πabc, where a, b, and c are the lengths of the semi-axes. 3

FIGURE 2.9.

51 (A) TORUS V = 2π2Rr2. Area of surface = S = 4π2Rr.

51 (B)  THEOREMS OF PAPPUS (a) If a plane area A is rotated about a line l in the plane of A and not cutting A, the volume of the solid generated is equal to the ­product of A and the distance traveled by the center of gravity of A. (b) If a plane curve C is rotated about a line l in the plane of C and not cutting C, the area of the surface generated is equal to the product of the length of C and the distance traveled by the center of gravity of C.

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CHAPTER

3

Trigonometry 52 ANGLE If two lines intersect, one line may be rotated about their point of intersection through the angle which they form until it coincides with the other line. The angle is said to be positive if the rotation is counterclockwise and negative, if clockwise. A complete revolution of a line is a rotation through an angle of 360°. Thus, A degree is l/360 of the plane angle about a point. A radian is an angle subtended at the center of a circle by an arc whose length is equal to that of the radius. 180° = p radians; 1°=

180 p radians; 1 rad. = degrees. 180 p

53 TRIGONOMETRIC FUNCTIONS OF AN ANGLE α Let α be any angle whose initial sidelines on the positive x-axis and whose vertex is at the origin, and (x, y) be any point on the terminal side of the angle. (x is positive if measured along OX to the right, from the y-axis; and negative, if measured along OX’ to the left from the y-axis. Likewise, y is positive if measured parallel to OY, and negative if measured parallel to OY’.) Let r be the positive

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28 • Mathematical Formulas and Scientific Data

FIGURE 3.1.

distance from the origin to the point. The trigonometric functions of an angle are defined as follows: sine α = sin α =

y . r

x . r y tangent α = tan α = . x cosine α = cos α =

cotangent α = ctn α = cot α = secant α = sec α =

r . x

x . y

r . y exsecant α = exsec α = sec α − 1. cosecant α = csc α = cosec α =

versine α = vers α = 1− cos α. coversine α = covers α = 1 − sin α. 1 haversine α = hav α = vers α 2

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Trigonometry • 29

54 SIGNS OF THE FUNCTIONS Quadrant

sin

cos

tan

ctn

sec

csc

I...................

+

+

+

+

+

+

II..................

+

−

−

−

−

+

III.................

−

−

+

+

−

−

IV.................

−

+

−

−

+

−

55 F  UNCTIONS OF 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360° 0°

30°

45°

60°

1 2

3 2

1 2

90°

l80°

270°

360°

1

0

−1

0

0

−1

0

1

sin

0

cos

1

3 2

tan

0

1 3

1

3

∞

0

∞

0

cot

∞

3

1

1 3

0

∞

0

∞

sec

1

2 3

∞

−1

∞

1

cosec

∞

1

∞

−1

∞

2

1 2

2

2

1 2

2 2 3

56 VARIATIONS OF THE FUNCTIONS Quadrant

sin

cos

tan

ctn

sec

csc

I

0 → +l

+ l →0

0→+∞

+∞→0

+l→+∞

+ ∞ → +l

II

+l → 0

0→−l

−∞→0

0→−∞

−∞→−l

+l→+∞

III

0 → −l

−l→0

0→+∞

+∞→0

−l→−∞

− ∞ → −l

IV

−l→0

0→+l

−∞→0

0→−∞

+ ∞ → +l

−l→−∞

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30 • Mathematical Formulas and Scientific Data

57 F  UNCTIONS OF ANGLES IN ANY QUADRANT IN TERMS OF ANGLES IN FIRST QUADRANT sin cos tan cot sec cosec

−α

90° ± α

180° ± α

270° ± α

−sin α

+ cos α

∓ sin α

−cos α

n (360)° ±α ± sin α

+cos α

∓ sin α

− cos α

±sin α

+ cos α

−tan α

∓ ctn α

± tan α

∓ ctn α

± tan α

−ctn α

∓ tan α

± ctn α

∓ tan α

± ctn α

+sec α

∓ csc α

− sec α

± csc α

+ sec α

−csc α

+ sec α

∓ csc α

− sec α

± csc α

where n denotes any integer.

58 FUNDAMENTAL IDENTITIES sin α =

1 1 1 sin a ; cos a = ; tan a = = . csc a sec a cot a cos a

cosec α =

1 1 1 cos a ; sec a = ;cot a = = . sin a cos a tan a sin a

sin2 α + cos2 α = 1; 1 + tan2 α = sec2 α; 1 + cot2 α = cosec2 α. sin 2 α = 2 sin α cos α cos 2 α = 2 cos2 α − 1 = 1 − 2 sin2 α = cos2 α − sin 2 − α,

2 tan a , 1 - tan 2 a sin 3 α = 3 sin α − 4 sin3 α,

tan 2 α =

cos 3 α = 4 cos3 α − 3 cos α, sin n α = 2 sin (n − l)α · cos α − sin (n − 2) α, cos n α = 2 cos (n − l)α · cos α − cos (n − 2) α, sin (α ± β) = sin α cos β ± cos α sin β, cos (α ± β) = cos α cosβ ∓ sin α sin β, tan (α ± β) =

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tan a ± tan b . 1 ∓ tan a tan b

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Trigonometry • 31

1 1 sin α + sin β = 2 sin (a + b) × cos (a - b). 2 2 1 1 sin α − sin β = 2 cos (a + b) × sin (a - b), 2 2 1 1 cos α + cos β = 2 cos (a + b) × cos (a - b), 2 2

1 1 cos α − cos β = -2 sin (a + b) × sin (a - b). 2 2 sin

a 1 - cos a positive if a in I or II quadrants, , = ± 2 2 2 negative otherwise.

1 + cos a a , positive if = ± 2 2 otherwise. 1 - cos a sin a tan a = =± = sin a 1 + cos a 2

cos

a in I or IV, negative 2 1 - cos a , 1 + cos a

a in I or III, negative otherwise. 2 1 1 sin2 α = (1 − cos 2 α); cos2 α = (1 + cos 2 α) 2 2 positive if

sin3 α =

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1 1 (3 sin α − sin 3 α); cos3 α = (cos3 α + 3 cos α). 4 4

sin α sin β =

1 1 cos (α − β) − cos (α + β). 2 2

cos α cos β =

1 1 cos (α − β) + cos (α + β), 2 2

sin α sin β =

1 1 sin (α + β) + sin (α − β). 2 2

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32 • Mathematical Formulas and Scientific Data

59 E  QUIVALENT EXPRESSIONS FOR SIN, COS, TAN, ETC. Function sin α sin α

sin α

cos α tan α cot α sec α cosec α

cos α

tan α

± 1 - cos2 a

± 1 - sin 2 a sin a

cos α

cot α

sec α

tan a

1

± 1 + tan 2 a

± 1 + cot 2 a

1

cot a

± 1 + tan 2 a

± 1 + cot 2 a

± 1 - sin 2 a

± 1 - cos2 a cos a

tan α

1 cot a

± 1 - sin 2 a sin a

cos a ± 1 - cos2 a

1 tan a

cot α

± 1 + tan 2 a

± 1 + cot 2 a cot a

± 1 + tan 2 a tan a

± 1 + cot 2 a

1 ± 1 - sin 2 a

1 cos a

1 sin a

± 1 - cos2 a

1

cosec α

± sec a - 1 sec a

1 cosec a

1 sec a

± cosec 2 a - 1 cosec a

2

± sec 2 a - 1 1 ± sec 2 a - 1

sec α sec a ± sec 2 a - 1

1 ± cosec 2 a - 1 ± cosec 2 a - 1 cosec a ± cosec 2 a - 1

cosec α

Note: The quadrant which terminates determines the sign used.

60 INVERSE OR ANTI-TRIGONOMETRIC FUNCTIONS The complete solution of the equation x = sin y is (in radians): y = (−l)nsin−1 x + n(π),   − π /2 £ sin−1 £ x /2, where sin−1 x is the principal value of the angle whose sine is x (n denotes an integer). Similarly, if x = cos y, y = ± cos−1 x + n(2π), 0 £ cos−1 x £ π. If x = tan y, y = tan−1 x + n(π), − π/2 < tan−1 x < π/2. Similar relations hold for x = cot y, x = sec y, and x = cosec y.

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Trigonometry • 33

61 C  ERTAIN RELATIONS AMONG INVERSE FUNCTIONS If the inverse functions are restricted as in Art. 60, the following formulae hold: sin−1 a = cos-1 1 - a2 = tan -1 = sec -1

1 1-a

2

tan−1 a = sin -1

1 - a2

= cosec -1

cos−1 a = sin -1 1 - a2 = tan -1 = sec -1

a

1 - a2 a

1 a.

a 1 - a2 = cot -1 a 1 - a2

1 1 = cosec -1 a 1 - a2 . a 1 1 = cos-1 = cot -1 2 2 a. 1+ a 1+ a

= sec -1 1 + a2 = cosec -1 cot−1 a = tan−1

= cot -1

1 ; sec−1 a = cos−1 a

1 + a2 a . 1 1 ; cosec−1 a = sin−1 . a a

62 SOLUTION OF TRIGONOMETRIC EQUATIONS To solve a trigonometric equation, reduce the given equation, by means of the relations expressed in Art. 58 to Art. 60 inclusive, to an equation containing only a single function of a single angle. Solve the resulting equation by algebraic methods (Art. 30) for the remaining function and from this find the values of the angle, by Art. 57 and Art. 59. All these values should then be tested in the original equation, discarding those which do not satisfy it.

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34 • Mathematical Formulas and Scientific Data

63 R  ELATIONS BETWEEN SIDES AND ANGLES OF PLANE TRIANGLES Let a, b, and c denote the sides and α, β, and γ, the corresponding opposite angles with r = radius of inscribed circle; 2 s = a + b + c; A = area; R = radius of circumscribed circle; hb = altitude on side b. α + β + γ = 180°. a b c = = (law of sines). sin a sin b sin g 1 tan (a + b) a+ b 2 = (law of tangents).∗ 1 a-b tan (a - b) 2 a2 = b2 + c2 − 2 bc cos (law of cosines).∗ a = b cos γ + c cos β.∗ cos α =

b2 + c2 - a2 ∗ 2 , sin a = s(s - a)(s - b)(s - c).∗ 2 bc bc

sin

a = 2

(s - b)(s - c) a , ∗ cos = bc 2

tan

a = 2

r (s - b)(s - c) , ∗ where r = (s - a)(s - b)(s - c) . = s(s - a) s-a s

A=

1 * 1 bhb = ab sin g * 2 2

s(s - a) ;∗ bc

= =

R=

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a* abc = . 2 sin a 4 A

a2 sin b sin g * 2 sin a s(s - a)(s - b)(s - c) = rs. hb = c sin a* = a sin g* =

2 rs b

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Trigonometry • 35

FIGURE 3.2.

64 SOLUTION OF A RIGHT TRIANGLE Given one side and any acute angle α, or any two sides, to find the remaining parts: a = (c + b)(c − a) = c sin α = b tan α. a b = (c + a)(c − a) = c cos α = . tan α ∗ Two more formulas may be obtained by replacing a by b, b by c, c by a, α by β, β by γ, γ by α.

sin α = c=

b a a , cos α = , tan α = , β = 90° − α. c c b a b = = sin α cos α

a 2 + b2 .

1 a2 b2 tan α c2 sin 2α A == = ab = . 2 2 tan α 2 4

FIGURE 3.3.

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36 • Mathematical Formulas and Scientific Data

65 SOLUTION OF OBLIQUE TRIANGLES The formulas of the preceding section Art. 63 will suffice to solve any oblique triangle. Use the trigonometric tables for numerical work. We give one method. Solutions should be checked by some other method.

FIGURE 3.4.

(a) Given any two sides b and c and included angle α. 1 1 1 (β + γ )= 90° − α;tan (β − γ)= 2 2 2 1 1 β= (β + γ) + (β − γ); γ= 2 2

b− c 1 tan (β + γ); 2 b+ c 1 1 b sin α (β + γ) − (β − γ); a= . 2 2 sin β

(b) Given any two angles α and β, and any side c. c sin α c sin β ;= . b sin γ sin γ (c) Given any two sides a and c, and an angle opposite one of these, say α. = γ 180° − (α + β);= a

c sin α a sin β β 180° − (α + γ),= b ,= . a sin α This case may have two sets of solutions, for γ may have two values, γ1 < 90° and γ2 = l80° − γ1 > 90°. If α + γ2 > l80°, use only γ1. γ sin=

(d) Given the three sides a, b, and c. 1 (s − a)(s − b)(s − c) (a + b + c), r= . s 2 1 r 1 r 1 r = = = tan α , tan β , tan γ . 2 s−a 2 s− b 2 s−c s=

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Trigonometry • 37

Spherical Trigonometry

66 THE RIGHT SPHERICAL TRIANGLE Let O be the center of the sphere, and a, b, and c the sides of a right triangle with opposite angles α, β, and γ = 90°, respectively, the sides being measured by the angle subtended at the center of the sphere. FIGURE 3.5.

sin a = sin a· sin c, sin a = tan b · cot β,

sin b = sin β · sin c. sin b = tan a · cot α.

cos α = cos a · sin β, cos α = tan b · cot c,

cos β = cos b · sin α. cos β = tan a · cot c.

cos c = cot α · cot β,

cos c = cos a · cos b.

67 NAPIER’S RULES OF CIRCULAR PARTS Let the five quantities a, b, co-α (complement of α), co-c, co-β, be arranged in order as indicated in the figure. If we denote any one of these quantities as a middle part, then two of the other parts are adjacent to it, and the other two parts are opposite to it. The above formulas may be remembered by means of the following rules : (a) The sine of a middle part is equal to the product of the tangents of the ­adjacent parts. (b) The sine of a middle part is equal to the product of the cosines of the o­ pposite parts.

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FIGURE 3.6.

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38 • Mathematical Formulas and Scientific Data

68 THE OBLIQUE SPHERICAL TRIANGLE Let a, b, and c denote the sides and α, β, and γ the corresponding opposite angles of the spherical triangle, Δ its area, E its spherical excess, R the radius of the sphere upon which the triangle lies, and α', β', γ', a, b, and c the corresponding parts of the polar triangle. 0° < a + b + c < 360°,

l80° < + + < 540°.

α = l80° − a¢,

β¢ = l80° − b¢,

γ = l80° − c¢,

a = l80° − α¢,

b = l80° − β¢,

c = l80° − γ¢.

sin a sin b sin g (law of sines). = = sin a sin b sin c cos a = cos b cos c + sin b sin c cos α. cos α = −cos β cos γ + sin β sin γ cos a (law of cosines). tan

a = 2

sin(s - b) × sin(s - c) , sin s × sin(s - a)

where s = 1 (a + b + c). 2

tan

a = 2

- cos s × cos(s - a) , cos(s - b) × cos(s - g)

1 where s = (a + b + g). 2

1 1 1 1 sin (a + b) tan c cos (a + b) tan c 2 2 2 2 , . = = 1 1 1 1 sin (a - b) tan (a - b) cos (a - b) tan (a + b) 2 2 2 2 1 1 1 1 sin (a + b) cot g cos (a + b) cot g 2 2 2 2 , . = = 1 1 1 1 sin (a - b) tan (a - b) cos (a - b) tan (a + b) 2 2 2 2 1 1 1 1 sin (a + b)cos c = cos (a - b)cos g. 2 2 2 2 1 1 1 1 cos (a + b)cos c = cos (a + b)sin g. 2 2 2 2

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Trigonometry • 39

1 1 1 1 sin (a - b)sin c = sin (a - b)cos g. 2 2 2 2 1 1 1 1 cos (a - b)sin c = sin (a + b)sin g. 2 2 2 2 tan

E 1 1 1 1 = tan s × tan (s - a) × tan (s - b) × tan (s - c). 4 2 2 2 2 2 pR E , D= a + b + g - 180° = E. 180 Hyperbolic Functions

69 DEFINITIONS (For definition of e see Art. 108) Hyperbolic sine of x = sinh x =

1 x -x (e - e ); 2

Hyperbolic cosine of x = cosh x =

1 x -x (e - e ); 2

Hyperbolic tangent of x = tanh x = cosech x =

e x - e- x sinh x = . e x + e- x cosh x

1 1 1 ; sech x = ;coth x = . sinh x cosh x tanh x

70 INVERSE OR ANTI-HYPERBOLIC FUNCTIONS If x = sinh y, is the inverse hyperbolic sine of x, write y = sinh−1 x or arc sinh x.

(

)

sinh -1 x = log e x + x2 + 1 , 1 æ1+ xö tanh -1 x = log e ç ÷, 2 è1- xø æ 1 + 1 - x2 sech -1 x = log e ç ç x è

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ö ÷, ÷ ø

(

)

cosh -1 x = log e x + x2 - 1 , 1 æ x +1ö coth -1 x = log e ç ÷, 2 è x -1 ø æ 1 + 1 + x2 cosech -1 x = log e ç ç x è

ö ÷ ÷ ø

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40 • Mathematical Formulas and Scientific Data

71 FUNDAMENTAL IDENTITIES sinh (−x) = − sinh x,

cosh2 x − sinh2 x

cosh (−x) = − cosh x,

sech2 x + tanh2 x

tanh (−x) = − tanh x,

cosech x − coth x = −1, 2

= 1, = 1,

2

sinh (x ± y) = sinh x cosh y ± cosh x sinh y. cosh (x ± y) = cosh x cosh y ± sinh x sinh y. tan (x ± y) =

tan x ± tanh y . 1 ± tanh x tanh y

sinh 2x = 2 sinh x cosh x, 2 sinh2

x = cosh x − 1, 2

cosh 2x = cosh2 x + sinh2 x. 2 cosh2

x = cosh x + 1. 2

72 CONNECTION WITH CIRCULAR FUNCTIONS e ix - e- ix , 2i sin x = −i sinh ix,

e ix + e- ix , 2 cos x = cosh ix,

tan x = −i tanh ix,

cos ix = cosh x

tan ix = i tanh ix,

sin x =

cos x =

sin ix = i sinh ix, sinh ix = i sin x,

i2 = −1.

cosh ix = cos x,

tanh ix = i tan x.

sin (x ± iy) = sinh x cos y ± i cosh x sin y. cos (x ± iy) = cosh x cos y ± i sinh x sin y. sinh (x + 2 iπ) = sinh x, sinh (x + iπ) = − sinh x, 1 ö æ sinh ç x + ip ÷ = i cosh x. 2 ø è

cosh (x + 2 iπ) = cosh x. cosh (x + iπ) = − cosh x. 1 ö æ cosh ç x + ip ÷ = i sinh x. 2 ø è

eθ = cosh θ + sinh θ,

e−θ = cosh θ − sinh θ.

eiθ = cosh θ + i sinh θ,

e−iθ = cosh θ − i sinh θ.

x + iy = reiθ , where r = +

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x2 + y2 , θ = arc cos

y x = arc sin . r r

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Trigonometry • 41

y 1 log e ( x ± iy) = log e ( x2 + y2 ) ± i arc tan 2 x n ( cos q + i sin q ) = cos nq + i sin nq. n

2 kp 2 kp ö æ + i sin ç cos ÷ = 1, k = 0, 1, ..., n - 1. n n ø è æ x+ yö æ x-yö sinh x + sinh y = 2 sinh ç cosh ç ÷ ÷. è 2 ø è 2 ø æ x+ yö æ x-yö sinh x - sinh y = 2 cosh ç sinh ç ÷ ÷. è 2 ø è 2 ø æ x+ yö æ x-yö cosh x + coshy = 2 cosh ç cosh ç ÷ ÷. è 2 ø è 2 ø æ x+ yö æ x-yö cosh x - cosh y = 2 sinh ç sinh ç ÷ ÷. è 2 ø è 2 ø

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4

CHAPTER

Analytic Geometry 73 RECTANGULAR COORDINATES Let X´X (x-axis) and Y´Y (y-axis) be two perpendicular lines meeting in the point O called the origin. The point P(x, y) in the plane of the x and y axes is fixed by the distances x (abscissa) and y (ordinate) from Y´Y and X´X, respectively, to P. x is positive to the right and negative to the left of the y-axis, and y is positive above and negative below the x-axis.

FIGURE 4.1.

74 POLAR COORDINATES Let OX (initial line) be a fixed line in the plane and O (pole or origin) a point on this line. The position of any point P(r, θ) in the plane is determined by the distance r (radius vector) from O to the point together with the angle (vectorial angle) measured from OX to OP. θ is positive if measured counterclockwise, negative if measured clockwise, r is positive if measured along the terminal side of θ and negative if measured along the terminal side of produced through the pole.

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44 • Mathematical Formulas and Scientific Data

75 RELATIONS BETWEEN RECTANGULAR AND POLAR COORDINATES

= x  = y

r x 2 + y2 , r cos θ, =   r sin θ, θ =tan −1 y , x 

y  sin θ = 2 x + y2   x cos θ =  x 2 + y2 

76 POINTS AND SLOPES Let P1(x1, y1) and P2(x2, y2) be any two points, and α1 be the angle measured counterclockwise from OX to P1P2: Distance between P1 and P2 = P1P2

d=

( x2 − x1 )

Slope of P1P2 = tan α1 = m =

2

+ ( y2 − y1 ) . 2

y2 − y1 . x2 − x1

 m x + m2 x1 m1 y2 + m2 y1  , Point dividing P1P2 in the ratio m1 : m2 is  1 2  m1 + m2   m1 + m2  x + x2 y1 + y2  Mid-point of P1 P2 is  1 , . 2   2 The angle β between lines of slopes m1 and m2, respectively, is given by tan β =

m2 − m1 . 1 + m1 m2

Two lines of slopes m1 and m2 are 1 perpendicular if m2 = − , and m1 parallel if m1 = m2.

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FIGURE 4.2.

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Analytic Geometry • 45

77 AREA OF TRIANGLE If the vertices are the points (x1, y1), (x2, y2), and (x3, y3), then the area is equal to the numerical value of x1 1 x2 2 x3

y1 y2 y3

1 1 1= ( x1 y2 − x1 y3 + x2 y3 − x2 y1 + x3 y1 − x3 y2 ) . 2 1

78 LOCUS AND EQUATION The set of all points which satisfy a given condition is called the locus of that condition. An equation is called the equation of the locus if it is satisfied by the coordinates of every point on the locus and by no other points. There are three common representations of the locus by means of equations: (a) Rectangular equation which involves the rectangular coordinates (x, y). (b) Polar equation which involves the polar coordinates (r, θ). (c) Parametric equations which express x and y or r and θ in terms of a third independent variable called a parameter.

79 TRANSFORMATION OF COORDINATES To transform an equation or a curve from one system of coordinates in x, y to another such system in x´, y´, substitute for each variable its value in terms of variables of the new system. (a) Rectangular System: Old axes parallel to new axes. The coordinates of new origin in terms of the old system are (h, k). x =  y =

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x′ + h, y′ + k

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46 • Mathematical Formulas and Scientific Data

(b) Rectangular System: Old origin coincident with a new origin and the x´-axis making an angle θ with the x-axis. x =  y =

x′ ⋅ cos θ − y′ ⋅ sin θ. x′ ⋅ sin θ + y′ ⋅ cos θ.

(c) Rectangular System: Old axes are not parallel with new ones. New origin at (h, k) in the old system. x =  y =

x′ ⋅ cos θ − y′ ⋅ sin θ + h. x′ ⋅ sin θ + y′ ⋅ cos θ + k.

80 STRAIGHT LINE The equations of the straight line may assume the following forms:

FIGURE 4.3.

(a)

y = mx + b.

(m = slope, b = intercept on y-axis).

(b)

y − y1 = m(x − x1).

[m = slope, line passes through point (x1, y1)].

(c)

y − y1 y2 − y1 = . x − x1 x2 − x1

[line passes thru points (x1, y1) and (x2, y2)].

(d)

x y + = 1. a b

(a and b are the intercepts on the x and y-axis, respectively).

(e)

(normal form, p is distance from x cos α + y sin α = p. ­origin to the line, α is angle which normal to the line makes with x-axis).

(f)

Ax + By + C = 0.

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(general form, slope = − A ÷ B).

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Analytic Geometry • 47

To reduce Ax + By + C = 0 to normal form (e), divide by ± A2 + B2 , where the sign of the radical is taken opposite to that of C when C ≠ 0. The distance from the line Ax + By + C = 0 to the point P2(x2, y2) is: d=

Ax2 + By2 + C ± A2 + B2

.

81 CIRCLE General equation of a circle with radius R and center at (h, k) is: (x − h)2 + (y − k)2 = R2.

82 CONIC The locus of a point P which moves so that its distance from a fixed Point F (focus) bears a constant ratio e (eccentricity) its distance from a fixed straight line (directrix) is a conic. If d is the distance from focus to directrix, and F is at the origin. x2 + y2 = e2 (d + x)2 . de . r= 1 − e cosq If

e = l, the conic is a parabola; if



e > l, a hyperbola; and if e < l, an ellipse.

FIGURE 4.4.

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48 • Mathematical Formulas and Scientific Data

83 PARABOLA

e = l.

FIGURE 4.5.

FIGURE 4.6.

(a) (y − k)2 = 4a(x − h). Vertex at (h, k), axis || OX (Figure 4.5).

Figure 4.5 is drawn for the case when a is positive.

(b) (x − h)2 = 4a(y − k). Vertex at (h, k) axis || OY (Figure 4.6).

Figure 4.6 is drawn for the case when a is negative.



Distance from vertex to focus = VF = a.



Distance from vertex to directrix = a.



Latus rectum = LR = 4a.

84 ELLIPSE

e < l.

FIGURE 4.7.

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FIGURE 4.8.

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Analytic Geometry • 49

(a)

(b)

( x − h)2 (y − k)2 = l. Center at (h, k), major axis || OX + a2 b2 (Figure 4.7). (y − k)2 ( x − h)2 = l. Center at (h, k), major axis || OY + a2 b2 (Figure 4.8). Major axis = 2a

Major axis = 2b.

Distance from center to either focus =

a2 − b2 .

Distance from center to either directrix = a/e. Eccentricity == e

a2 − b2 / a.

Latus rectum = 2b2/a. Sum of distances from any point P on the ellipse to foci, PF´ + PF = 2a.

85 HYPERBOLA

e > l.

FIGURE 4.9.

FIGURE 4.10.

Center at (h, k), transverse axis 2 || OX. ( y − k ) ( x − h) (a) − = 1. 2 2 a b Slopes of asymptotes = ± b/a (Figure 4.9). 2

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50 • Mathematical Formulas and Scientific Data

(b)

Center at (h, k), transverse axis || OY.

(y − k) ( x − h) − = 1. a2 b2 Slopes of asymptotes = ± a/b (Figure 4.l0). 2

2

Transverse axis = 2a.

Conjugate axis = 2b.

Distance from center to either focus =

a2 + b2 . Distance from center to either directrix = a/e. Difference of distances of any point on hyperbola from the foci = 2a. a2 + b2 . a Latus Rectum = 2b2/a. Eccentricity = e =

86 SINE CURVE = y a= sin (bx c).

FIGURE 4.11.

y = a cos (bx + c´) = a sin (bx + c), where c = c´ +

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π . 2

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Analytic Geometry • 51

y = p sin bx + q cos bx = a sin (bx + c),  q ,a p2 + q2 . where c = tan −1  =  p a = amplitude = maximum height of wave. 2p = wave length = distance from any point on wave to the correb sponding point on the next wave. c x = − = phase, indicates a point on x-axis from which the positive b half of the wave starts.

87 TRIGONOMETRIC CURVES (1) y = a tan bx, or (1)  y = a sec bx, or, 1 1  y  y x = tan −1   . x = sec −1   . b b a a (2) y = a cot bx, or, 1  y x = cot −1   . b a

FIGURE 4.12.

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(2)  y = a cosec bx, or, 1  y x = cosec −1   . b a

FIGURE 4.13.

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52 • Mathematical Formulas and Scientific Data

88 LOGARITHMIC AND EXPONENTIAL CURVES y = logax or x = ay.

y = ax or x = logay.

Logarithmic Curve

Exponential Curve

FIGURE 4.14

FIGURE 4.15

89 PROBABILITY CURVE (FIGURE 4.16)

y = e− x

2

FIGURE 4.16.

90 OSCILLATORY WAVE OF DECREASING AMPLITUDE (FIGURE 4.17) y = e–ax sin bx.

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Analytic Geometry • 53

FIGURE 4.17.

91 CATENARY (FIGURE 4.18) = y

x −  a  ax a + e e  . 2 

A curve made by a cord of uniform weight suspended freely between two points. Length of

 2  2 d 2 2  s l 1 +  arc ==  ,  3  l  

FIGURE 4.18.

approximately, if d is small in comparison with l.

92 CYCLOID  x=  y =

a ( φ − sin φ ) . a (1 − cos φ ) .

A curve is described as a point on the circumference of a circle which rolls along a fixed straight line.

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FIGURE 4.19.

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54 • Mathematical Formulas and Scientific Data

Area one arch = 3πa2. Length of the arc of one arch = 8a.

93 PROLATE AND CURTATE CYCLOID aφ − b sin φ,  x=   y = a − b cos φ. A curve is described by a point on a circle at a distance b from the center of the circle of radius a which rolls along a fixed straight line. Prolate Cycloid

Curtate Cycloid

FIGURE 4.20.

FIGURE 4.21.

94 EPICYCLOID   x=   = y 

a+b  φ , (a + b)cos φ − a cos   a  a+ b  φ . (a + b)sin φ − a sin   a 

A curve is described as a point on the circumference of a circle that rolls along the outside of a fixed circle.

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Analytic Geometry • 55

95 CARDIOID

FIGURE 4.22.

FIGURE 4.23.

96 CARDIOID r = a (1 + cos θ ) . An epicycloid in which both circles have the same radii is called a cardioid.

97 HYPOCYCLOID a−b  φ , (a − b)cos φ − b cos   b  a−b  (a − b)sin φ − b sin  φ  b 

  x=    y= 

A curve is described as a point on the circumference of a circle that rolls along the inside of a fixed circle.

98 HYPOCYCLOID OF FOUR CUSPS 2

2

2

x 3 + y3 = a3 . x = a cos3 φ,

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y = a sin3 φ.

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56 • Mathematical Formulas and Scientific Data

The radius of a fixed circle is four times the radius of a rolling circle.

FIGURE 4.24.

FIGURE 4.25.

99 INVOLUTE OF THE CIRCLE = x  = y

a cos φ + a φ sin φ a sin φ + a φ cos φ.

A curve generated by the end of a string is kept taut while being unwound from a circle.

100 LEMNISCATE r2 = 2 a2 cos 2θ.

FIGURE 4.26.

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FIGURE 4.27.

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Analytic Geometry • 57

101 N-LEAVED ROSE (l) r = a sin nθ

(2) r = a cos nθ.

If n is an odd integer there are n leaves, and if n is even, 2n leaves, of which the figure shows one.

102 SPIRALS

FIGURE 4.28.

r = a θ.

FIGURE 4.29.

r θ = a.

FIGURE 4.30.

r = eaθ , or a θ = loge r.

Solid Analytic Geometry

103 COORDINATES (FIGURE 4.31) (a) Rectangular system: The position of a point P(x, y, z) in space is fixed by its three distances x, y, and z from three coordinate planes XOY, XOZ, and ZOY, which are mutually perpendicular and meet in a point O (origin). (b) Cylindrical system: The position of any point P(r, θ, z) is fixed by (r, θ), the polar coordinates of the projection of P in the XOY plane, and by z, its distance from the XOY plane.

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FIGURE 4.31.

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58 • Mathematical Formulas and Scientific Data

(c) Spherical (or polar or geographical) system: The position of any point P(ρ, θ, φ) is fixed by the distance ρ = OP , the angle θ = ∠XOM and the angle φ = ∠ZOP . θ is called the longitude and the co-latitude. The following relations exist between the three coordinates systems:  x = ρ sin φ cos θ,   y = ρ sin φ sin θ,  z = ρ cos φ. 

 r = ρ sin φ,   z = ρ cos φ,  

ρ r 2 + z2 . =   z . cos φ = 2 r + z2 

104 POINTS, LINES, AND PLANES Distance between two points P1(x1, y1, z1) and P2(x2, y2, z2), is d=

( x2 − x1 )

2

+ ( y2 − y1 ) + ( z2 − z1 ) . 2

2

Direction cosines of a line are the cosines of the angles α, β, and γ which the line or any parallel line makes with the coordinate axes. The direction cosines of the line segment P1 (x1, y1, z1) to P2(x2, y2, z2) are: cos α =

y − y1 x2 − x1 z −z , cos β = 2 , cos γ = 2 1 . d d d          

cos2 α + cos2 β + cos2 γ =1. Angle θ between two lines, whose direction angles are α1, β1, γ1, and α2, β2, γ2, is given by cos= θ cos α1 cos α 2 + cos β1 cos β2 + cos γ 1 cos γ 2 . The equation of a plane is: Ax + By + Cz + D = 0, where A, B, and C are proportional to the direction cosines of a normal (a line perpendicular to the plane) to the plane.

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Analytic Geometry • 59

Angle between two planes is the angle between their normals. Equation of a straight line through the point P1(x1, y1, z1) is:

x − x1 y − y1 z − z1 = = , a b c where a, b, and c are proportional to the direction cosines of the line. a, b, and c are called the direction numbers of the line.

105 FIGURES IN THREE DIMENSIONS Plane∗

Elliptic Cylinder∗

FIGURE 4.32.

FIGURE 4.33.

x y z + + = 1. a b c ∗

x2 y2 + = 1. a2 b2

Only a portion of the figure is shown.

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Sphere

Ellipsoid

FIGURE 4.34.

FIGURE 4.35.

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60 • Mathematical Formulas and Scientific Data

x 2 y2 z 2 + + = 1. a2 b2 c2

x 2 + y2 + z 2 = r2. Elliptic Paraboloid

Portion of Cone

Hyperboloid of One Sheet

FIGURE 4.36.

FIGURE 4.37.

FIGURE 4.38.

x 2 y2 + = cz. a2 b2

x2 y2 z2 + − = 0. a2 b2 c2

Hyperboloid of Two Sheets

Hyperbolic Paraboloid

FIGURE 4.39.

FIGURE 4.40.

x2 y 2 z 2 − − = 1. a 2 b2 c2

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x2 y 2 z 2 1. + − = a 2 b2 c2

x2 y 2 − = cz. a 2 b2

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5

CHAPTER

Differential Calculus 106 DEFINITION OF FUNCTION A variable y is said to be a function of the variable x, if, for every x (taken at will on its range), y is determined. The symbols f(x), F(x), g(x), φ(x), etc., are used to represent various functions of x. The symbol f(a) represents the value of f(x) when x = a.

107 DEFINITION OF DERIVATIVE AND NOTATION Let y = f(x) be a single-valued (continuous) function of x. Let Δx be any increment (increase or decrease) given to x, and let Δy be the corresponding increment in y. The derivative of y with respect to x is the limit, (if it exists), of the ratio of Δy to Δx as Δx approaches zero in any manner whatsoever; that is, dy ∆y f ( x + ∆x) − f ( x) =lim =lim =f ′( x) =Dx y =y′. ∆ → 0 ∆ → 0 x x dx ∆x ∆x Higher derivatives are defined as follows:



d 2 y d  dy  d = = f ′( x) = f ″( x). dx2 dx  dx  dx



d3 y d  d2 y  d = f ″( x)= f ′″( x).  = dx3 dx  dx2  dx

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(second derivative)

(third derivative)

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62 • Mathematical Formulas and Scientific Data

d n y d  d n−1 y  d ( n−1) = = f ( x) f ( n) ( x).  n −1  = n dx dx  dx  dx

(nth derivative)

The symbol f(n) (a) represents the value of f(n) (x) when x = a.

108 CERTAIN RELATIONS AMONG DERIVATIVES dy dx = 1÷ . dx dy dy dy du If  y = f(u), and u = F(x), then = ⋅ dx du dx If  x = f(y),      then

If  x = f(α), y = φ(α),  then dy φ′(α) d 2 y , = = dx f ′(α) dx2



f ′(α) ⋅ φ″(α) − φ′(α) ⋅ f ″(α)

[ f ′(α)]

3

.

109 TABLE OF DERIVATIVES In this table, u and v represent functions of x: a, n, and e represent constants (e = 2.7l83 ...), and all angles are measured in radians: d ( x) = 1 dx

d (a) = 0. dx

d du dv ( u ± v ± ...)= ± ± .... dx dx dx d du (au) = a . dx dx d  u  = dx  v 

v

du dv −u dx dx . 2 v

d n du . u ) = nun−1 ( dx dx

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d dv du (= uv) u + v . dx dx dx d du sin u = cos u . dx dx d du cos u = − sin u . dx dx

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Differential Calculus • 63

log a e du d log a u = . x u dx

d du tan u = sec 2 u . dx dx

d 1 du log e u = . dx u dx

d du cot u = − cosec 2 u . dx dx

d u du a =⋅ a u log e a ⋅ . dx dx

d du sec u = sec u tan u . dx dx

d u du e = eu . dx dx

d du cosec u = − cosec u cot u . dx dx

d du d v du dv = + u v log e u vers u = sin u . u vu v−1 dx dx dx dx dx     sin x 1 1 =1, lim(1 + x)1/ x =e =2.71828 =1 + 1 + + + . x →0 x →0 2! 3! x

lim

d du 1 sin −1 u = , 2 dx 1 − u dx

−π π ≤ sin −1 u≤ . 2 2

d du 1 cos−1 u = , 2 dx 1 − u dx

0 ≤ cos−1 u ≤ π.

d 1 du tan −1 u = . dx 1 + u2 dx

d 1 du cot −1 u = − . dx 1 + u2 dx

d 1 du π π −1 sec u , − π≤ sec −1 u < − , 0≤ sec −1 u < . = 2 dx 2 2 u u − 1 dx d 1 du π π cosec −1 = − , − π < csc −1 u≤ − , 0 < csc −1 u ≤ . 2 dx dx 2 2 u u −1 d 1 du vers−1 u = , v 2 dx 2 u − u dx

0 ≤ vers−1 u ≤ π.

d du sinh u = cosh u . dx dx

d du cosh u = sinh u . dx dx

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64 • Mathematical Formulas and Scientific Data

d du tanh u = sech 2 u . dx dx

d du coth u = − cosech 2 u . dx dx

d du sech u = − sech u tanh u . dx dx

d du cosech u = − cosech u coth u . dx dx

d sinh −1 u = dx

1

du . u + 1 dx 2

d = cosh −1 u dx

1

du , u > 1. u − 1 dx 2

d 1 du tanh −1 u = . dx 1 − u2 dx

d 1 du coth −1 u = − 2 . dx u − 1 dx

d du 1 sech −1 u = − , u > 0. 2 dx u 1 − u dx

d du 1 cosech −1 u = − . 2 dx u u + 1 dx

110 SLOPE OF A CURVE: EQUATION OF TANGENT AND NORMAL (RECTANGULAR COORDINATES) The slope of the curve y = f(x) at the point P(x, y) is defined as the slope of the tangent line to the curve at P: Slope = m = tan α =

dy = f´(x). dx

Slope at x = x1 is m1 = f´ (x1). Equation of tangent line to curve at P1 (x1, y1) is y − y1 = m1 (x − x1). Equation of normal line to curve at P1 (x1, y1) is 1 − y − y1 = ( x − x1 ) . m1 Angle (θ) of intersection of two curves whose slopes are m1 and m2 at the common point is given by tan θ =

m2 − m1 . 1 + m1 m2

The sign of tan θ determines whether the acute or obtuse angle is meant.

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Differential Calculus • 65

FIGURE 5.1

111 DIFFERENTIAL If y = f(x), Δx is an increment of x, and f´ (x) is the derivative of f(x) with respect to x, then the differential of x equals the increment of x, or dx = Δx; and the differential of y, dy is the product of f´(x) and the increment of x; dy df ( x) dy =f ′( x)dx = dx = dx, and dx dx

dy f ′( x) = . dx

If x = f (t), y = φ(t), then dx = f´ (t) dt and dy = φ´(t) dt. Every derivative formula has a corresponding differential formula. d(sin u) = cos u . du;

d(u . v) = u dv + vdu.

112 MAXIMUM AND MINIMUM VALUES OF A FUNCTION A maximum (minimum) value of a function f(x) in the interval (a, b) is a value of the function which is greater (less) than the ­values of the function in the immediate vicinity. The values of x which give a maximum or minimum value to y = f (x) are found by solving the equations f´ (x) = 0 or ∞. If a is a root of f´ (x) = 0 and if f´ (a) < 0, f (a) is a maximum; if f´ (a) > 0, f (a) is a minimum. If f ˝ (a) = 0, f ˝´ (a) 0, f (a) is neither a maximum nor a minimum, but if f ˝ (a) = f ˝´ (a) = 0, f (a) is maximum or minimum according as f IV (a) < > 0.

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66 • Mathematical Formulas and Scientific Data

In general, if the first derivative which does not vanish for x = a is of odd order, f (a) is neither a maximum nor a minimum; but if it is of even order, the 2nth, say, then f (a) is a maximum or minimum according as f (2n) (a) < > 0. To find the largest or smallest values of a function in an interval (a, b), find f(a), f(b), and compare with the maximum and minimum values as found in the interval.

113 POINTS OF INFLECTION OF A CURVE The curve is said to have a point of inflection at x = a if f ˝(a) = 0 and f ˝(x) < 0 on one side of x = a and f ˝(x) > 0 on the other side of x = a. Wherever f ˝(x) < 0, the curve is concave downward, and wherever f ˝(x) > 0, the curve is concave upward.

114 DERIVATIVE OF ARC LENGTH: RADIUS OF CURVATURE Let s be the length of arc measured along the curve y = f(x), [or in polar coordinates r = φ(θ)], from some fixed point to any point P(x, y), and α be the angle of inclination of the tangent line at P with OX. Then 1

dx = cos α= ds

 dy  1+   dx 

2

,

dy = sin α= ds

 dy  ds = dx + dy = 1 +    dx  2

ds =

dr + r dθ= 2

2

2

 dr  r +   dθ  2

 dx  1+   dy 

2

,

2

 dx  dx = 1 +   dy.  dy  2

2

d= θ

 dθ  1 + r   dr.  dr  2

x = r cos θ, y = r sin θ,

If

2

2

1

dx = cos θ · dr − r sin θ · dθ, dy = sin θ · dr + r cos θ · dθ

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Differential Calculus • 67

The radius of curvature R at any point P(x, y) of the curve y = f(x) is 3

  dy 2  2 1 +    ds   dx   R = = = dα  d2 y   2  dx 

{

1 + [ f ′( x)]

2

}

f ″( x)

3 2

,

3

 2  dr 2  2 r +     dθ   . R=  2 d2 r  dr  2 r + 2  − r 2 dθ  dθ  1 . R The center of curvature corresponding to the point (x1, y1) on y = f(x) The curvature (K) at (x, y) is K = is (h, k), where h= x1 − k y1 + =

{

f ′ ( x1 ) 1 +  f ′ ( x1 )  f ″ ( x1 )

1 +  f ′ ( x1 ) 

2

},

2

f ″ ( x1 )

.

115 MEAN ROLLE’S THEOREM If y = f (x) and its derivative f´(x) be continuous on the interval (a, b), there exist a value of x somewhere between a and b such that f (b= ) f (a) + b − a) f ′( x1 ), a < x1 < b. Rolle’s Theorem is a special case of the theorem of the mean with f (a) = f (b) = 0; that is, there exists at least one value of x between a and b for which f´ (x1) = 0.

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68 • Mathematical Formulas and Scientific Data

116 EVALUATION OF INDETERMINATE FORMS If f(x) and F(x) be two continuous functions of x having continuous derivatives, f´(x) and F´(x), then: (a) If lim f ( x) = 0 and lim F( x) = 0 and lim F′( x) ≠ 0, x→ a

x→ a

x→ a

[or if lim f ( x) = lim F( x) = ∞ ], then



x→ a

x→ a

lim

f ( x) f ′( x) = lim . x → a F( x) F′( x)

x→ a

(b) If lim f ( x) = 0 and lim F(x) = ∞ [i.e., F(x) becomes infinite as x→ a x→ a x approaches a as a limit], then lim[ f ( x).F( x)] may often be x→ a determined by writing f ( x).F( x) =

f ( x) , 1 / F( x)

thus expressing the functions in the form of (a).

(c) If lim f ( x) = ∞ and lim F( x) = ∞, then lim[ f ( x) − F( x)] may x→ a

x→ a

often be determined by writing

x→ a

[1 / F( x)] − [1 / f ( x)] f ( x) − F( x) = 1 / [ f ( x).F( x)]



and using (a).

(d) The lim  f ( x)F ( x)  may frequently be evaluated upon writing x→ a f ( x)F ( x) = eF ( x).log e f ( x) When one factor of the last exponent approaches zero and the other becomes infinite, the exponent is of the type considered in (b). Thus, we are led to the indeterminate forms which are symbolized by 0 / 0, ∞ / ∞, 0°, 1∞ , ∞ 0 .

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Differential Calculus • 69

117 TAYLOR’S AND MACLAURIN’S THEOREM Any function (continuous and having derivatives) may, in general, be expanded into a Taylor’s series: f ( x= ) f (a) + f ′(a) ⋅ + f ′″(a) ⋅

( x − a) ( x − a)2 + f ″(a ) ⋅ 1! 2!

( x − a)3 ( x − a)n−1 + …+ f ( n−1) (a) ⋅ + Rn , 3! (n − 1)!

where a is any quantity for which f (a), f´ (a), f” (a), ... are finite. If the series is to be used for approximating f(x) for some value of x, then a should be picked so that the difference (x − a) is numerically very small, and thus only a few terms of the series need to be used. The remainder, after n terms, is Rn = f (n)(x1) · (x − a)n/n!, where x1 lies between a and x. Rn gives the limits of error in using n terms of the series for the approximation of the function: n! = n = 1 ⋅ 2 ⋅ 3 ⋅ 4 n. If a = 0, the above series is called Maclaurin’s series: ) f (0) + f ′(0) f ( x=

x x2 x3 x n −1 + f ″(0) + f ′″(0) + …+ f ( n−1) (0) + Rn . 1! 2! 3! (n − 1)!

118 SERIES The following series may be obtained through the expansion of the functions by Taylor’s or Maclaurin’s Theorems. The expressions following a series indicate the region of convergence of the series, that is, the values of x for which Rn approaches zero as n becomes infinite, so that an approximation of the function may be obtained by using a number of terms of the series. If the region of convergence is not indicated, the series converges for all finite values of x. (n! = l·2·3...n). Loge u ≡ Log u. (a) Binomial Series n(n − 1) n− 2 2 a x 2! n(n  1)(n  2) n 3 3  a x  , x2  a2 . 3!

(a + x)n =a n + na n−1 x +

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70 • Mathematical Formulas and Scientific Data

If n is a positive integer, the series consists of (n + l) terms; otherwise, the number of terms is infinite.  1 bx b2 x2 b3 x3 (a − bx)−1 =  1 + + 2 + 3 + … , b2 x2 < a2 . a a a a  (b) Exponential, Logarithmic, and Trigonometric Series. e = 1+

1 1 1 1 + + + + . 1! 2! 3! 4!

ex = 1 + x +

x2 x3 x 4 + + + . 2! 3! 4!

ax = 1 + x log a + 2

e− x = 1 − x2 +

( x log a)2 ( x log a)3 + + . 2! 3!

x 4 x6 x8 − + − . 2! 3! 4!

1 1 2 3 0 < x ≤ 2. log x = ( x − 1) − ( x − 1) + ( x − 1) −…, 2 3        log x =

2 3 1 x −1 1  x −1 1  x −1 x> . +  +    + , 2 x 2 x  3 x       

 x − 1 1  x − 1 3 1  x − 1  5  +  log x = 2   +   +  ,   x > 0.  x + 1 3  x + 1  5  x + 1      log (1 + x) = x −

x2 x3 x 4 + − +, 2 3 4

−1 < x ≤ 1.             

3 5  x  1 x  1 x  log (1 + x) = log a + 2  +  +    +  ,  2 a + x 3  2 a + x  5  2 a + x   a > 0, −a < x< + ∞.

  x 3 x 5 x7 1+ x  x2 < 1. log   = 2  x + 3 + 5 + 7 + ...  , 1− x              

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Differential Calculus • 71

1 1  1  1  1  1  1    x+1 log   = 2  +   +   +   +  ,  x −1   x 3  x  5  x  7  x   3

5

7

    x2 >

1.

1 1  1   x +1 + + + … ,     x > 0. log   = 2 3 5 5(2 x + 1)  x   2 x + 1 3(2 x + 1) 

(

)

log x + 1 + x2 = x −

1 x 3 1 ⋅ 3 x 5 1 ⋅ 3 ⋅ 5 x7 + − +, 2 3 2⋅4 5 2⋅4⋅6 7

      x2

< 1.

sin x = x −

x 3 x 5 x7 + − + . 3! 5! 7!

cos x = 1 −

x2 x 4 x6 + − + 2! 4! 6!

tan x = x +

x3 2 x 5 17 x7 62 x9 π2 + + + + , x2 < . 3 15 315 2835 4       

sin −1 x = x +

x 3 1 3 x 5 1 3 5 x7 + ⋅ ⋅ + ⋅ ⋅ ⋅ + ,       x2 < 1. 6 2 4 5 2 4 6 7

1 1 1 tan −1 x = x − x3 + x 5 − x7 + ,            x2 < 1. 3 5 7 =

π 1 1 1 − + 3 − 5 + ,             x2 > 1. 2 x 3x 5x

log sin x = log x − log cos x = −

x2 x 4 x6 17 x8 − − − − , 2 12 45 2520

log tan x = log x

x2 7 x 4 62 x6 + + + , 3 90 2835

e sin x = 1 + x +

MFSDH.CH05_2PP.indd 71

x2 x4 x6 − − − ,           x2 1 2 2 x2 2 ⋅ 4 4 x 4 2 ⋅ 4 ⋅ 6 6 x6

cosh−1 x = log 2 x −

1 1 1⋅3 1 1⋅3⋅5 1 − − −. 2 4 2 2x 2 ⋅ 4 4x 2 ⋅ 4 ⋅ 6 6 x6

tanh−1 x = x +

x 3 x 5 x7 + + + ,              x2 < 1. 3 5 7

119 PARTIAL DERIVATIVES: DIFFERENTIALS If z = f(x, y), is a function of two variables, then the derivative of z with respect to x, as x varies while y remains constant, is called the ∂z first partial derivative of z with respect to x and is denoted by . ∂x Similarly, the derivative of z with respect to y, as y varies while x remains constant, is called the first partial derivative of z with ∂z respect to y and is denoted by . ∂y Similarly, if z = f (x, y, u, ...), then the first derivative of z with respect to x, as x varies while y, u, ... remain constant, is called the

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Differential Calculus • 73

∂z . Likewise, ∂x the second partial derivatives are defined as indicated below: first partial of z with respect to x and is denoted by

∂ 2 z ∂  ∂z  ∂ 2 z ∂  ∂z  ∂ 2 z ∂  ∂z  ∂  ∂z  ∂ 2 z ; . = =  =   =  = ; 2   2 ∂x ∂x  ∂x  ∂y ∂y  ∂y  ∂x ∂y ∂x  ∂y  ∂y  ∂x  ∂y ∂x If z = f(x, y, ..., u), and x, y, ..., u are functions of a single variable t, then dz ∂z dx ∂z dy ∂z du = + + …+ , dt ∂x dt ∂y dt ∂u dt ∂z ∂z ∂z dz = dx + dy +  + du. ∂x ∂y ∂u If F(x, y, z, ..., u) = 0, then ∂F dx + ∂F dy +  + ∂F du = 0. ∂x ∂y ∂u

120 SURFACES: SPACE CURVES (SEE ANALYTIC GEOMETRY ART. 97-98) The tangent plane to the surface F(x, y, z) = 0 at the point (x1, y1, z1) on the surface is  ∂F  ∂F   ∂F  0,  + ( z − z1 )   + ( y − y1 )   =  ∂x 1  ∂z 1  ∂y 1

( x − x1 ) 

 ∂F  ∂F where  at (x1, y1, z1), etc.  is the value of  ∂x 1 ∂x The equations of the normal to the surface at (x1, y1 z1) are y − y1 x − x1 z − z1 = = .  ∂F   ∂F   ∂F       ∂x 1  ∂y   ∂z 1 1

The direction cosines of the normal to the surface at the point (x1, y1, z1) are proportional to

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74 • Mathematical Formulas and Scientific Data

 ∂F   ∂F   ∂F    ,  ,  .  ∂x 1  ∂y 1  ∂z 1 Given the space curve x = x(t), y = y(t), z = z(t). The direction cosines of the tangent line to the curve at any point are proportional to dx dy dz , , , or to dx, dy, dz. dt dt dt The equations of the tangent line to the curve at (x1, y1, z1) on the curve are: y − y1 x − x1 z − z1 = =  dx   dy   dz         dt 1  dt 1  dt 1  dx  dx where   is the value of at (x1, y1, z1), etc. dt  1 dt

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CHAPTER

6

Integral Calculus—I 121 INTRODUCTION F(x) is said to be an indefinite integral of f(x), if the derivative of F(x) is f(x), or the differential of F(x) is f(x) dx; symbolically: F( x)   f ( x)dx if

dF( x)  f ( x), or dF( x)  f ( x)dx. dx

In general:  f ( x)dx  F( x)  C, where C is an arbitrary constant.

122 FUNDAMENTAL THEOREMS ON INTEGRALS. SHORT TABLE OF INTEGRALS (u and v denote functions of x; a, b and C denote constants). 1.

 df (x)  f (x)  C.

2. d  f ( x)dx  f ( x)dx.

MFSDH.CH06_2PP.indd 75

3.

 0  dx  C.

4.

 a f (x)dx  a f (x)dx.

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76 • Mathematical Formulas and Scientific Data

5.

 (u  v)dx   udx   vdx.

6.

 u dv  uv   v du.

u dv du dx  uv   v dx. dx dx ( fy)dy 8.  f (y)dx   . dy dx 7.



9.

n  u du 

10. 

u n 1  C, n  1. n1

du  log e u  C, u > 0; or, log e | u | C, u  0. u

11.  e u du  e u  C. 12.  bu du 

bu  C, b  0, b  1. log e b

13.  sin u du   cos u  C. 14.  cos u du  sin u  C. 15.  tan u du  log e sec u  C   log e cos u  C. 16.  cot u du  log e sin u  C   log e cosec u  C.  u  17.  sec u du  log e (sec u  tan u)  C  log e tan     C. 2 4 u 18.  cosec u du  log e (cosec u  cot u)  C  log e tan  C. 2 1 1 19.  sin 2 u du  u  sin u cos u  C. 2 2 1 1 20.  cos2 u du  u  sin u cos u  C. 2 2 21.  sec 2 u du  tan u  C.

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Integral Calculus—I • 77

22.  cosec 2 u du   cot u  C. 23.  tan 2 u du  tan u  u  C. 24.  cot 2 u du   cot u  u  C. 25. 

du 1 u  tan 1  C. 2 u a a a

26. 

du 1 1  ua 1  u  2 2 log e     C   ctnh    C, if u  a , 2 u a 2a a  ua a 1 1  a u 1 u 2 2  log e    C   tanh    C, if u  a . 2a a u a a     

27.  28.  29.  30. 

2

2

 u   u  sin 1    C, a  0; or, sin 1    C, | a | 0. a a 2  u2 | a| du

du u2  a 2



 log e u  u2  a2

  C. 

 a u  cos1    C.  a  2 au  u du

2

1 1 a  u  sec 1    C  cos1  C. a u a u u2  a 2 a du

 a  a 2  u2 1 31.    log e   a u u a 2  u2  du

*

   C.  

1 u 32.  a2  u2  du   u a2  u2  a2 sin 1   C. 2 a * 1 33.  u2  a2 du   u u2  a2  a2 log e u  u2  a2   C.  2 





34.  sinh u du  cosh u  C. 35.  cosh u du  sinh u  C. 36.  tanh u du  log e (cosh u)  C.

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78 • Mathematical Formulas and Scientific Data

37.  coth u du  log e (sinh u)  C. 38.  sech u du  sin 1 (tan u)  C. u  39.  cosech u du  log e  tanh   C. 2  40.  sech u . tanh u . du   sech u  C. 41.  cosech . coth u . du   cosech u  C. *

 u  u2  a 2 log e   a   u  u2  a 2 a 

  a  a 2  u2  u   sinh 1   ; log e    u a     a  a 2  u2  u   cosh 1   ; log e    u a  

* log e 

  u   sech 1   ;  a 

  u   cosech 1   .  a 

Definite Integrals

123 DEFINITION OF DEFINITE INTEGRAL Let f(x) be continuous for the interval from x = a to x = b inclusive. Divide this interval into n equal parts by the points a, x1, x2, ..., xn–1, b such that Δx = (b − a)/n. The definite integral of f(x) with respect to x between the limits x = a to x = b is:



b



b

a

a

f ( x)dx  lim  f (a)x  f  x1  x  f  x2  x  ...  f  xn1  x . n 

b

b f ( x)dx    f ( x)dx    F( x)a  F(b)  F(a),  a

where F(x) is a function whose derivative with respect to x is f(x).

124 APPROXIMATE VALUES OF DEFINITE INTEGRAL Approximate values of the above definite integral are given by the rules of § 4l, where y0, y1, y2 ..., yn–1, yn are the values of f(x) for x = a, x1, x2, ..., xn–1, b, respectively, and h = (b − a)/n.

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Integral Calculus—I • 79

125 SOME FUNDAMENTAL THEOREMS

  f (x)  f (x)  ...  f (x) dx   b

1

a

∫

b

∫

b

∫

b

∫

b







b





a

2

n

b

a

b

b

a

a

f1 ( x)dx   f2 ( x)dx  ...   fn ( x)dx.

b

k f ( x)dx = k ∫ f ( x)dx , if k is a constant.

a

a

a

a

a

f ( x)dx =   f ( x)dx. b

f ( x)dx =

c

a

b

f ( x)dx   f ( x)dx. c

f ( x)dx = (b − a) f (x1), where x1 lies between a and b.

a







t

f ( x)dx = lim  f ( x)dx. t  a

b

f ( x)dx = lim  f ( x)dx. t   t

f ( x)dx =



c





f ( x)dx   f ( x)dx. c

If f(x) has a singular point∗ at x = b, b ≠ a,



b

a

f ( x)dx  lim  e0

b e

a

f ( x)dx.

The mean value of the function f (x) on the interval (a, b) is: 1 b f ( x)dx. b  a a Some Applications of the Definite Integral

126 PLANE AREA The area bounded by y = f (x), y = 0, x = a, x = b, where y has the same sign for all values of x between a and b, is: b

A   f ( x)dx, dA  f ( x)dx. a

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80 • Mathematical Formulas and Scientific Data

The area bounded by the curve r = f(θ) and the two radii θ = α, θ = β, is A

1  1 2 f () d, dA  r 2  d.    2 2

vvv

dr  rd

r

Y P(x, y)

d



y

a

P (r, )

dx

O



X

b

O

 

Figure 6.1. 

Figure 6.2. 

127 LENGTH OF ARC The length (s) of the arc of a plane curve f(x, y) = 0 from the point (a, c) to the point (b, d) is s

b

a

2

2

d  dx   dy  1     dx   1     dy. c  dx   dy 

If the equation of the curve is x = f(t), y = f(t), the length of the arc from t = a to t = b is s

b

a

2

2

 dx   dy        dt.  dt   dt 

If the equation of the curve is r = f (θ), then s

2

1

MFSDH.CH06_2PP.indd 80

2

2

r2  dr   d  r 2     d   r 2    1  dr. r1 d     dr 

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Integral Calculus—I • 81

128 VOLUME BY PARALLEL SECTIONS If the plane perpendicular to the x-axis at (x, 0, 0) cuts from a given solid a section whose area is A(x), then the volume of that part of the solid between x = a and x = b is:

∫

b

A( x)dx.

a

129 VOLUME OF REVOLUTION The volume of a solid of revolution generated by revolving that portion of the curve y = f(x) between x = a and x = b b

(a) about the x-axis is   y2 dx ; a

d

(b) about the y-axis is   x2 dy , where c and d are the values of c

y corresponding to the values a and b of x.

130 AREA OF SURFACE OF REVOLUTION The area of the surface of a solid of revolution generated by revolving the curve y = f (x) between x = a and x = b (a) about the x-axis is 2  

b

a

2

 dy  y 1     dx,  dx 

(b) about the y = axis is 2  

d

c

2

 dx  x 1     dy.  dy 

131 PLANE AREAS BY DOUBLE INTEGRATION (a) Rectangular coordinates A

b

a



f ( x)

( x )

d

dy dx or 

c



( y )

 ( y)

dx dy;

(b) Polar coordinates A

2

1

MFSDH.CH06_2PP.indd 81



f2 (  ) f1 ( )

r dr d or 

r2

r1



2 ( r )

1 ( r )

r d dr.

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82 • Mathematical Formulas and Scientific Data

132 VOLUMES BY DOUBLE INTEGRATION If z = f(x, y),

V=

b  ( x)

 a

 ( x)

f ( x, y) dy dx or 

d

c



 ( y)

 ( y)

f ( x, y) dx dy.

133 VOLUMES BY TRIPLE INTEGRATION (a) Rectangular coordinates V     dx dy dz; (b) Cylindrical Coordinates V     r dr d dz; (c) Spherical Coordinates V     2 sin  d d d., where the limits of integration must be supplied. Other formulas may be obtained by changing the order of integration. Area of Surface z = f(x, y) 2

A  

2  z   z        1  dy dx,  x   y 

where the limits of integration must be supplied.

134 MASS The mass of a body of density δ is m   dm, dm    dA, or   ds, or   dV , or   dS, where dA, ds, dV, and ds are, respectively, the elements of area, length, volume, and surface of Art. l25 to Art. l32.

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Integral Calculus—I • 83

135 DENSITY If δ is a variable (or constant) density (mass per unit of element), and δ is the mean density of a solid of volume V, then 

  dV .  dV

136 MOMENT The moments Myz, Mxz, and Mxy, of a mass m with respect to the coordinate planes (as indicated by the subscripts) are: Myz   x dm, M xz   y dm, M xy   z dm.

137 CENTROID OF MASS OR CENTER OF GRAVITY The coordinates  x , y, z  of the centroid of a mass m are: x

 x dm ,  dm

y

 y dm ,  dm

z

 z dm .  dm

NOTE In the above equations x, y, z are the coordinates of the center of gravity of the element dm.

138 CENTROID OF SEVERAL MASSES The x-coordinate ( x ) of the centroid of several masses m1, m2, ..., mn, having x1 , x2 ... , xn , respectively, as the x-coordinates of their centroids, is: x

m1 x1  m2 x2  ...  m n xn . m1  m2  ...  m n

Similar formulas hold for the other coordinates y , z .

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84 • Mathematical Formulas and Scientific Data

139(A)  MOMENT OF INERTIA (SECOND MOMENT) The moments of inertia (I). (a) for a plane curve about the x-axis, y-axis, and origin, respectively, are: I x   y2 ds,



Iy   x2 ds,

I0    x2  y2  ds;

(b) for a plane area about the x-axis, y-axis, and origin, respectively, are: I x   y2 dA,



Iy   x2 dA,

I0    x2  y2  dA;

(c) for a solid of mass m about the yz, xz, and xy-planes, x-axis, etc., respectively, are:

Iyz   y2 dm,

I xz   y2 dm,

I xy   z2 dm,



The limits of integration are to be supplied.

I x  I xz  I xy , etc.

139(B) THEOREM OF PARALLEL AXES Let L be any line in space and Lg a line parallel to L, passing through the centroid of the body of mass m. If d is the distance between the lines L and Lg, then IL  ILg  d 2 m, where IL and ILg are the moments of inertia of the body about the lines L and Lg, respectively.

140 RADIUS OF GYRATION If I is the moment of inertia of a mass m, and K is the radius of gyration, I = mK2. Similarly for areas, lengths, volumes, etc.

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Integral Calculus—I • 85

If masses (or areas, etc.) m1, m2, ..., mn, have, respectively, the radii of gyration k1 k2, ..., kn, with respect to a line or plane, then with respect to this line or plane, the several masses taken together have the radius of gyration K, where

K2 

m1 k12  m2 k22  ...m n kn2 . m1  m2  ...  m n

141 WORK The work W done in moving a particle from s = a to s = b by a force whose component expressed as function of s in the direction of motion is Fs, is W

s b

s a

Fs ds, dW  Fs ds.

142 PRESSURE The pressure (P) against an area vertical to the surface of a liquid and between the depths a and b is: P

y b

y a

wly dy, dp  wly dy,

where w is the weight of liquid per unit volume, y is the depth beneath surface of liquid of a horizontal element of area, and l is the length of the horizontal element of area expressed in terms of y.

143 CENTER OF PRESSURE The depth y of the center of pressure against an area vertical to the surface of the liquid and between the depths a and b is: y b

 y 

ydp

y a y b y a

MFSDH.CH06_2PP.indd 85

dp

.

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CHAPTER

7

Integral Calculus—II Certain Elementary Processes

144 TO INTEGRATE ∫ R( x ) . dx , WHERE R(X) IS RATIONAL FUNCTION OF X. Write R(x) in the form of a fraction whose terms are polynomials. If the fraction is improper, (i.e., the degree of the denominator is less than or equal to the degree of the numerator), divide the numerator by the denominator and thus write R(x) as the sum of a quotient Q(x) and a proper fraction P(x). (In a proper fraction the degree of the numerator is less than the degree of the denominator.) The polynomial Q(x) is readily integrated. To integrate P(x) separate it into a sum of partial fractions (as indicated below), and integrate each term of the sum separately. To separate the proper fraction P(x) into partial fractions, write P(x) = f(x)/ψ(x), where f(x) and ψ(x) are polynomials, ( x)  ( x  a)p ( x  b)q ( x  c)r ..., and the constants a, b, and c, are all different. By algebra, there exist constants A1, A2, ..., B1, B2,..., such that Ap f ( x) B1 A1 A2    ...   p 2 ( x) ( x  a) ( x  a) ( x  a) ( x  b) Bq B2   ...   ... 2 ( x  b) ( x  b)q

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88 • Mathematical Formulas and Scientific Data

The separate terms of this sum may be integrated by the formulas dx

 (x  )

t



1 dx , t  1,   log( x  ). t 1 (t  1)( x  ) ( x  )

If f(x) and ψ(x) have real coefficients and ψ(x) = 0 has imaginary roots the above method leads to imaginary quantities. To avoid this, separate P(x) in a different way. As before, corresponding to each p-fold real root a of ψ(x), use the sum Ap A1 A2   ...  . ( x  a) ( x  a)2 ( x  a)p To each λ-fold real quadratic factor x2 + αx + β, of ψ(x) which does not factor into real linear factors, use, instead of the two sets of terms occurring in the first expansion of R(x) and dependent on the conjugate complex roots of x2 + αx + β = 0, sums of the forms D x  E D1 x  E1 D2 x  E2   ...  ,  2 2  x  x     x 2  x     x 2  x    where the quantities D1, D2, ..., E1, E2, ..., are real constants. These new sums may be separately integrated by means of Integral formulas l49 to l56, etc.

145 TO INTEGRATE AN IRRATIONAL ALGEBRAIC FUNCTION If no convenient method of integration is apparent the integration may frequently be performed by means of a change of variable. For example, if R is a rational function of two arguments and n is an integer, then to integrate 1   (a)  R  x,(ax  b) n  dx , let (ax + b) = yn, whence the integral

reduces to

∫ P(y) dy, where P(y) is a rational function of y

and may be integrated as in Art. 143;

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Integral Calculus—II • 89

1 1   (b)  R  x,  x2  bx  c  2  dx, let  x2  bx  c  2  z  x, reduce   R to a rational function of z, and proceed as in Art. 143;

(c) ∫ R(sin x, cos x)dx, let tan sin x 

x = t, whence 2

2t 1  t2 2 dt , cos x  , dx  , 2 2 1 t 1 t 1  t2

reduce to a rational function of t and proceed as in Art. 143.

146 TO INTEGRATE EXPRESSIONS CONTAINING a2  x 2 , x 2  a2 Expressions containing these radicals can frequently be integrated after making the following transformations: (a) if

a2 − x2 occurs, let x = a sin t;

(b) if

x2 − a2 occurs, let x = a sec t;

(c) if

x2 + a2 occurs, let x = a tan t;

147 TO INTEGRATE EXPRESSIONS CONTAINING TRIGONOMETRIC FUNCTIONS The integration of such functions may frequently be facilitated by the use of the identities of Art. 58.

148 INTEGRATION BY PARTS The relation

 u dv  u v   v du is often effective in reducing a given integral to one or more simpler integrals.

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90 • Mathematical Formulas and Scientific Data

149 TABLE OF INTEGRALS In the following table, the constant of integration, C, is omitted but should be added to the result of every integration. The letter x represents any variable; u represents any function of x; the remaining letters represent arbitrary constants, unless otherwise indicated; all angles are in radians. Unless otherwise mentioned loge u ≡ log u. Expressions Containing (ax + b) 42.  (ax  b)n dx  43. 

1 (ax  b)n1 , n  1. a(n  1)

dx 1  log e (ax  b). ax  b a

dx 1 .  2 a(ax  b) (ax  b) dx 1  . 45.  3 (ax  b) 2 a(ax  b)2 1 46.  x(ax  b)n dx  2 (ax  b)n 2 a (n  2) b  2 (ax  b)n1 , n   1,  2. a (n  1) 44. 

47. 

xdx x b   2 log(ax  b). ax  b a a

48. 

xdx b 1  2  2 log(ax  b). 2 (ax  b) a (ax  b) a

49. 

xdx b 1 .  2  2 3 2 (ax  b) 2 a (ax  b) a (ax  b)

n 2 50.  x (ax  b) dx 

MFSDH.CH07_2PP.indd 90

1  (ax  b)n 3 (ax  b)n 2  2 b  n2 a3  n  3 n 1 (ax  b)   b2 , n  1,  2,  3. n  1 

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Integral Calculus—II • 91

51. 

x2 dx 1 1   (ax  b)2  2 b(ax  b)  b2 log (ax  b). ax  b a3  2 

52. 

b2  x2 dx 1  ax b b ax b      ) . ( ) 2 log (  ax  b  (ax  b)2 a3 

53. 

x2 dx 1  3 3 (ax  b) a

  b2 2b log( ax b )    .  2  ax  b 2(ax  b)  

54.  x m (ax  b)n dx 

1 a(m  n  1)

 x m (ax  b)n1  mb x m 1 (ax  b)n dx  ,    1  x m 1 (ax  b)n  nb x m (ax  b)n1 dx  ,    m  n1 m  0, m  n  1  0. 55. 

dx 1 x .  log x(ax  b) b ax  b

56. 

dx 1 a ax  b   2 log . bx b x x (ax  b)

57. 

dx 2 ax  b a2 x   3 log . 3 2 2 ax  b x (ax  b) 2 b x b

58. 

dx 1 1 ax  b   2 log . 2 b(ax  b) b x x(ax  b)

2

2 dx 1  1  ax  2 b  x   3   59.  .   log 3 ax  b  x(ax  b) b  2  ax  b 

60. 

MFSDH.CH07_2PP.indd 91

dx b  2 ax 2a ax  b  2  3 log . 2 x x (ax  b) b x(ax  b) b 2

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92 • Mathematical Formulas and Scientific Data

2 (ax  b)3 . 3a

61.  ax  b dx 

2(3 ax  2 b) (ax  b)3 . 15 a2

62.  x ax  b dx 

63.  x

ax  b dx 

2

64.  x3 ax  b dx 

67.  68. 

69. 

70.  71.  72. 

MFSDH.CH07_2PP.indd 92

105 a3

.

2  35 a3 x3  30 a2 bx2  24 ab2 x  16 b3  (ax  b)3 315 a4

65.  x n ax  b dx  66. 

2 15 a2 x2  12 abx  8 b2  (ax  b)3

2 a

 u u 2

n 1

2

.

 b  du, u  ax  b. n

ax  b dx . dx  2 ax  b  b x x ax  b dx ax  b xdx ax  b x2 dx ax  b x3 dx ax  b x n dx ax  b



2 ax  b . a



2(ax  2 b) ax  b. 3 a2



 

dx x ax  b

2  3 a2 x2  4 abx  8 b2  15 a3

ax  b.

2  5 a3 x3  6 a2 bx2  8 ab2 x  16 b3  35 a4 2 a



n 1

1 b

u log

2

ax  b.

 b  du, u  ax  b. n

ax  b  b ax  b  b

, for b  0.

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Integral Calculus—II • 93



73.

dx 2 ax  b  tan 1 , b  0; b b ax  b

x

2 ax  b , b  0. tan h 1 b b 74. 

dx x

2

ax  b dx

ax  b a dx  .  bx 2 b x ax  b



ax  b 3 a ax  b 3 a2   2 2 bx2 4 b2 x 8b

75. 

x

76. 

dx 1   m  n 1 n m x (ax  b) b

3

ax  b



dx

x

ax  b

.

ax  b ( u  a)m  n 2 du ,u . m  x u 2n

2(ax  b) 2 77.  (ax  b) dx  . a(2  n)  2n

2 78.  x(ax  b) dx  2 a  2n

79. 

dx x(ax  b) x m dx

80.  81. 

ax  b

n 2



dx x

n

ax  b n



1 b

 (ax  b) 2 b(ax  b)   2n  4  n 4n

dx x(ax  b)

MFSDH.CH07_2PP.indd 93



a b

dx (ax  b)

n 2

 .  .

x m 1 dx 2 x m ax  b 2 mb  . (2 m  1)a (2 m  1)a  ax  b 

 ax  b (2 n  3)a dx  . n 1  n 1 (2 n  2)b x (n  1)bx ax  b

(ax  b) 2 82.  dx  a  (ax  b) x 83. 

n2 2

2n 2

n2 2

(ax  b) dx  b x

n2 2

dx.

dx 1 cx  d log , bc  ad  0.  (ax  b)(cx  d) bc  ad ax  b

2/15/2023 12:18:10 PM

94 • Mathematical Formulas and Scientific Data

84. 

c dx 1  1  cx  d     log   , (ax  b)2 (cx  d) bc  ad  ax  b bc  ad  ax  b   bc  ad  0.

85.  (ax  b)n (cx  d)m dx 

1 (ax  b)n1 (cx  d)m  m(bc  ad) (m  n  1)a

 (ax  b) (cx  d) n

86.

m 1

dx  . 

dx

 (ax  b) (cx  d) n



m

1 1   a(m  n  2)  n 1 (m  1)(bc  ad)  (ax  b) (cx  d)m 1 dx

 (ax  b) (cx  d) n

m 1

 , 

m  1, n  0, bc  ad  0. 87. 

(ax  b)n dx (cx  d)m  

88. 



89.

 (ax  b)n1 1 (ax  b)n dx  m n  (   ) a 2  (cx  d)m 1  , (m  1)(bc  ad)  (cx  d)m 1

 (ax  b)n 1 (ax  b)n1   n ( b c  ad )  (cx  d)m dx  . (m  n  1)c  (cx  d)m 1

xdx 1 d b   log(ax  b)  log(cx  d) , (ax  b)(cx  d) bc  ad  a c  bc  ad  0. xdx

1

 (ax  b) (cx  d)  bc  ad 2

b d cx  d     a(ax  b)  bc  ad log ax  b  , bc  ad  0.   90. 

MFSDH.CH07_2PP.indd 94

cx  d ax  b

dx 

2 (3 ad  2 bc  acx) ax  b. 3 a2

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Integral Calculus—II • 95

91.





ax  b 2 ax  b 2  dx  cx  d c c

ad  bc tan 1 c

c(ax  b) , c  0, ad  bc. ad  bc 92. 

ax  b 2 ax  b 1  dx  cx  d c c log

93. 

c(ax  b)  bc  ad c(ax  b)  bc  ad dx

(cx  d) ax  b

dx

94.  (cx  d)

bc  ad c , c  0, bc  ad.

2



c ad  bc



tan 1

1

ax  b c bc  ad c  0, bc  ad.

log

c(ax  b) c  0, ad  bc. ad  bc

c(ax  b)  bc  ad c(ax  b)  bc  ad

,

Expressions Containing ax 2  c , ax n  c , x 2  p2 , and p2 − x 2 . 95. 

 x x dx 1 1  tan 1 , or  ctn 1   . 2 p p p p x  p

96. 

 x p x dx 1 1  log , or tanh 1   . 2 p x p 2p p x  p

97. 

 a dx 1 tanh 1  x   , a and c  0. 2 ax  c ac  c

98. 

dx 1 x a  c  log , a  0, c  0. 2 ax  c 2  ac x a  c

2

2



MFSDH.CH07_2PP.indd 95

1 2  ac

log

c  x a c  x a

, a  0, c  0.

2/15/2023 12:18:56 PM

96 • Mathematical Formulas and Scientific Data

99. 

 ax

dx 2

 c

1 x  2 2(n  1)c  ax  c n1



n

2n  3 dx , n  1.  2(n  1)c  ax2  c n1



100.  x  ax  c  2

101.  102. 

2 1  ax  c  dx  2a n1

n

n 1

, n  1.

x 1 dx  log  ax2  c  . 2a ax2  c dx

x  ax2  c  dx



1 x2 log 2 . 2c ax  c 1 a dx   2 . cx c ax  c

103. 

x

104. 

x2 dx x c dx    2 . 2 ax  c a a ax  c

105. 

x n dx x n 1 c x n1 dx   , n  1. ax2  c a(n  1) a  ax2  c

106.  107. 

2

 ax

2

 c

x2 dx

 ax

2

 c

dx

x

2

 ax

2

n





 c

n

1 x 1 dx   . n 1  2 2 2(n  1)a  ax  c  2(n  1)a  ax  c n1



1 dx a dx   . n 1  2 2 2 c x  ax  c  c  ax  c n





108.  x2  p2 dx 

1 x x2  p2  p2 log x  x2  p2 .    2

109.  p2  x2 dx 

 1 2 2 2 1  x   x p  x  p sin    . 2   p  

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Integral Calculus—II • 97

110. 

111. 

dx x p 2

2



 x  x  sin 1   or  cos1   . p x  p  p dx

2

2

112.  ax2  cdx 



115. 

dx ax  c 2

dx ax  c 2





 a  x c ax2  c  sin 1  x  , a  0. c  2 2 a 

1 a



 a  sin 1  x  , a  0. c  a  3 1 2 ax  c  2 . 3a

117.  x2 ax2  c dx 

x 4a 

118.  x2 ax2  c dx 

MFSDH.CH07_2PP.indd 97

xdx ax  c 2





log x a  ax2  c , a  0.

1

116.  x ax2  c dx 

119. 



x c ax2  c  log x a  ax2  c , a  0. 2 2 a

113.  ax2  c dx  114. 



 log x  x2  p2 .

x 4a

 ax

2

c2 8 a

3

 c  3



cx ax2  c 8a



log x a  ax2  c , a  0.

cx ax2  c 8a  a  c2  sin 1  x  , a  0. c  8a a 

 ax

2

 c  3

1 ax2  c . a

2/15/2023 12:20:07 PM

98 • Mathematical Formulas and Scientific Data

120. 

x2 dx ax  c 2



x 1 ax2  c   ax2  c dx. a a

121. 

ax2  c  c ax2  c dx  ax2  c  c log , c  0. x x

122. 

ax2  c dx  ax2  c   c tan 1 x

123.  124.  125.  126.  127. 

128. 

dx



x p2  x2 dx



x x2  p2 dx x ax2  c dx x ax2  c





dx x2 ax2  c x n dx ax2  c



 p  p2  x2 1 log   p x 

ax2  c c

, c  0.

 .  

1 1  p  p cos1   , or  sin 1  . p p  x  x 1 c

ax2  c  c , c  0. x

log

 a sec 1  x   , c  0. c c 

1



ax2  c . cx

x n1 ax2  c (n  1)c x n 2 dx  , n  0. na na  ax2  c 3

129.  x n ax2  c dx 

x n1  ax2  c  2 (n  2)a

x 130. 

MFSDH.CH07_2PP.indd 98

2

ax xn

 ax c dx  

2

n2

(n  1)c (n  2)a

ax2  c dx, n  0. 3

 c2

c(n  1) x



n2



(n  4)a ax2  c dx, n  1. (n  1)c  x n 2

2/15/2023 12:20:31 PM

Integral Calculus—II • 99

131. 

dx



x n ax2  c 3

132.   ax2  c  2 dx 

ax2  c dx (n  2)a  , n  1. n 1  n  2 (n  1)c x c(n  1) x ax2  c

x  2 ax2  5c  ax2  c 8 3 c2 log x a  ax2  c , a  0.  8 a



3

133.   ax2  c  2 dx 

134. 

dx

 ax

2

 c

3 2



x 2 ax2  5 c  ax2  c  8  a  3 c2 sin 1  x   , a  0. c  8 a  x

c ax2  c

3



135.  x  ax2  c  2 dx 

.

5 1 2 2 . ax  c   5a

3

136.  x2  ax2  c  2 dx 

3 x3 c ax2  c  2   x2 ax2  c dx.  6 2

3

3 2

137.  x n  ax2  c  dx  138.  139. 

MFSDH.CH07_2PP.indd 99

xdx

 ax2  c 

3 2

x2 dx

 ax2  c 

3 2





x n1  ax2  c  2 n4 1

a ax2  c x

a ax  c 2



3c x n ax2  c dx.  n4

.



1 a a





log x a  ax2  c , a  0.

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100 • Mathematical Formulas and Scientific Data

140.  141.  142.  143. 

144. 

145. 

x2 dx

 ax

 c

2

3 2

x3 dx

 ax

 c

2

3 2

dx

x  ax n  c 

 ax

dx n

 c

m

dx x ax n  c dx x ax n  c





x a ax  c 2

x2 a ax  c 2



 a  sin 1  x  , a  0. c  a a 



2 a2

1

ax2  c .



1 xn . log n cn ax  c



1 dx a x n dx  .  1 m c   ax n  c  c   ax n  c m 1



n c

log

2



n c

ax n  c  c ax n  c  c

sec 1

, c  0.

 ax n , c  0. c

146.  x m 1  ax n  c  dx p

1  x m  ax n  c  p  npc x m 1  ax n  c  p1 dx  .   m  np  1   x m  ax n  c  p1  (m  np  n) x m 1  ax n  c  p1 dx  .    cn(p  1)  



1  x m  n  ax n  c  p1  (m  n)c x m  n1  ax n  c  p dx  .   a(m  np) 



p 1 p 1  m x  ax n  c   (m  np  n)a  x m  n1  ax n  c  dx  .    mc

147.  148. 

MFSDH.CH07_2PP.indd 100

x m dx

 ax

n

 c dx

x

m

 ax

n

p



 c

1 x m  n dx c x m  n dx  . a   ax n  c  p1 a   ax n  c  p

p



1 dx a dx   . p 1   m n m n c x  ax n  c  p c x  ax  c 

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Integral Calculus—II • 101

Expressions Containing (ax 2 + bx + c ). 1

2 ax  b  b2  4 ac

149. 

dx  ax  bx  c

150. 

dx 2 2 ax  b  tan 1 , b2  4 ac. 2 2 ax  bx  c 4 ac  b 4 ac  b

151. 

dx 2  , b2  4 ac. 2 ax  b ax  bx  c

152. 

2

b  4 ac 2

log

2 ax  b  b  4 ac 2

, b2  4 ac.

2

2

 ax

dx

2

 bx  c 

n 1



2 ax  b

n  4 ac  b2   ax2  bx  c  

2(2 n  1)a

n  4 ac  b

2



 ax

n

dx

2

 bx  c 

n

.

153. 

b xdd 1 dx  log  ax2  bx  c   . 2  2 a ax  bx  c ax  bx  c 2 a

154. 

x2 dx x b   2 log  ax2  bx  c  2 ax  bx  c a 2 a b2  2 ac dx  . 2 2  2a ax  bx  c

155.  156. 

2

x n dx x n 1 c x n 2 dx b x n1 dx    . ax2  bx  c (n  1)a a  ax2  bx  c a  ax2  bx  c

 ax

xdx

2

 bx  c 

n 1



(2 c  bx)

n  4 ac  b2   ax2  bx  c  

MFSDH.CH07_2PP.indd 101

b(2 n  1)

n  4 ac  b

2



 ax

n

dx

2

 bx  c 

n

.

2/15/2023 12:21:25 PM

102 • Mathematical Formulas and Scientific Data

157. 

158. 

159. 

 ax

x m dx

2

 bx  c 

dx



n 1

x  ax2  bx  c 



x m 1

a(2 n  m  1)  ax2  bx  c 

n



(n  m  1) b x m 1 dx   (2 n  m  1) a  ax2  bx  c n1



c x m 2 (m  1)   . (2 n  m  1) a  ax2  bx  c n1

  b 1 x2 dx  log  .  2 2   ax  bx  c   2 c  ax  bx  c  2c  

dx



x2  ax2  bx  c 

 ax2  bx  c  1 b log   x2 2 c2   cx

 b2 a dx .  2   2 c   ax  bx  c   2c 160. 

161. 

x

m

MFSDH.CH07_2PP.indd 102

 bx  c 

n 1



1

(m  1)cx

m 1

 ax

2

 bx  c 



(n  m  1) b dx   m  1 2 m 1 c x  ax  bx  c n1



dx (2 n  m  1) a   . m  2 2 m 1 c x  ax  bx  c n1 dx

x  ax2  bx  c  

162. 

 ax

dx

2

n



n

1

2 c(n  1)  ax2  bx  c 

n 1

b dx 1 dx  . 2 c   ax2  bx  c n c  x  ax2  bx  c n1 dx

ax  bx  c 2



1 a





log 2 ax  b  2 a ax2  bx  c , a  0.

2/15/2023 12:21:38 PM

Integral Calculus—II • 103

163. 

164. 

165. 

dx ax2  bx  c xdx ax2  bx  c x n dx

ax  bx  c x n1 dx



a

sin 1

2 ax  b

, a  0.

b2  4 ac

ax2  bx  c b dx  . 2 2 a ax  bx  c a





2

1



x n 1 am 

ax2  bx  c

b(2 n  1) 2 an

ax2  bx  c 

c(n  1) x n 2 dx . an  ax2  bx  c

2 ax  b ax2  bx  c 4a 4 ac  b2 dx .   2 8a ax  bx  c

166.  ax2  bx  c dx 

3

167.  x ax2  bx  c dx 

 ax2  bx  c  2 3a



b ax2  bx  c dx.  2a 3

2 5 b   ax  bx  c  2  2 2 168.  x ax  bx  c dx   x   6a  4a  2  5b  4 ac  ax2  bx  c dx.   16 a2

169.  170.  171. 

MFSDH.CH07_2PP.indd 103

dx x ax2  bx  c



dx x ax  bx  c 2

dx x ax2  bx





 ax2  bx  c  c b  log    , c  0.  x c 2 c  

1

1 c

sin 1

bx  2 c x b2  4 ac

, c  0.

2 ax2  bx , c  0. bx

2/15/2023 12:22:00 PM

104 • Mathematical Formulas and Scientific Data

172. 

dx

173. 

dx

ax2  bx  c c(n  1) x n1



x n ax2  bx  c b(3  2 n) dx a(2  n) dx   .   n 1 2  n  2 2 c(n  1) x ax  bx  c c(n  1) x ax2  bx  c

174. 

 ax

2

 bx  c 

3 2

dx

 ax

2

 bx  c 

3 2





b

2

2(2 ax  b)

 4 ac  ax  bx  c 2

1 2 a

3

 x  b / 2a

2

, b2  4 ac.

, b2  4 ac.

Miscellaneous Algebraic Expressions 175.  2 px  x2 dx 

176. 

177. 

1 ( x  p) 2 px  x2  p2 sin 1 [( x  p) / p].  2

 p x  cos1  . 2 px  x  p  dx

2

dx ax  b. cx  d or

2 ac



tanh 1

2  ac

tan 1

 c(ax  b) , a(cx  d)

c(ax  b) . a(cx  d)

178.  ax  b. cx  d dx 

(2 acx  bc  ad) ax  b. cx  d 4 ac (ad  bc)2 dx  .  8 ac ax  b. cx  d

179. 

ax  b. cx  d (ad  bc) cx  d dx dx  .   ax  b a 2a ax  b. cx  d

180. 

xb dx  x  d . x  b  (b  d) log  x  d  x  b  . xd

MFSDH.CH07_2PP.indd 104

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Integral Calculus—II • 105

181. 

1 x dx  sin 1 x  1  x2 . 1 x

182. 

p x dx  p  x . q  x  (p  q) sin 1 q x

183. 

p x dx   p  x . q  x  (p  q) sin 1 q x

184. 

dx x  p. q  x

 2 sin 1

xq . p q q x . p q

xp . q p

Expressions Containing sin ax 185.  sin u du   cos u, where u is any function of x. 1 186.  sin ax dx   cos ax. a 187.  sin 2 ax dx 

x sin 2 ax  . 2 4a

1 1 188.  sin 3 ax dx   cos ax  cos3 ax. a 3a 189.  sin 4 ax dx 

3 x 3 sin 2 ax sin 3 ax cos ax   . 8 16 a 4a

sin n1 ax cos ax n  1  sin n 2 ax dx,  na n (n pos. integer).

190.  sin n ax dx   191. 

MFSDH.CH07_2PP.indd 105

dx 1 ax 1  log tan  log(csc ax  ctn ax). sin ax a 2 a

2/15/2023 12:22:57 PM

106 • Mathematical Formulas and Scientific Data

192. 

dx 1   csc 2 ax dx   ctn ax. 2 a sin ax

193. 

n2 dx cos ax dx 1 ,   n n 1 n2  a(n  1) sin ax n  1 sin ax sin ax

n integer > 1. 194. 

dx 1   ax    tan    . 1  sin ax a 4 2 

195. 

 bc 2 dx   ax    tan 1  tan     , b2  c2 . 2 2 b  c sin ax a b  c  4 2   b c

196. 

1 dx c  b sin ax  c2  b2 cos ax 2 , c  b2 .  log b  c sin ax a c2  b2 b  c sin ax

197.  sin ax sin bx dx 

sin(a  b) x sin(a  b) x 2  , a  b2 . 2(a  b) 2(a  b)

x x  198.  1  sin x dx  2  sin  cos  ; use + sign 2 2    when (8 k  1)  x(8 k  3) , otherwise −, k an integer. 2 2 x x  199.  1  sin x dx   2  sin  cos  ; use + sign 2  2   when (8 k  3)  x(8 k  1) , otherwise −, k an integer. 2 2 Expressions Involving cos ax 200.  cos u du  sin u, where u is any function of x. 201.  cos ax dx 

MFSDH.CH07_2PP.indd 106

1 sin ax. a

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Integral Calculus—II • 107

202.  cos2 ax dx 

x sin 2 ax  . 2 4a

203.  cos3 ax dx 

1 1 sin ax  sin 3 ax. a 3

204.  cos4 ax dx 

3 3 x 3 sin 2 ax cos ax sin ax   . 8 16 a 4a

205.  cos n ax dx 

cos n1 ax sin ax na



n1 cos n 2 ax dx. n 

206. 

dx 1  ax   1  log tan     log(tan ax  sec ax). cos ax a  2 4 a

207. 

dx 1  tan ax. 2 cos ax a

208. 

n2 dx sin ax dx 1 ,   n n 1  cos ax a(n  1) cos ax n  1 cos n 2 ax

n integer > 1. 209. 

dx 1 ax  tan . 1  cos ax a 2

210. 

dx 1 ax   cot . 1  cos ax a 2

x x dx  2 2 sin . 2 2 Use + when (4k − 1) π< x ≦ (4k +1) π, otherwise −, k an integer.

211.  1  cos x dx   2  cos

MFSDH.CH07_2PP.indd 107

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108 • Mathematical Formulas and Scientific Data

x x 212.  1  cos x dx   2  sin dx  2 2 cos . 2 2 Use top signs when 4kπ < x ≦ (4k +2)π, otherwise bottom signs. 213. 

 b2  c2 .sin ax  2 dx 1  tan 1   , b  c2 .  c  b cos ax  b  c cos ax a b2  c2  

214. 

 c2  b2 .sin ax  2 1 dx 2  tanh 1  , c  b . cos b  c cos ax a c2  b2 c b ax   

215.  cos ax  cos bx dx 

sin(a  b) x sin(a  b) x 2 , a  b2 .  2(a  b) 2(a  b)

Expressions Containing sin ax and cos ax 1  cos(a  b) x cos(a  b) x  2 216.  sin ax cos bx dx    , a  b2 .  2 ab a  b  217.  sin n ax cos ax dx 

1 sin n1 ax, n  1. a(n  1)

218.  cos n ax sin ax dx  

1 cos n1 ax, n  1. a(n  1)

219. 

sin ax 1 dx   log cos ax. a cos ax

220. 

cos ax 1 dx  log sin ax. a sin ax

221.  (b  c sin ax)n cos ax dx 

1 (b  c sin ax)n1 , n  1. ac(n  1)

222.  (b  c cos ax)n sin ax dx  

MFSDH.CH07_2PP.indd 108

1 (b  c cos ax)n1 , n  1. ac(n  1)

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Integral Calculus—II • 109

223. 

cos ax dx 1  log(b  c sin ax). b  c sin ax ac

224. 

sin ax 1 dx   log(b  c cos ax). b  c cos ax ac

225. 

dx 1  2 b sin ax  c cos ax a b  c2

226. 

dx 1  sin 1 U , 2 b  c cos ax  d sin ax a b  c2  d 2

1  1 c    log tan 2  ax  tan b   .   

 c2  d 2  b(c cos ax  d sin ax)  1  ; or  log V , U 2 2 2  c  d (b  c cos ax  d sin ax)  a c  d 2  b2  c2  d 2  b(c cos ax  d sin ax)  c2  d 2  b2 (c sin ax  d cos ax)    V  2 2     c d b c cos s ax  d sin ax     b2  c2  d 2 ,   ax   . 227. 





228. 

dx b  c cos ax  d sin ax 1  b  (c  d) cos ax  (c  d) sin ax  2 , b  c2  d 2 .   ab  b  (c  d) cos ax  (c  d) sin ax  sin 2 ax dx b  c cos ax 2



1 ac

sin ax cos ax dx

 x  b c b tan 1   tan ax   . b  b c  c 1 log  b cos2 ax  c sin 2 ax  . 2 a(c  b)

229. 

b cos ax  c sin ax

230. 

dx 1 b cos ax  c sin ax  . log b cos ax  c sin ax b2 cos2 ax  c2 sin 2 ax 2 abc

MFSDH.CH07_2PP.indd 109

2

2



2/15/2023 12:24:41 PM

110 • Mathematical Formulas and Scientific Data

231. 

dx 1  c tan ax  tan 1   . 2 2 b cos ax  c sin ax abc  b  2

2

232.  sin 2 ax cos2 ax dx 

x sin 4 ax  . 8 32 a

233. 

dx 1  log tan ax. sin ax cos ax a

234. 

dx 1   tan ax  cot ax  . 2 sin ax cos ax a

235. 

sin 2 ax 1  ax    dx    sin ax  log tan     . cos ax a  2 4 

236. 

ax  cos2 ax 1 dx  cos ax  log tan  . sin ax a 2

2

sin m 1 ax cos n1 ax a(m  n) m 1  sin m  2 ax cos n ax dx, m, n  0. mn

237.  sin m ax cos n ax dx  

m n 238.  sin ax cos ax dx 

sin m 1 ax cos n1 ax n  1  a(m  n) mn

 sin 239. 

m

ax cos n 2 ax dx, m, n  0.

mn2 sin m ax sin m 1 ax dx   n n 1 n1 a(n  1) cos ax cos ax sin m ax  cosn2 ax dx, m, n  0, n  1.

mn2 cos n ax  cos n1 ax dx   240.  m m 1 (m  1) a(m  1) sin ax sin ax cos n ax  sin m 2 ax dx, m, n,  0, m  1.

MFSDH.CH07_2PP.indd 110

2/15/2023 12:25:07 PM

Integral Calculus—II • 111

241. 

dx 1 1 mn2   n m 1 n 1 (n  1) sin ax cos ax a(n  1) sin cos ax m

 sin 242. 

m

dx . ax cos n 2 ax

dx 1 1  n m 1 a(m  1) sin sin ax cos ax ax cos n1 ax mn2 dx .  m 2  (m  1) sin ax cos n ax m

1  cos2 ax  dx. (Expand, divide, and sin 2 n ax 243.  dx   cos ax cos ax n

use Art. 205). 1  sin 2 ax   cos2 n ax 244.  dx   dx. (Expand, divide, and sin ax sin ax n

use Art. 190). 1  cos2 ax   sin 2 n1 ax 245.  dx   sin ax dx. cos ax cos ax n



(Expand, divide, and use Art. 218).

1  sin 2 ax   cos2 n1 ax 246.  dx   cos ax dx. sin ax sin ax n



(Expand, divide, and use Art. 217). Expressions Containing tan ax or cot ax (tan ax=1/cot ax)

247.  tan u du   log cos u, or log sec u, where u is any function of x. 1 248.  tan ax dx   log cos ax. a

MFSDH.CH07_2PP.indd 111

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112 • Mathematical Formulas and Scientific Data

249.  tan 2 ax dx 

1 tan ax  x. a

250.  tan 3 ax dx 

1 1 tan 2 ax  log cos ax. a 2a

251.  tan n ax dx 

1 tan n1 ax   tan n 2 ax dx, n integer  1. a(n  1)

252.  cot u du  log sin u, or −log cosec u, where u is any function of x. 253.  cot 2 ax dx  

dx 1   cot ax  x. 2 a tan ax

254.  cot 3 ax dx  

1 1 cot 2 ax  log sin ax. a 2a

255.  cot n ax dx  

dx 1  ax   cot n 2 ax dx, n a(n  1) tan ax

n integer > 1. 256. 

cot ax dx dx 1   2 b  c tan ax b cot ax  c b  c2 c 1   bx  log(b cos ax  c sin ax)  2 2  a b c  

257. 

tan ax dx dx  b  c cot ax b tan ax  c c 1    2 bx  log(c cos ax  b sin ax) . 2  a b c  

258. 

dx b  c tan 2 ax



 bc  sin 1  sin ax  , b a bc   1

b pos., b2 > c2.

MFSDH.CH07_2PP.indd 112

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Integral Calculus—II • 113

Expressions Containing sec ax = 1/cos ax or cosec ax = 1 sin ax.  u  259.  sec u du  log(sec u  tan u)  log tan    , where u is 2 4 any function of x. 1  ax   260.  sec ax dx  log tan    . a  2 4 2 261.  sec ax dx 

262.  sec 3 ax dx 

1 tan ax. a 1   ax    tan ax sec ax  log tan     . 2 a   2 4 

n 263.  sec ax dx 



1 sin ax n2 sec n 2 ax dx,  n 1  a(n  1) cos ax n  1

n integer > 1.

u 264.  cosec u du  log(cosec u  cot u)  log tan , where u is 2 any function of x. 265.  cosec ax dx 

ax 1 log tan . a 2

1 266.  cosec 2 ax dx   cot ax. a 267.  cosec 3 ax dx 

ax  1   cot ax cosec ax  log tan  .  2a  2

268.  cosec n ax dx  

cos ax n2 1  cosec n 2 ax dx, n 1 a(n  1) sin ax n  1 

n integer > 1.

MFSDH.CH07_2PP.indd 113

2/15/2023 12:26:09 PM

114 • Mathematical Formulas and Scientific Data

Expressions Containing tan ax and sec ax or cot ax and cosec ax. 269.  tan u sec u du  sec u, where u is any function of x. 270.  tan ax sec ax dx 

1 sec ax. a

271.  tan n ax sec 2 ax dx  272.  tan ax sec n ax dx 

1 tan n1 ax, n  1. a(n  1)

1 sec n ax, n  0. an

273.  cot u cosec u du   cosec u, where u is any function of x. 1 274.  cot ax cosec ax dx   cosec ax. a 275.  cot n ax co sec 2 ax dx   276.  cot ax cosec n ax dx  

277. 

1 cot n1 ax, n  1. a(n  1)

1 cosec n ax, n  0. an

cosec 2 ax dx 1   log cot ax. cot ax a

Expressions Containing Algebraic and Trigonometric Functions 278.  x sin ax dx 

1 1 sin ax  x cos ax. 2 a a

279.  x2 sin ax dx 

x2 2x 2 ax  ax  cos ax. sin cos a a2 a3

280.  x3 sin ax dx 

x3 3 x2 6 6x ax  ax  sin sin cos ax  3 cos ax. a a2 a4 a

MFSDH.CH07_2PP.indd 114

2/15/2023 12:26:34 PM

Integral Calculus—II • 115

281.  x sin 2 ax dx 

x2 x sin 2 ax cos 2 ax   4 4a 8 a2

282.  x2 sin 2 ax dx 

x3  x2 1   3 6  4a 8a

 x cos 2 ax  sin 2 ax  4 a2 

283.  x3 sin 2 ax dx 

x 4  x3 3x   3 8  4a 8a

284.  x sin 3 ax dx 

x cos 3 ax sin 3 ax 3 x cos ax 3 sin ax    . 12 a 4a 36 a2 4 a2

285.  x n sin ax dx  

  3 x2 3 sin ax 2    2  16 a4   8a

n 1 n x cos ax   x n1 cos ax dx. a a

286. 

sin ax dx (ax)3 (ax)5  ax    . x 3  3! 5  5!

287. 

sin ax dx x

m



cos ax dx 1 sin ax a .  m 1  (m  1) x (m  1) x m 1

288.  x cos ax dx 

1 1 cos ax  x sin ax. 2 a a

289.  x2 cos ax dx 

290.  x

3a x cos ax dx 

291.  x cos2 ax dx 

MFSDH.CH07_2PP.indd 115

x2 2x 2 ax  ax  sin ax. cos sin a a2 a3 2

3

  cos 2ax. 

2

 6  cos ax a4

a x  2

3

 6 x  sin ax a3

.

x2 x sin 2 ax cos 2 ax   . 4 4a 8 a2

2/15/2023 12:26:59 PM

116 • Mathematical Formulas and Scientific Data

292.  x2 cos2 ax dx 

x3  x2 1  x cos 2 ax   3  sin 2 ax  . 6  4a 8a  4 a2

4  3   2  293.  x3 cos2 ax dx  x   x  3 x3  sin 2 ax   3 x 2  3 4  cos 2ax. a

8

4

8a 

 8a

16 a 

294.  x cos3 ax dx 

x sin 3 ax cos 3 ax 3 x sin ax 3 cos ax    . 12 a 4a 36 a2 4 a2

295.  x n cos ax dx 

1 n n x sin ax   x n1 sin ax dx, n pos. a a

296. 

cos ax dx (ax)2 (ax)4  log ax    ..... x 2  2! 4  4!

297. 

sin ax dx cos ax 1 cos ax a dx    m 1  . m  (m  1) x (m  1) x x m 1

Expressions Containing Exponential and Logarithmic Functions 298.  e u du  e u , where u is any function of x. 299.  bu du 

bu , where u is any function of x. log b

300.  eax dx 

1 ax e , a

301.  xeax dx 

ax

dx 

bax . a log b

xbax bax eax ax ax xb dx (  1 ),   .  a log b a2 (log b)2 a2

302.  x2 eax dx 

MFSDH.CH07_2PP.indd 116

b

eax 2 2  a x  2 ax  2  . a3

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Integral Calculus—II • 117

303.  x n eax dx  304.  x n eax dx 

1 n ax n n1 ax x e   x e dx, n pos. a a

eax (ax)n  n(ax)n1  n(n  1)(ax)n 2  ...  (1)n n !, a n 1 

n pos. integ.

305.  x n e ax dx  

e ax (ax)n  n(ax)n1  n(n  1)(ax)n 2  .....  n ! , a n 1 

n pos. integ. x n bax n  306.  x n bax dx   x n1 bax dx, n pos.  a b a b log log  307. 

(ax)2 (ax)3 eax dx  log x  ax    ....... x 2  2! 3  3!

308. 

 eax eax 1  eax dx  , n integ. > 1. a dx     n n 1 n 1  n1  x x x 

309. 

dx 1  ax  log  b  ceax   .  ax  ab  b  ce

310. 

eax dx 1  log  b  ceax  . ax ac b  ce

311. 

 dx 1 b  tan 1  eax  , b and c pos.  ax c be  ce a bc  ax

312.  eax sin bx dx 

MFSDH.CH07_2PP.indd 117

eax (a sin bx  b cos bx). a2  b2

2/15/2023 12:27:48 PM

118 • Mathematical Formulas and Scientific Data

eax [(b  c) sin(b  c) x  a cos(b  c) x] 2 2  a2   b  c   eax [(b  c) sin(b  c) x  a cos(b  c) x]  . 2  a2  (b  c)2 

313.  eax sin bx sin cx dx 

314.  eax cos bx dx 

eax (a cos bx  b sin bx). a2  b2

315.  eax cos bx cos cx dx  

eax [(b  c) sin(b  c) x  a cos(b  c) x] . 2  a2  (b  c)2 

316.  eax sin bx cos cx dx  

eax [(b  c) sin(b  c) x  a cos(b  c) x] 2  a2  (b  c)2 

eax [a sin(b  c) x  (b  c) cos(b  c) x] 2  a2  (b  c)2 

eax [a sin(b  c) x  (b  c) cos(b  c) x] . 2  a2  (b  c)2 

317.  eax sin bx sin(bx  c)dx 

eax cos c eax [a cos(2 bx  c)  2 b sin(2 bx  c)]  . 2a 2  a2  4 b2 

318.  eax cos bx cos(bx  c)dx. 

eax cos c eax [a cos(2 bx  c)  2 b sin(2 bx  c)]  2a 2  a2  4 b2 

319.  eax sin bx cos(bx  c)dx. 

MFSDH.CH07_2PP.indd 118

eax sin c e ax [a sin(2 bx  c)  2 b cos(2 bx  c)]  2a 2  a2  4 b2 

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Integral Calculus—II • 119

320.  eax cos bx sin(bx  c)dx. 

eax sin c e ax [a sin(2 bx  c)  2 b cos(2 bx  c)]  2a 2  a2  4 b2 

ax 321.  xe sin bx dx 



a

eax 2



2 2

b

322.  xeax cos bx dx  

a

eax 2



2 2

b

323.  eax cos n bx dx  

n(n  1)b2 a2  n2 b2

ax n 324.  e sin bx dx 



n(n  1)b2 a2  n2 b2

xeax (a sin bx  b cos bx) a2  b2  a2  b2  sin bx  2 ab cos bx  .  

xeax (a cos bx  b sin bx) a2  b2  a2  b2  cos bx  2 ab sin bx  .   eax cos n1 bx(a cos bx  nb sin bx) a2  n2 b2

e

ax

cos n 2 bx dx.

eax sin n1 bx(a sin bx  nb cos bx) a2  n2 b2

e

ax

sin n 2 bx dx.

325.  log ax dx  x log ax  x. 326.  x log ax dx 

x2 x2 log ax  . 2 4

327.  x2 log ax dx 

MFSDH.CH07_2PP.indd 119

x3 x3 log ax  . 3 9

2/15/2023 12:28:21 PM

120 • Mathematical Formulas and Scientific Data

328.  (log ax)2 dx  x(log ax)2  2 x log ax  2 x. 329.  (log ax)n dx  x(log ax)n  n (log ax)n1 dx, n pos. 1   log ax 330.  x n log ax dx  x n1   , n  1. 2   n  1 (n  1)  331.  x n (log ax)m dx 

x n 1 m (log ax)m  x n (log ax)m 1 dx.  n1 n1

(log ax)n (log ax)n1 332.  dx  , n  1. x n1 333. 

dx  log(log ax). x log ax

334. 

dx 1  . n x(log ax) (n  1)(log ax)n1

335. 

x n dx  x n 1 n1 x n dx   , m  1.. (log ax)m (m  1)(log ax)m 1 m  1  (log ax)m 1

x n dx 1 336.   n 1 log ax a

ey dy  y , y  (n  1) log ax.

 (n  1)2 (log ax)2 log log  (  ) log  ax n 1 ax  2  2!  3 3  (n  1) (log ax)  ... .  3  3! 

x n dx 1  n 1 337.  log ax a

338. 

MFSDH.CH07_2PP.indd 120

 (log ax)2 (log ax)3 dx 1   log log ax  log ax    ... . log ax a  2  2! 3  3! 

2/15/2023 12:28:52 PM

Integral Calculus—II • 121

x 339.  sin(log ax)dx  [sin(log ax)  cos(log ax)]. 2 x 340.  cos(log ax)dx  [sin(log ax)  cos(log ax)]. 2 341.  eax log bx dx 

1 ax 1 eax e log bx   dx. a a x

Expressions Containing Inverse Trigonometric Functions 342.  sin 1 ax dx  x sin 1 ax 

1 1  a2 x 2 . a

343.   sin 1 ax  dx  x  sin 1 ax   2 x  2

2

344.  x sin 1 ax dx 

x2 x 1 sin 1 ax  2 sin 1 ax  1  a2 x 2 . 4a 2 4a

345.  x n sin 1 ax dx  346. 

347. 

x n 1 a x n1 dx sin 1 ax  , n  1. n  1  1  a2 x 2 n1

sin 1 ax dx 1 13  ax  (ax)5 (ax)3  x 233 2455 135  (ax)7  ..., a2 x2  1. 2467 7 sin 1 ax dx x2

1 1  1  a2 x 2   sin 1 ax  a log . x ax

348.  cos1 ax dx  x cos1 ax 

MFSDH.CH07_2PP.indd 121

2 1  a2 x2 sin 1 ax. a

1 1  a2 x 2 . a

2/15/2023 12:29:16 PM

122 • Mathematical Formulas and Scientific Data

349.   cos1 ax  dx  x  cos1 ax   2 x  2

2

350.  x cos1 ax dx 

x2 x 1 cos1 ax  2 cos1 ax  1  a2 x 2 . 4a 2 4a

351.  x n cos1 ax dx 

352. 

353. 

x n 1 a x n1 dx cos1 ax  , n  1. n  1  1  a2 x 2 n1

cos1 ax dx  1 13  log ax  ax  (ax)5 (ax)3       2 2 3 3 2 4 5 5 x 135  (ax)7  ..., a2 x2  1. 2467 7 cos1 ax dx x2

1 1  1  a2 x2   cos1 ax  a log . x ax

354.  tan 1 ax dx  x tan 1 ax  355.  x n tan 1 ax dx 

356. 

tan 1 ax dx x

2

1 log 1  a2 x2  . 2a

x n 1 a x n1 dx tan 1 ax  , n  1. n1 n  1  1  a2 x 2

 1  a2 x 2 1 a   tan 1 ax  log  2 2 x 2  a x

357.  cot 1 ax dx  x cot 1 ax  358.  x n cot 1 ax dx 

MFSDH.CH07_2PP.indd 122

2 1  a2 x2 cos1 ax. a

 . 

1 log 1  a2 x2  . 2a

x n 1 a x n1 dx cot 1 ax  , n  1. n1 n  1  1  a2 x 2

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Integral Calculus—II • 123

359. 

cot 1 ax dx x

2

 1  a2 x 2  1 a   cot 1 ax  log  2 2  . x 2  a x 

360.  sec 1 ax dx  x sec 1 ax 





1 log ax  a2 x2  1 . a

361.  x n sec 1 ax dx 

x n 1 x n dx 1 sec 1 ax  n  1. n1 n  1  a2 x 2  1



   sec 1 ax  ; sign when 0  sec 1 ax  . 2 2

Use + sign when

362.  cosec 1 ax dx  x cosec 1 ax  363.  x n cosec 1 ax dx 





1 log ax  a2 x2  1 . a

x n 1 x n dx 1 cosec 1 ax  , n  1. n1 n  1  a2 x 2  1

 Use + sign when 0  cosec 1 ax  ; − sign when 2    cosec 1 ax  0. 2



Definite Integrals 364. 



0 

adx    , if a > 0: 0, if a = 0; , if a < 0. 2 a x 2 2 2

1

n 1

365.  x n1 e x dx   log e (1 / x) 0 0

dx (n).



(n  1)  n  (n), if n > 0.

(2)  (1)  1.



(n  1)  n !, if n is an integer.

1     . 2



(n)  (n  1) .

Z(y) = Dy [loge Γ(y)].



Z(1) = −0.5772157…(See integral 418).

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124 • Mathematical Formulas and Scientific Data



366.  e zx  zn  x n1 dx  (n), z  0. 0

1



0

0

367.  x m 1 (1  x)n1 dx   368. 



0

369.



 2

0

x m 1 dx (m)(n)  . mn (m  n) (1  x)

 x n1 dx  , 0  n  1. 1 x sin n  2

sin n x dx   cos n x dx 0

n 1    1 2 2  , if n  1;   2 n    1 2  1  3  5...(n  1)    , if n is an even integer; 2  4  6...(n) 2 2  4  6...(n  1) = ,if n is an odd integer. 1  3  5  7...n 370. 



371. 



372. 



0

0

0

sin 2 x  dx  . 2 2 x sin ax  dx  , if a  0. x 2 sin x cos ax dx  0, if a  1, or a  1; x   , if a  1, or a  1; 4   , if  1  a  1. 2

   373.  sin 2 ax dx   cos2 ax dx  . 0 0 2

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Integral Calculus—II • 125

374. 

/ a

0



sin ax  cos ax dx   sin ax  cos ax dx  0. 0





0

0

375.  sin ax sin bx dx   cos ax cos bx dx  0, a  b.

376. 



377. 



0

0

2a , if a  b is odd; a  b2 =0, if a  b is even.

sin ax cos bx dx 

2

sin ax sin bx 1 dx  a, if a  b. 2 x 2

378.  cos  x2  dx   sin  x2  dx  



0

0



379.  e a x dx  2 2

0

1  . 2 2

 1 1     , if a  0. 2a 2a  2  (n  1) , a n 1 n!  n1 , if n is a positive integer, a  0. a



380.  x n  e ax dx  0



381.  x2 n e ax dx  2

0

382. 



383. 



xe ax dx 

0

0



MFSDH.CH07_2PP.indd 125

1  . 2a 2

e ax  dx  . a x

384.  e 0

1  3  5...(2 n  1)  . 2 n 1 a n a

  x 2  a2 / x 2 

dx 

1 2 a e  , if a  0. 2

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126 • Mathematical Formulas and Scientific Data



385.  e ax cos bx dx  0



386.  e ax sin bx dx  0

387. 



0



a , if a  0. a  b2 2

b , if a  0. a  b2 2

e ax sin x dx  ctn 1 a, a  0. x

388.  e

 a2 x2 cos bx dx 

2 / 4 a2

  e b 2a

, if a  0.

0

389.   log x  dx  (1)n .n !, n pos. integ. 1

n

0

log x 2 dx   . 0 1 x 6

390. 

1

log x 2 dx   . 0 1 x 12

391. 

1

392. 

1

log x

0

1  x2

393. 

1

log x

0

dx  

2 . 8

 dx   log 2. 2 1 x 2

2 1  1  x  dx  394.  log  . .   0 4 1 x x

  ex  1  2 395.  log  x dx  .  0 4  e 1 

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Integral Calculus—II • 127

396. 

dx  . log(1 / x)

1

0

1



0

0

397.  log log x dx   e x log x dx    0.577 2157....

 2

 2

 398.  log sin x dx   log cos x dx   log e 2. 2 0 0 

399.  x log sin x dx   0

2 log e 2. 2

1

1

1 2   400.   log  dx  . x 2 0 1

1  401.   log  x 0



1 2

dx   . n

1

1 (n  1)  402.  x  log  dx  , if m + 1 > 0, n + 1 > 0. x (m  1)n1  0 m

  a  a2  b2 403.  log  a  b cos x  dx   log   2 0 

  , ab.  

log 1  sin a cos x  dx  a. cos x 0



404.  1

405.  0

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xb  xa 1 b . dx  log log x 1 a

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128 • Mathematical Formulas and Scientific Data



406.  0

dx  , if a > b > 0.  2 a  b cos x a  b2

 2

dx  407.  a  b cos x 0

 b cos1    a  , a  b. a2  b2



cos ax dx   a   e , if a  0;  ea , if a  0. 2 1 x 2 2 0

408.  



cos xdx sin xdx  .   2 x x 0 0

409.  

410.  0

e ax  e bx b dx  log . x a



 a tan 1 ax  tan 1 bx dx  log . x 2 b 0

411.  

412.  0

 2

413.  0

cos ax  cos bx b dx  log . x a dx   . 2 2 a cos x  b sin x 2 ab 2

2

 2

414.  0



415.  0

a

dx

2

cos2 x  b2 sin 2 x 

(a  b cos x)dx a  2 ab cos x  b2 2

2

4 a3 b3

.

 0, if a2  b2 ;  

MFSDH.CH07_2PP.indd 128



  a2  b2 

 , if a2  b2 ; a  , if a  b. 2a

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Integral Calculus—II • 129

.

1

1  x2  2. dx  4 1 x 4 0

416.  1

417.  0

log(1  x) 1 1 1 1 2 . dx  2  2  2  2  ...  12 x 1 2 3 4

1

e xu x2 x3 x4    ..., where du    log x  x  2  2! 3  3! 4  4! u 

418. 

1 1 1     lim  1    ...   log t   0.5772157..., 0  x  . t  t 2 3   1

cos xu x2 x4 x6 du    log x     ..., 2  2! 4  4! 6  6! u 

419. 

where   0.5772157..., 0  x  . 1

420.  0

  e xu  e xu x3 x5 du  2  x    ...  , 0  x  . 3  3! 5  5! u  

1

1  e xu x2 x3 x4 du  x     ..., 0  x  . u 2  2! 3  3! 4  4! 0

421.   2

422.  0

 1  13  4  1    K 2    K 2 2  24  1  K sin x 2   2  dx

2

2

2   135  6 2   K  ... , if K 1.  246    2

2 2 423.  1  K sin x dx  0

   1  2  1  3  K4 1    K    2   2   24  3 2

2

2 6   13 5  K   ... , if K 2  1.   246  5 

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130 • Mathematical Formulas and Scientific Data

424. f ( x) 

1 x 2 x x a0  a1 cos  a2 cos  ...  b1 sin 2 c c c 2 x  b2 sin  ...,  c  x   c, c

where am 

c

c

mx mx 1 1 f ( x)cos dx, bm  f ( x)sin dx.  c c c c c c

(Fourier Series) 

a , 0  | b | a. a  b2

425.  e ax cosh bx dx 

2

0



b , 0  | b |  a. a2  b2

426.  e ax sinh bx dx  0



427.  xe ax sin bx dx  0

2 ab

 a  b2  2



428.  xe ax cos bx dx  0



429.  x e

2  ax



430.  x e

2  ax

sin bx dx 

cos bx dx 

0



431.  x3 e ax sin bx dx  0



432.  x e 0

MFSDH.CH07_2PP.indd 130

3  ax

a2  b2

a

0

cos bx dx 

2

2

 b2 

2

, a  0.

, a  0.

2 b  3 a2  b2 

a

 b2 

2

3

, a  0.

2 a  a2  3 b2 

a

2

 b2 

3

, a  0.

24 ab  a2  b2 

 a2  b2 

4

, a  0.

6  a4  6 a2 b2  b4 

a

2

 b2 

4

, a  0.

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Integral Calculus—II • 131



433.  x n e ax sin bx dx  0



434.  x n e ax cos bx dx  0

i  n ! (a  ib)n1  (a  ib)n1  2  a2  b2 

n 1

n! (a  ib)n1  (a  ib)n1  2  a2  b2 

n 1

, a  0.

, a  0.



435.  e x log x dx     0.577 2157....

(Also See Art. 418.)

0



1  1 436.     e x dx    0.577 2157... x x 1e 0 

1 1  437.    e x  dx    0.577 2157... x1 x  0 1    x x 1  e  e  1   dx    0.577 2157... 438.   x 0

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CHAPTER

8

Vectors 150 DEFINITIONS: ANALYTIC REPRESENTATION A vector V is a quantity which is completely specified by a magnitude and a direction. A vector  may  be represented geometrically by a directed line segment V = OA. A scalar S is a quantity which is completely specified by a magnitude. Let i, j, and k represent three vectors of unit magnitude along the three mutually perpendicular lines OX, OY, and OZ, respectively. Let V be a vector in space, and a, b, and c the magnitudes of the projections of V along the three lines OX, OY, and OZ, respectively. Then V may be represented by V = ai + bj + ck. The magnitude of V is |V| = + a 2 + b 2 + c 2 , and the direction cosines of V are such that cos α: cos β: cos γ = a : b : c.

151 VECTOR SUM V OF n VECTORS Let V1, V2, ... Vn be n vectors given by V1 = a1i + b1 j + c1 k, etc. Then the sum is V = V1 + V2 + ... + Vn = (a1+ a2 + .... + an) i + (b1+ b2 + .... + bn) j + (c1+ c2 + .... + cn) k.

152 PRODUCT OF A SCALAR S AND A VECTOR V SV = (Sa) i + (Sb) j + (Sc) k. (S1 + S2)V = S1V + S2V · (V1 + V2)S = V1S + V2S.

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134 • Mathematical Formulas and Scientific Data

153 SCALAR PRODUCT OF TWO VECTORS: V1.V2. V1 · V2 = |V1| |V2| cos φ,  where φ is the angle from V1 to V2. V1 · V2 = a1a2 + b1b2 + c1c2 = V2 · V1  V1 ·V1 = |V1|2. (V1 + V2) · V3 = V1 · V3 + V2 · V3. V1 · (V2 + V3) = V1 · V2 + V1 · V3. i · i = j · j = k · k = 1.    i · j = j · k = k · i = 0.

154 V  ECTORS PRODUCT OF TWO VECTORS: V1 × V2. V1 × V2 = |V1| |V2| sin f 1, where f is the angle from V1 to V2 and 1 is a unit vector perpendicular to the plane of V1 and V2 and so directed that a right-handed screw driven in the direction of 1 would carry V1 into V2. V1 × V2 = − V2 × V1 = (b1 c2 − b2 c1) i + (c1 a2 − c2 a1) j

+ (a1 b2 − a2 b1) k.

(V1 + V2) × V3 = V1 × V3 + V2 × V3. V1 × (V2 + V3) = V1 × V2 + V1 × V3. V1 × (V2 × V3) = V2(V1 · V3) − V3(V1 · V2). i × i = j × j = k × k = 0,  i × j = k, j × k = i, k × i =j,

V1 · (V2 × V3) = (V1 × V2) · V3 = V2 · (V3 × V1) a1 a2 a3 = [V1 V2 V3] = b1 b2 b3 . c1 c2 c3

155 DIFFERENTIATION OF VECTORS V = ai + bj + ck. If V1, V2 ….. are functions of a scalar variable t, then

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Vectors • 135

dV dV d (V1 + V2 + ...) = 1 + 2 + ..., dt dt dt dV1 da1 db dc i + 1 j + 1 , etc. = dt dt dt dt

where

dV dV d (V1 ×V2 ) = 1 ×V2 + V2 × 2 . dt dt dt dV dV d (V1 ´ V2 ) = 1 ´ V2 + V1 ´ 2 . dt dt dt V×

dV dV dV =V . If |V| is constant, V × = 0. dt dt dt

grad S ≡ ÑS º

¶S ¶S ¶S i+ j + k , where S is a scalar. ¶x ¶y ¶z

div V ≡ Ñ × V º

¶a ¶b ¶c + + . (divergence of V). ¶x ¶y ¶z

i ¶ curl V ≡ rot V º ¶x a 2 div grad S ≡ Ñ S º

j ¶ ¶y b

k ¶ º Ñ ´V . ¶z c

¶ 2S ¶ 2S ¶ 2S + + . ¶x2 ¶y2 ¶z2

2 2 2 Ñ 2V ≡ iÑ a + jÑ b + k Ñ c.

curl grad S = 0. div curl V = 0. curl curl V = grad div V − Ñ 2V .

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136 • Mathematical Formulas and Scientific Data

156 GREEN’S THEOREM Let F be a vector and V be a volume bounded by a surface S. Then

òòò div F dV = òòò Ñ × F dV = òò F × d S , (V )

(V )

(S )

where the integrations are to be carried out over the volume V and the surface S.

157 STOKES’ THEOREM Let F be a vector and dr = dx i + dy j + dz k, and S be a surface bounced by a simple closed curve C. Then

ò F × dr = òò curl F × d S = òò Ñ ´ F × d S.

(C )

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(S )

(S )

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PART II

Scientific Data for Engineers Constants e = Base of natural logarithms ≈ 2.71828 log10e ≈ 0.434294  loge 10 ≈ 2.30259 log10 N ≈ loge N × 0.4343 loge N ≈ log10 N × 2.3026 1-radian ≈ 57° · 2958 ≈ 57° 17′ 45″  π ≈ 3 · 14159265 log10 π ≈ 0.49715

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p 1 ≈ 0 · 31831 ≈ 0 · 01745 π3 ≈ 9 · 8696 180 p

2/15/2023 12:45:31 PM

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CHAPTER

1

SI Units When making measurements of a physical quantity, the final result is expressed as a number followed by the unit. The number expresses the ratio of the measured quantity to some fixed standard and the unit is the name or symbol for the standard. Over the years, a large number of standards have been defined for physical measurement and many systems of units have evolved. Recently, there has been an attempt to simplify the language of science by the adoption of a system of units, the System International Units, which is intended to be used universally. This system of units, SI, was the outcome of a resolution of the 9th General Conference of Weights and Measures (CGPM) in 1948, which instructed an international committee to “study the establishment of a complete set of rules for units of measurement.” The constants in this book are given in SI except where stated otherwise. SI contains three classes of units: (i) base units, (ii) derived units, and (iii) supplementary units. Base Units: There are seven base units: (i)

the meter, the standard of length,

(ii)

the kilogram, the standard of mass,

(iii) the second, the standard of time, (iv) the ampere, the standard of electric current, (v)

MFSDH.Part II_2PP.indd 139

the kelvin, the standard of temperature,

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140 • Mathematical Formulas and Scientific Data

(vi) the candela, the standard of luminous intensity, and (vii) the mole, the standard of amount of substance. Derived Units: Derived units can be formed by combining base units. Thus the unit of force can be produced by combining the first three base units. Often derived units are given names, for example, the unit of force is the newton. Supplementary Units: Two supplementary units are at present defined, the radian and the steradian, which are the units for plane and solid angles, respectively. SI Prefixes and Multiplication Factors: To obtain multiples and submultiples of units, standard prefixes are used as shown below: Multiplication factor 1 000 000 000 000 = 1012 1 000 000 000 = 109 1 000 000 = 106 1 000 = 103 100 = 102 10 = 101 0.1 = 10–1 0.01 = 10–2 0.001 = 10–3 0.000 001 = 10–6 0.000 000 001 = 10–9 0.000 000 000 001 = 10–12 0.000 000 000 000 001 = 10–15 0.000 000 000 000 000 001 = 10–18

Symbol T G M k h da d c m µ n p f a

Prefix tera giga mega kilo hecto deca deci centi milli micro nano pico femto atto

It should be noted that masses are still expressed as multiples of the gram, although the base unit is the kilogram. Thus 10–6 kg should be written as 1 mg.

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SI Units • 141

Other systems of units: Some other systems of units are still in common use. Thus for mechanical measurements, the British or fps system is still largely used, while for electrical measurements, the electrostatic and electromagnetic cgs systems are by no means obsolete. In the pages which follow, these systems of units are discussed and tables are included to help in conversion from one ­system to another.

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CHAPTER

2

The Fundamental Mechanical Units (A) SI UNITS In any system of measurement in mechanics, three fundamental units are required. These are the units of length, mass, and time. The base units as used in SI are the meter, kilogram, and second. The meter (m) This is defined as 1 650 763-73 of the wavelength, in vacuo of the orange light emitted by 86 in the transition 2p10 to 5d5. 38 Kr The kilogram (kg) This is defined as the mass of a platinum-iridium cylinder kept at Sévres. Originally intended to be the mass of a cubic decimeter of water at its maximum density, the cylinder was subsequently found to be 28 parts per million too large. The cylinder was then taken as an arbitrary standard of mass, while the volume of water which had the same mass (at maximum density) was defined to be 1 liter (l).Thus 1 litre = 1000.028 cm3. The 1964 General Conference on Weights and Measures redefined the liter to be a cubic decimeter but recommended that this unit should not be used in work of high precision. The second (s) This is the time taken by 9 192 631 770 cycles of the radiation from the hyperfine transition in cesium when unperturbed by external

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144 • Mathematical Formulas and Scientific Data

fields. Alternatively the ephemeris second is defined as 1/31 556 925-974 7 of the tropical year for 1900. Derived units of length, mass, and time Through common usage, certain multiples and submultiples of the three fundamental units have been given names. A list of the more common ones is given below as they have been in frequent use. None of them is a recognized SI unit. Length and area Micron (µm) = 10–6 m Angstrom (Å) = 10–10 m

Mass Tonne (t) = 106 g     = 1000 kg

Time Minute (min) = 60 s Hour (h) = 3 600 s

Fermi (fm) = 10–15 m

Day (d) = 86 400 s

Are (a) = 100 m

Year (a) ≃ 3.1557 × 107 s

2

Barn (b) = 10–28 m2

The radian (rad) is the plane angle between two radii of a circle that cut off on the circumference of an arc equal in length to the radius. The steradian (sr) is the solid angle which, having its vertex in the center of a sphere, cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere. Other units of angular measures are: The degree (°) is a unit of angle equal to (π/180) rad. The minute of arc (′) is (1/60) degree and thus is equal to (π/10 800) rad. The second of arc (″) is (1/60) minute and thus is equal to (π/648 000) rad.

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CHAPTER

3

The Fundamental Electrical and Magnetic Units When units were first required for the measurement of electrical quantities it was natural to define them in terms of the three fundamental units, centimeter, gram, and second, which were already commonly used in mechanics. Electrical phenomena are related to mechanical phenomena by two effects: (a) the force between static electric charges (Coulomb’s law) and (b) the force between electric currents (Ampere’s law). Correspondingly, two distinct systems of cgs electrical units were introduced: the electrostatic and electromagnetic systems. Neither of these systems has units of convenient size in practical applications. Consequently, a practical set of electrical units, defined as decimal multiples of the electromagnetic units was established by various International Congresses of Electricians meeting between 1881 and 1889. The first two units defined were the ohm (109 emu), chosen to be similar to the Siemens unit of resistance, and the volt (108 emu), chosen to be similar to the emf of the Daniell cell. From these, six other units, the ampere, coulomb, joule, watt, henry, and farad were derived. These practical units were not made into a complete system, since no magnetic units were defined, the unmodified magnetic units of the electromagnetic system (e.g., oersted and gauss) being used in practical applications.

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146 • Mathematical Formulas and Scientific Data

In 1901, Giorgi showed that if the meter, kilogram, and second were used as fundamental units instead of the centimeter, gram and second, a single, consistent and comprehensive system of electrical and magnetic units could be built up incorporating the already firmly-established practical units. This is because, using the meter, kilogram and second, the unit of mechanical power becomes 107 erg s–1, which is the appropriate practical electrical unit, that is, the watt. The use of the Giorgi system, also known as the mks system, or the Absolute Practical System was approved by an International Electro-technical Commission in 1935. The Absolute Practical System, with the ampere as the electrical base unit was adopted by the CGPM for SI.

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CHAPTER

4

Relations Between the Systems of Electrical Units Coulomb’s law for the force F between charges Q1 and Q2, distance r apart in vacuo, may be expressed in the form

F=

Q1Q2 ...(1.1) e 1r2

where ε1 is a constant, called the permittivity of free space. In the cgs electrostatic system, ε1 is chosen to be unity. This choice of the value of ε1, together with the use of the centimeter, gram, and second uniquely determines the system of units. Ampere’s law for the force between two parallel current elements I1ds1 and I2ds2 distance r apart in vacuo may be expressed in the form

F = m1

I1 ds1 I2 ds2 sin q ...(1.2) r2

where µ1 is a constant, called the permeability of free space. In the cgs electromagnetic system, µ1 is chosen to be unity. This choice of the value of µ1, together with the use of the centimeter, gram, and second, again determine the system of units uniquely. It may be noted that these two systems of units, defined by ε1 = 1 and µ1 = 1, cannot be combined directly to form a single

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148 • Mathematical Formulas and Scientific Data

consistent system. It can be shown from Maxwell’s electromagnetic theory that, in any consistent system of units, µ1ε1 = 1/c2, where c is the velocity of electromagnetic radiation (e.g., light) in free space, measured in the appropriate units of length and time (e.g., c ≃ 3 × 108 ms–1).

In SI, neither ε1 nor µ1 is chosen to be unity. The fundamental units chosen are the meter, kilogram, and second and ampere which are sufficient to determine the complete system uniquely. In particular, µ1 may be shown to have the value 10–7 newton ampere–2, where the newton is the SI unit of force. This value of µ1 is readily derived from equation (2). The appropriate value of ε1 is then calculated, knowing the experimentally determined value of the velocity of light.

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CHAPTER

5

Rationalization of MKS Units It is found that many formulae are simplified if the permeability of free space is re-defined as µ0 = 4πµ1. Ampere’s law for current ­elements in free space is then expressed in “Rationalized mks units” as

F=

m 0 I1 ds1 I2 ds2 sin θ ...(1.3) 4p r 2

where µ0 = 4π × 10–7 newton ampere–2 (or henry meter–1). Similarly, the permittivity of free space is re-defined as ε0 = ε1/4π, and Coulomb’s law, for charges in free space, is expressed in rationalized mks units as

F=

Q1Q2 4pe 0 r 2 ...(1.4)

The value of ε0, given by 1/c2µ0, is approximately 8.85 × 10–12 farad meter–1. For an isotropic, homogeneous medium other than free space, µ0 in equation (3) is replaced by µ0 = µrµ0, where µr, is the relative permeability of the medium; and ε0 in equation (4) is replaced by ε = εr ε0, where εr, is the relative permittivity (dielectric coefficient) of the medium.

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CHAPTER

6

Definitions of Electric and Magnetic Quantities in SI The base unit Current (I): The unit of current is the ampere (A) defined as that constant current which, if maintained in each of two infinitely long straight parallel wires of negligible cross-section placed 1 meter apart, in a vacuum, will produce between the wires a force of 2 × 10–7 newtons per meter length. Derived units Charge (Q): The unit of charge (quantity) is the coulomb (C) defined as the quantity of electricity transported per second by a current of 1 ampere. Potential Difference (V): The unit of potential difference is the volt (V), defined as the difference of electrical potential between two points of a wire carrying a constant current of 1 ampere when the power dissipation between those points is 1 watt. Resistance (R): The unit of resistance is the ohm (Ω), defined as the electrical resistance between two points of a conductor when a constant potential difference of 1 volt applied between these points produces in the conductor a current or 1 ampere.

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152 • Mathematical Formulas and Scientific Data

Conductance (G): The unit of conductance is the siemens (S), defined as the electrical conductance between two points of a conductor when a constant potential difference of 1 volt applied between these points produces in the conductor a current of 1 ampere. Inductance (L): The unit of inductance is the henry (H), defined as the inductance of a closed circuit in which an electromotive force of 1 volt is produced when the current in the circuit varies uniformly at the rate of 1 ampere per second. Capacitance (C): The unit of capacitance is the farad (F), defined as the capacitance of a capacitor between the plates of which there appears a difference of potential of 1 volt when it is charged by 1 coulomb. Magnetic Intensity (H): is defined through Ampere’s theorem for the intensity due to a current element. In the usual notaI.ds.sin θ. tion H = Unit ampere meter –1. 4p r 2 Magnetic Flux (φ) of the induction B: is defined as Ú B.n dA where n is a unit vector perpendicular to an element of area dA. Unit, weber (Wb). Magnetic Flux Density or Induction (B): is defined through the equation for the force on a current element placed in a ­magnetic field, viz F = B.I.ds.sinθ, in the usual notation. Unit, Tesla (T). B = µ0µrH where µr is the relative permeability of the medium with respect to free space and µ0 is the permeability of free space. µ0 = 4π × 10–7 henry meter–1. Magnetic Moment (m)∗: is the couple exerted on a magnet placed at right angles to a uniform field with unit flux density. Unit, ampere meter2. Intensity of Magnetisation (M)∗: is the magnetic moment per unit volume of a magnet. Unit, ampere meter–1. Pole Strength (P)∗: On the mks system the hypothetical concept of an isolated magnetic pole is abandoned by many writers on the grounds that all magnetism arises from electrical effects, hence the definitions of H and B (above). Other writers use the idea of a magnetic pole as a simple and convenient concept in

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Definitions of Electric and Magnetic Quantities in SI • 153

magnetometry. In this connection, we define a unit magnetic, pole as one which when situated 1 meter distant in vacuum from an equal pole experiences a force of µ0/4π newtons. Alternatively, it can be defined as that pole strength which when placed in a unit induction experiences a force of 1 newton. Unit, ampere-meter. NOTE

For these definitions, we adopt the Sommerfeld system of units in which the magnetic moment of a current loop is the product of the area of the loop and the current flowing around the edge of the loop: m = IA. An alternative system due to Kennelly uses the relation m = µ0IA.

Line of Force: A line of force is a curve in a magnetic field, such that the tangent at every point is the direction of the magnetic force at that point. Magnetomotive Force (Fm): is defined as the line integral Ú H.dl evaluated for a closed path. It is equal to the total conduction current linked. Unit, ampere. Coulomb’s Magnetic Law: states that the force between two m m PP poles P1 and P2 situated distance d apart is given by F = r 0 12 2 , 4p d where µr is the permeability of the medium and µ0 the permeability of the free space = 4π × l0–7 henry meter–1. µrµ0 replaces the permeability µ of the cgs system. Electrical Intensity (X or E): The electrical intensity at a point in an electric field is the force exerted on unit charge (1 coulomb) placed at that point, assuming that the field is not disturbed by so doing. Unit, volt meter–1 (which is equivalent to the newton coulomb–1). Coulomb’s Electrostatic Law: appears in the form Q1Q2 , where Q1 and Q2 are the two charges situated a F= 4pe 0e r d 2 distance d apart in a medium whose permittivity relative to that of free space is εr. The permittivity of free space ε0 ≃ (l/36π) × 10−9 farad meter−1. εr is a pure number. (The product ε0εr, is analogous to the dielectric constant as defined in the cgs system.)

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CHAPTER

7

Definitions in the CGS Electromagnetic System of Units Magnetic Pole (P): When two like magnetic poles are placed 1 cm apart in vacuo, they repel one another with a force of 1 dyne. Magnetic Field Strength or Intensity (H): is the force experienced by a unit North pole when placed at the given point in a magnetic field, it is assumed that the introduction of the pole does not disturb the field. Unit, oersted. The intensity is one ­oersted when a unit North pole experiences a force of 1 dyne on being placed at the given point in the field. The field strength in vacuum is represented as the number of lines of force passing perpendicularly through 1 cm2 placed at the point in question. On this convention 4π lines of force leave a unit North pole. Magnetic Flux (φ): through any area at right angles to a magnetic field is the product of the area and the field strength. Unit, maxwell. One maxwell is the flux through unit area (1 cm2) placed perpendicularly to a unit uniform field. Hence one line of force is equivalent to one maxwell. Magnetic Moment (m): of a magnet, is the couple exerted on the magnet when placed at right angles to a unit uniform field. For a bar magnet, it is equivalent to the product of the pole strength and the distance between the poles. Unit, pole cm.

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156 • Mathematical Formulas and Scientific Data

Magnetic Potential (Ω): is the work done in bringing a unit North pole from infinity or a point of zero potential to the point in question. Unit, gilbert. 1 gilbert is that potential that requires the expenditure of 1 erg of work in bringing a unit North pole from infinity to the point. Intensity of Magnetisation (M): of a sample of material is the magnetic moment per unit volume. Magnetic Susceptibility (χ): of a material is the ratio of the intensity of magnetization produced in the sample to the magnetic M field which produced the magnetization, c = . H N.B.: χ is not a constant but is a function of H. Magnetic Induction (B): in any material is the number of lines of magnetic force (often called lines of induction) passing perpendicularly through the unit area. Unit, gauss. One gamma = 10–5 gauss. Magnetic Permeability (µ): of any material is the ratio of the magnetic induction in the sample to the magnetic field producing it, that is, µ = B/H. Although µ is so defined, B is not proportional to H, for B = H + 4πM. Also B/H = 1 +4π M/H or µ = 1 + 4πχ. Hence µ is not a constant but a function of H. (see χ above). Coulomb’s Law of Force: states that the force F between two poles of strength P1 and P2 is proportional to the product of the pole strengths and inversely proportional to the square of their PP distance apart (d). Thus F = 1 22 where 1/µ is the constant of promd portionality, µ is the permeability of the medium in which the poles are located. In this system, as already stated, the permeability of free space is defined to be unity. Current (I): The electromagnetic unit (emu) of current is that which when flowing around 1 cm arc of a circle of radius 1 cm, produces a magnetic field of 1 oersted at the center. Unit, emu of current. Charge (Q): The emu of charge (quantity) is that delivered in 1 second by the passage of unit current. Unit, biot.

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Definitions in the CGS Electromagnetic System of Units • 157

Potential Difference (PD): When unit current flows between two points in a circuit and unit work (1 erg) is done per second, the PD between the two points is unity. Unit, emu of PD. Electromotive Force (emf): When lines of magnetic force cut a conductor an emf is created which is given numerically (in emu) by the number of lines cut per second. Emf = dn/dt. Resistance (R): A conductor has unit resistance when applying unit PD, unit current flows. Unit, emu of resistance. Self Inductance (L): A conductor possesses unit self-inductance if unit emf is developed across it when the rate of change of current is unity. Unit, emu of self-inductance. Mutual Inductance (M): Two conductors possess unit mutual inductance when unit emf is developed in one by unit rate of change of current in the other. Unit, emu of mutual inductance. Capacitance (C): A conductor has unit capacitance when the addition of unit charge raises its potential by unity. Unit, cm. Dielectric Constant or Specific Inductivity Capacity (εr): of a material is the ratio of the capacity of a condenser with the material as dielectric to that of the same condenser in vacuum without a material dielectric. Coulomb’s Electrostatic Law of Force: states that the force F between two charges Q1 and Q2 is proportional to the product of the charges and inversely proportional to the square of their disQQ tance apart d. Thus F = 1 22 , where 1/εr is the constant of proe rd portionality. εr is the dielectric constant of the medium in which the charges are located. On the electrostatic system of units, the dielectric constant of free space is unity.

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CHAPTER

8

Heat Units and Definitions Temperature (t, θ or T). In SI, temperatures are measured on the thermodynamic scale with the Absolute Zero as zero and the triple point of water (i.e., the temperature at which ice, water, and water vapor are in equilibrium) as the upper fixed point. The thermodynamic scale is that given by a theoretical Carnot heat engine and is equal to the perfect gas scale. SI base unit, the Kelvin (K). The kelvin (K) unit of thermodynamic temperature, is the fraction 1/273-16 of the thermodynamic temperature of the triple point of water. The Degree Celsius (°C). The centigrade scale of temperature used the ice point as zero and the boiling point of water at 1 standard atmosphere as the upper fixed point set to be 100°C. The Celsius scale of temperature is defined to be the same as the thermodynamic scale with the zero shifted to the ice point, which is 273.15 K, and thus: q / °= C T / K − 273.15

The International Practical Scale of Temperature (IPST). In view of the difficulty of measuring on the thermodynamic scale, a scale of temperature based on fixed points was suggested by the 7th CCPM in 1927. The scale has been revised since so as to make temperatures on this scale agree as nearly as possible with the thermodynamic Celsius scale. A list of the fixed points and other important temperatures can be found in the book.

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160 • Mathematical Formulas and Scientific Data

The Degree Fahrenheit (°F). On the Fahrenheit scale, the ice point is 32°F and the steam point, 212°F. Thus t/°F = 32 + 1.8 (θ/°C). The Degree Reaumur (°R). On the Reaumur scale, the ice point is 0°R and the steam point, 80°R. Thus t/°R = 0.8 (θ/°C). Quantity of Heat (Q). Quantities of heat are measured in joules (J) in SI. Other units have been used, notably the calorie. The calorie is the amount of heat required to raise the temperature of 1 gram of water by l°C. This definition is not very precise ­however as the specific heat capacity of water varies with temperature. The 15°calorie was defined as the heat required to raise the temperature of 1 g of water from 14.5°C to 15.5°C. The mean ­calorie was defined as one-hundredth of the heat required to raise the temperature of 1 g of water from 0°C to 100°C. The k­ ilocalorie (1 000 calories) has also been used and written Calorie. Where quantities of heat are expressed in calories, it is recommended that the conversion factor to convert to joules be stated. In the fps system, the British thermal unit (Btu) is used. This is the quantity of heat required to raise the temperature of 1 lb of water through 1°F. The therm is 105 Btu. Specific Heat Capacity (cρ, c0). This is the amount of heat required to raise the temperature of 1 kg of a substance 1 K. Unit, J kg–1 K–1. Molar Heat Capacity (Cm). This is the amount of heat required to raise the temperature of 1 mol of substance through IK. Unit, Jmol–1 K–1. Heat Capacity (C). The amount of heat required to raise the temperature of a body through 1 K. Units, J K–1. Water Equivalent. The mass of water having the same total heat capacity as the given body. Thermal Conductivity (λ). The rate of flow of heat (dQ/dt) through a surface of area, A, in a medium is given by: dQ dT = −l A , dt dx

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Heat Units and Definitions • 161

where (dT/dx) is the temperature gradient, measured in the direction normal to the surface. The quantity λ is the thermal conductivity of the medium. Units, Jm–1s–1K–1, or Wm–1K–1. Specific Latent Heat (l). The specific latent heat of fusion (specific enthalpy Change on fusion) of a body is the heat required to convert 1 kg of the solid at its melting point into liquid at the same temperature. Unit, J kg–1. The specific latent heat of vaporization (enthalpy change on vaporization) of a liquid is the heat required to convert 1 kg of the liquid at its boiling point into vapor at the same temperature. Unit, J kg–1. Linear Expansivity (α). The increase in length per unit rise in temperature. Unit, K–1. Cubic Expansivity (γ). The increase in volume per unit rise in temperature. Unit, K–1. Critical Temperature (Tc) of a gas or vapor is that temperature above which it is not possible to liquefy the gas by the application of pressure alone. To liquefy a gas it must be cooled below its critical temperature before being compressed. Critical Pressure (pc): That pressure which just liquefies a gas at its critical temperature. Critical Volume (Vc): The volume of unit mass of gas at its critical temperature and pressure, that is, it is the reciprocal of the critical density. It is often taken as the volume of one mole of a gas at its critical temperature and pressure. Radiation. Stefan-Boltzmann Law: The total energy, E, of all wavelengths radiated per second per square meter by a full radiator at temperature T to surroundings at T0 is given by E = σ(T4 – T04 ), where σ is Stefan’s constant, σ = 5.669 7 × 10–8 W m–2 K–4. Planck’s Radiation Law: The energy density of radiation in an enclosure at temperature T having wavelengths in the range λ to λ + dλ is uλ dλ, where uλ dλ = 8πchλ–5 (exp hc/λkT – 1)–1 dλ = c1λ–5(exp c2/λT –1)–1 dλ c1 = 4.992 1 × 10–24 Jm

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c2 = 1.43879 × 10–2 m K

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162 • Mathematical Formulas and Scientific Data

The corresponding relation for radiation of frequency, v, is uv dv = (8πh/c3) (exp hv/kT – 1)–1 v3 dv. h = Planck’s constant; c = speed of light; k = Boltzmann’s constant; T = temperature of the enclosure. Wien’s Displacement Law: The wavelength of the most strongly emitted radiation in the continuous spectrum from a full radiator is inversely proportional to the absolute temperature of that body, that is, λT = b, where b is Wien’s constant = 2.898 × l0–3 m K. The Energy (E) of a quantum of radiation of frequency v is E = hv where h is Planck’s constant.

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CHAPTER

9

Photometric and Optical Units and Definitions Luminous intensity. In SI, the unit of luminous intensity is the candela. This unit replaces the International candle which was defined in terms of the light emitted per second in all directions by a specified electric lamp. SI base unit, the candela (cd). The candela is the luminous intensity, in the perpendicular direction, of a surface of 1/600 000 square meters of a full radiator at the temperature of freezing platinum under a pressure of 101 325 newtons per square meter. 1 candela = 0.982 international candles. Luminous flux: The unit of luminous flux, the lumen (1m) is defined as the light energy emitted per second within, unit solid angle by a uniform point source of unit luminous intensity. Thus, 1 cd = 1 1m sr–1. Illuminance of a surface is defined as the luminous flux reaching it perpendicularly per unit area. The British unit is the lumen ft–2, formerly called the foot-candle (fc). The metric unit is the lumen m–2 or lux (1x). Lambert’s Cosine law: For a surface receiving light obliquely, the illumination is proportional to the cosine of the angle which the light makes with the normal to the surface.

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164 • Mathematical Formulas and Scientific Data

Brightness of a surface is that property by which the surface appears to emit more or less light in the direction of view. This is a subjective quantity. The corresponding physical measurement of the light actually emitted is called luminance. Luminance of a surface is the measure of the light actually emitted (i.e., the luminous intensity) per unit projected area of surface, the plane of projection being perpendicular to the direction of view. Unit, cd ft–2 or cd m–2. In engineering, the luminance of an ideally diffusing surface emitting or reflecting one-lumen ft–2 is called one foot-lambert (ft–L). The Refractive index of a material (n) is the ratio of the velocity of light in free space to that in the material. Snell’s law: For light incident on a boundary between two media, the ratio of the sine of the angle of incidence (the angle between the light ray in the first medium and the normal to the boundary surface) to the sine of the angle of refraction (the angle between the refracted ray in the second medium and the normal) is a constant, being equal to the inverse ratio of the refractive indices of the two media. Dioptre is the unit of measure of the power of a lens and is given numerically by the reciprocal of the focal length expressed in meters.

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CHAPTER

10

Acoustical Units and Definitions Pressure: The unit of sound pressure is the pascal usually quoted as the root mean square (r.m.s.) pressure for a pure sinusoidal wave. Frequency: The unit of frequency is the cycle per second, now designated the hertz (Hz). Threshold of Hearing is, for a normal (average) observer, the sound level or intensity which is just audible. For a pure sinusoidal note of frequency 1000 Hz, it is close to a root mean square pressure of 2 × 10–5 Pa. Power Ratio: The unit of acoustical (or electrical) power measurement with respect to a standard level, is one bel. The interval between two powers W1 and W0 in bels is log10 (W1/W0). In practical work, the decibel (dB) is used. The interval between two powers W1 and W0 is 10 log10 (W1/W0) dB. In some instances, it is more convenient to employ natural logarithms. The power ratio so obtained is called the neper and is defined as follows. The power 1 interval between W1 and W0 is loge (W1/W0) nepers. Hence 2 1 neper = 8.686 dB. Intensity (I) of a sound wave in a given direction is the sound energy transmitted per second in this direction through unit area placed perpendicularly to the specified direction. Unit, W m–2. For a sinusoidal plane or spherical wave, the intensity is proportional to

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166 • Mathematical Formulas and Scientific Data

the mean square pressure exerted on an area at right angles to the given direction. Hence the interval between two intensities is given by 10 log10 (I1 /I0) dB or 20 log10 (ρ1/ρ0) dB where ρ1 and ρ0 are the r.m.s. pressures corresponding to the intensities I1 and I0. Loudness is the physiological counterpart of acoustical intensity. It is a function of the intensity but also varies with frequency and composition of the note being heard. The Weber–Fechner Law states that the sensation (loudness) is proportional to the logarithm of the stimulus (intensity). Loudness level of a sound is judged by comparison in free air with a standard sinusoidal note whose frequency is 1000 Hz. The unit is the phon. If an average observer decides that a sound is equally loud as the standard 1000 Hz note of intensity n dB above the standard reference level corresponding to a r.m.s. pressure of 2 × 10–5 Pa (i.e., the threshold of hearing), then the sound is said to have an “equivalent loudness” of n British Standard phons. Reverberation in an enclosure is the persistence of sound due to multiple reflections from the walls, etc. of the enclosure. Reverberation time is the time required, from the moment of cessation of a sound for the intensity to, drop by 60 dB, that is, to one-millionth of its original value. Unit, second. Absorption Coefficient of a surface is the ratio of the sound energy absorbed to the total sound energy incident on the surface. The ideal absorber is one from which no sound is reflected or scattered. For unit area of various substances, the coefficient is expressed in terms of equivalent area of open window (diffraction effects excluded). Unit, ft2 of open window or Sabine. The coefficient varies with frequency. Sabine’s Relation: For an auditorium whose walls, etc. consist of areas S1 S2, etc. of absorption coefficient α1 α2, etc., the reverberation time t (in seconds) is given by t = 0.05V (unit of length, Σa S 0.16V ft) or t = (unit of length, meter) where V is the volume of Σa S the auditorium and Σα S = α1S1 + α2S2 + etc.

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CHAPTER

11

Properties of the Elements The following table lists the elements with atomic number up to 92 alphabetically by name. Columns 1-4 and 13-16 are self-­ explanatory. Column 5 gives the crystal structures of the elements in their solid state. Where a change in structure occurs, the transition temperature is indicated (in K) under the crystal structures. The ­following abbreviations are used: bcc = body-centered cubic cubic (diam) = diamond structure fcc = face-centered cubic hcp = hexagonal close-packed

hex = hexagonal mon = monoclinic ortho = orthorhombic tetra = tetragonal

Column 6 lists the atomic radii of the elements in pm (10–12 m). These radii are calculated as half the distance of closest approach of atomic centers in the crystalline state. Column 7 gives the principal oxidation numbers and column 8, the corresponding ionic radii. Columns 9 and 10 give the energies (eV) required to remove the first and second electrons from the atom multiply by 96.49 to convert to kJ mol–1. Column 11 gives the energy required to remove an electron from the negative ion formed by the atom with an extra electron. These “electron affinities” are difficult to measure and there are few reliable results. Column 12 gives the electronegativities assigned to the elements by Pauling. These are numbers between 0 and 4 which may be used in determining the contribution of the ionic and covalent components of the bonds between different atoms.

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17

24

Argon

Arseni

Barium

Beryllium

Bismuth

Boron

Bromine

Cadmium

Caesium

Calcium

Carbon

Chlorine

Chromium

Cobalt

Copper

Ar

As

MFSDH.Part II_2PP.indd 168

Ba

Be

Bi

B

Br

Cd

Cs

Ca

C

Cl

Cr

Co

Cu

29

27

6

20

55

48

35

5

83

4

56

33

18

51

Antimony

13

Aluminum

Atomic number, Z

Al

Name

Sb

Symbols

63.55

58.93

52.00

35.45

12.01

40.08

132.90

112.40

79.90

10.81

208.98

9.01

137.34

74.92

39.95

121.75

26.98

Atomic weight (M/g mol−1)

1.8

0.9

0.98

3.615

1.25

—

> 0.19

—

3.363

0.33

> 0.7

0.30

—

—

−1.0

> 2.0

0.5

Electron affinities (E1/eV)

1.9

1.8

1.6

3.0

2.5

1.0

0.7

1.7

2.8

2.0

1.9

1.5

0.9

2.0

—

1.9

1.5

Electronegativities

8 930

8 900

7 200

(273 K)

3.21

2 300

1 540

1 870

8 650

(298 K)

3 100

2 500

9 800

1 800

3 600

5 730

1.66

6 700

2 700

Density (ρ/kg m−3)

(sub)

1550

1 356

1 765

2 160

172.1

>3 800

1 120

301.6

594.2

265.9

2 600

544.4

2 868

3170

2 755

238.5

5 100

1 760

960

1 038

331.9

(sub)

2 820

1 830

3 243

1 910

(28 atm) 1 000

886

87.4

1 650

2 740

Boiling point (Tb/K)

1 090

83.7

903.7

933.2

Melting point (Tm/K)

168 • Mathematical Formulas and Scientific Data

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49

53

36

57

82

3

Gold

Hafnium

Helium

Hydrogen

Indium

Iodine

Iridium

Iron

Krypton

Lanthanum

Lead

Lithium

Magnesium

Manganese

Mercury

Molybdenum

Au

Hf

He

H

In

I

Ir

Fe

Kr

La

Pb

Li

Mg

Mn

Hg

Mo

9

Germanium

MFSDH.Part II_2PP.indd 169

42

80

25

12

26

77

1

2

72

79

32

Fluorine

F

Ge

95.94

200.59

54.94

24.31

6.94

207.19

138.91

83.80

55.85

192.2

126.90

114.82

1.00797

4.003

178.49

196.97

72.59

19.00

1.0

1.54

—

− 0.32

0.82

—

—

—

0.6

—

3.070

—

0.76

− 0.53

—

2.1

—

3.448

1.8

1.9

1.5

1.2

1.0

1.8

1.1

—

1.8

2.2

2.5

1.7

2.1

—

1.3

2.4

1.8

4.0

10 200

(273 K)

13 590

7 440

1 741

534

11 340

6 150

3.49

7 870

22 420

4 940

7 310

(273 K)

0.08987

0.166

13 300

19 300

5 400

1.7 (273 K)

53.5

2 880

234.3

1 517

924

452

600.4

1190

116.5

1 808

2 716

386.6

429.8

14.01

0.95 (26 atm)

2 423

1 336.1

1 210.5

5 830

629.7

2 370

1 380

1 590

2 017

3 742

120.8

3 300

4 800

457.4

2 300

20.4

4.21

5 700

3 239

3 100

85.01

Properties of the Elements • 169

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Neon

Nickel

Niobium

Nitrogen

Oxygen

Palladium

Phosphorus

Platinum

Polonium

Potassium

Radium

Radon

Rhenium

Rhodium

Samarium

Selenium

Silicon

Silver

Ni

Nb

N

O

Pd

P

Pt

Po

K

Ra

Rn

Re

Rh

Sm

Se

Si

Ag

Name

Ne

Symbols

MFSDH.Part II_2PP.indd 170

47

14

34

62

45

75

86

88

19

84

78

15

46

8

7

41

28

10

Atomic number, Z

107.87

28.09

78.96

150.35

102.91

186.2

222

226

39.10

209

195.09

30.97

106.4

16.00

14.01

92.91

58.71

20.18

Atomic weight (M/g mol−1)

2.5

1.5

3.7

—

—

0.15

—

—

0.82

—

—

0.8

—

1.471

0.05

—

1.3

− 0.57

Electron affinities (E1/eV)

1.9

1.8

2.4

1.1

2.2

1.9

—

0.9

0.8

2.0

2.2

2.1

2.2

3.5

3.0

1.6

1.8

—

Electronegativities

10 500

2 300

4 810

7 500

12 440

20 500

(273 K)

9.73

5 000

860

9 400

21 450

1 800 (y)

2 200 (r)

12 000

1.33

1.165

8.570

8 900

0.839

Density (P/kgm−3)

1 234

1 680

490

1 345

2 230

3 450

202

970

336.8

527

2 042

317.2

1 825

54.7

63.3

2 741

1 726

24.5

Melting point (Tm/K)

2 485

2 628

958

2 200

4 000

5 900

211.3

1 410

1 047

1 235

4 100

552

3 200

90.2

77.3

5 200

3 005

27.2

Boiling point (Tb/K)

170 • Mathematical Formulas and Scientific Data

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Sodium

Strontium

Sulfur

Tantalum

Tellurium

Thallium

Thorium

Tin

Titanium

Tungsten

Uranium

Vanadium

Xenon

Yttrium

Zinc

Zirconium

Na

Sr

S

Ta

MFSDH.Part II_2PP.indd 171

Te

Tl

Th

Sn

Ti

W

U

V

Xe

Y

Zn

Zr

40

30

39

54

23

92

74

22

50

90

81

52

73

16

38

11

91.22

65.37

88.91

131.30

50.94

238.03

183.85

47.90

118.69

232.04

204.37

127.60

180.95

32.06

87.62

22.99

—

—

—

—

—

0.94

0.5

0.39

—

—

—

3.6

—

2.07

—

0.84

1.4

1.6

1.2

—

1.6

1.7

1.7

1.5

1.8

1.3

1.8

2.1

1.5

2.5

1.0

0.9

970

6 500

7 140

4 600

5.50

6 100

19 050

19 320

4 540

7 300

11 500

11 860

6 240

16 600

2 070

2 600

371

2 125

692.6

1 768

161.2

2 160

1 405.4

3 650

1 948

505.1

2 000

576.6

722.6

3 269

386

1 042

3 851

1 180

3 200

166.0

3 300

4 091

6 200

3 530

2 540

4 500

1 730

1 260

5 698

717.7

1 657

1 165

Properties of the Elements • 171

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CHAPTER

12

Properties of Metallic Solids (at 293 K) Values quoted for Tensile Strength and Yield Stress are in units of 104 N m−1 (= MPa). Values of Young’s Modulus are in units of 109 N m−l (= GPa). These values are typical observations and are approximate only. The elastic properties vary somewhat between specimens depending on the manufacturing process and the previous history of the sample. The Shear Modulus (G) and Bulk Modulus (K) can be calculated from the relations: G = ½E/(l + v) and K = ½E/ (l − 2v), where E is Young’s Modulus and v is Poisson’s Ratio.

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MFSDH.Part II_2PP.indd 174

Aluminum

Aluminum strong alloy

Antimony

Bismuth

Brass (70Cu/30Zn)

Bronze (90Cu/10Sn)

Cobalt

Constantan

Copper

1.

2.

3.

4.

5.

6.

7.

8.

9.

Name

Density ρ/kg m−3

8 930

8 880

8 900

8 800

8 500

9 800

6 680

2 800

2 710

Melting point Tm/K 1356

1360

1765

1300

1300

544

904

800

932

Specific latent heat of fusion I/J kg−1 21

25

5

16

39

38

× 104

Specific heat capacity Cp/J kg −1 K−1 385

420

420

360

370

126

205

880

913

Linear expansivity α/K −1 17

17

12

17

18

13

10

23

23

× 10−6

Thermal conductivity λ/W m −1 K −1 385

23

69

180

110

8

18

180

201

Electrical resistivity ρ/Ω m 1.7

47

6

30

~8

115

40

5

2.65

× 10−8

Temperature coefficient of resistance (1/ρ0) (dρ /dT) K−1 39

± 0.4

66

~ 15

45

~ 50

16

40

× 10−4

Tensile strength (σT/MPa) 150

~ 500

260

550

600

80

Yield strength (σγ/MPa) 75

140

450

550

50

Elongation (e/%) 45

10

8

10

43

Young’s Modulus (E/GPa) 117

170

100

32

78

71

Poisson’s ratio 0.35

0.33

0.35

0.33

0.34

9

8

7

6

5

4

3

2

1

174 • Mathematical Formulas and Scientific Data

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German silver (60Cu/25Zn/15Ni)

Gold

Invar (64Fe/36Ni)

Iron, pure

Iron, cast grey

Iron, cast white

Iron, wrought

Lead

10.

11.

12.

MFSDH.Part II_2PP.indd 175

13.

14.

15.

16.

17.

1810 600

11 340

1420

1500

1810

7 850

7 700

7 150

7 870

1800

1340

19 300

8 000

1300

8 700

2.6

14

14

10

27

7

126

480

500

106

503

132

400

29

12

11

11

12

0.9

14

18

35

60

75

75

80

16

296

29

21

14

10

10

10

81

2.4

33

43

60

65

20

34

4

15

~ 370

230

100

300

480

120

450

12

150

165

280

50

45

~0

45

40

40

18

197

110

206

145

71

130

0.44

0.28

0.27

0.29

0.26

0.44

0.33

17

16

15

14

13

12

11

10

Properties of Metallic Solids (at 293 K) • 175

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MFSDH.Part II_2PP.indd 176

Magnesium

Manganin

Monel (70 Ni/30Cu)

Nickel

Nickel, a strong alloy

Phosphor bronze

Platinum

Silver

Sodium

18.

19.

20.

21.

22.

23.

24.

25.

26.

Name

Density ρ/kg m−3

970

10 500

21 450

8 500

8 900

8 800

8 500

1 740

Melting point Tm/K 371

1230

2042

1320

1726

1600

924

Specific latent heat of fusion I/J kg−1 12

10

11

31

41

38

× 104

Specific heat capacity Cp/J kg−1 K−1 1240

235

136

380

460

400

246

Linear expansivity α/K−1 71

19

9

17

13

14

18

25

× 10−6

Thermal conductivity λ/W m−1 K−1 134

419

69

59

210

22

150

Electrical resistivity (ρ/Ω m) 4.5

1.6

11

7

59

42

45

4

× 10−8

Temperature coefficient of resistance (1/ρ0) (dρ /dT) K−1 44

40

38

60

60

20

± 0.1

43

× 10−4

Tensile strength σT/MPa 150

350

560

1300

300

520

190

Yield strength σγ/MPa 180

420

1200

60

240

95

Elongation e/% 45

10

30

40

5

Young’s modulus E/GPa 70

150

120

110

207

120

44

Poisson’s ratio 0.37

0.38

0.38

0.38

0.36

0.33

0.29

26

25

24

23

22

21

20

19

18

176 • Mathematical Formulas and Scientific Data

2/15/2023 12:45:34 PM

Solder, soft (50Pb/50Sn)

Stainless Steel (18Cr/δNi)

Steel, mild

Steel, piano wire

Tin

Titanium

Zinc

27.

28.

MFSDH.Part II_2PP.indd 177

29.

30.

31.

32.

33.

7 140

4 540

7 300

7 800

7 860

7 930

9 000

693

1950

505

1700

1700

1800

490

10

6.0

190

385

523

226

420

510

176

31

9

23

15

16

111

23

65

50

63

150

5.9

53

11

15

96

40

38

50

50

6

150

620

30

3000

460

600

45

480

300

230

50

20

35

60

50

110

40

210

210

0.25

0.36

0.36

0.29

0.29

33

32

31

30

29

28

27

Properties of Metallic Solids (at 293 K) • 177

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MFSDH.Part II_2PP.indd 178

2/15/2023 12:45:34 PM

CHAPTER

13

Properties of Non-Metallic Solids (at 293 K) The following table lists materials which do not readily conduct electricity. In many cases, the physical constants cannot be specified accurately as the values observed depend so much on the manufacture and life history of the specimen. The values given are to be taken as representative only.

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MFSDH.Part II_2PP.indd 180

2/15/2023 12:45:34 PM

Alumina, ceramic

Bone

Brick, building

Brick, fireclay

Brick, paving

1.

2.

3.

4.

5.

Name

2 500

2 100

2 300

1 850

3 800

Density ρ/kg m−3

 

 

2 300

Melting point Tm/K

 

 

800

Specific heat capacity cp/J kg−1 K−1

4.0

4.5

9

9

× 10−4

Linear expansivity α/K−1

0.8

0.6 

 

29

Thermal conductivity λ/W m−1 K−1

~5

140

~ 150

Tensile strength σγ/MPa

 

 

 

Elongation e/%

 

28

345

Young’s modulus E/GPa

5

4

3

2

1

180 • Mathematical Formulas and Scientific Data

Brick, silica

Carbon, graphite

diamond

Concrete

Cork

Cotton

Epoxy resin

Fluon (PTFE)

Glass (crown)

Glass (flint)

Glass (wool)

Ice

Kapok

Magnesium oxide

6.

7.

MFSDH.Part II_2PP.indd 181

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

3 600

50

920

50

4 200

2 600

2 200

1 120

1 500

240

2 400

3 300

2 300

1 750

3 200

273

1 400

1 500

1 400

3 800

960

2 100

670

500

670

1050

1 400

1 400

2 050

3 350

525

710

12

51

8

9

55

39

12

~0

7.9

0.03

2.0

0.04

0.8

1.0

0.25

0.05

0.1

900

5.0

0.8

~ 100

22

50

400

~4

50–75

2–6

207

80

71

0.34

4.5

14

1 200

207

19

18

17

16

15

14

13

12

11

10

9

8

7

6

Properties of Non-Metallic Solids (at 293 K) • 181

2/15/2023 12:45:34 PM

1 050 1 250

Naphthalene

Nylon

Paraffin wax

Perspex

Phenol formaldehyde

Polyethylene (low den)

(high den)

Polypropylene

Polystyrene

Polyvinylchloride ­ (non-rigid)

Polyvinylchloride (rigid)

22.

MFSDH.Part II_2PP.indd 182

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

1 700

900

955

920

1 300

1 190

900

1 150

1 150

1 500

Melamine formaldehyde

21.

2 600

Marble

20.

485

485

510

450

410

410

350

330

470

350

1 000

1 800

1 300

2 100

2 300

2 300

1 700

1 500

2 900

1 700

1 310

1 700

880

55

150

70

62

250

250

40

85

110

100

107

40

10

0.08

0.2

0.2

0.25

0.25

0.4

0.3

2.9

60

15

50

35

26

13

50

50

70

70

5–25

200–400

1–3

>220

100–300

400–800

0.4–0.8

2–7

60–300

2.8

0.01

3.1

1.2

0.43

0.18

6.9

3

9

32

31

30

29

28

27

26

25

24

23

22

21

20

182 • Mathematical Formulas and Scientific Data

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MFSDH.Part II_2PP.indd 183

2/15/2023 12:45:34 PM

Polyvinylidene chloride

Quartz fiber

Rubber (polyisoprene)

33.

34.

35.

Name

910

2 660

Density ρ/kg m−3

300

2 020

470

Melting point (Tm/K)

1 600

788

Specific heat capacity cp/J kg−1 K−1

220

0.4

190

× 10−4

Linear expansivity α/K−1

0.15

9.2

Thermal conductivity λ/W m−1 K−1

17

30

Tensile strength σγ/MPa

480–510

160–240

Elongation e/%

0.02

73

Young’s modulus E/GPa

35

34

33

Properties of Non-Metallic Solids (at 293 K) • 183

MFSDH.Part II_2PP.indd 184

650

600

Titanium carbide

Wood, oak (with grain)

Wood Spruce (with grain)

Wood Spruce (across grain)

38.

39.

40.

41.

4 500

2 070

Sulfur

37.

3 170

Silicon carbide

36.

386

730

7

64

4.5

0.15

28

0.26

0.5

14

12

345

41

40

39

38

37

36

184 • Mathematical Formulas and Scientific Data

2/15/2023 12:45:34 PM

CHAPTER

14

Properties of Liquids (at 293 K)

MFSDH.Part II_2PP.indd 185

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MFSDH.Part II_2PP.indd 186

Acetic acid (C2H4O2)

Acetone (C3H6O)

Benzene (C6H6)

Bromine (Br)

Carbon disulfide (CS2)

Carbon tetrachloride (CCl4)

Chloroform (CHCl3)

Ether, diethyl (C4H10O)

1.

2.

3.

4.

5.

6.

7.

8.

Name

Density ρ/kg m−3

714

1490

1632

1293

3100

879

780

1049

Melting point Tm/K 157

210

250

162

266

279

178

290

Boiling point Tb/K 308

334

350

319

332

353

330

391

Specific latent heat of vaporization I/J kg−1 35

25

19

36

18.3

40

52

39

× 10−4

Specific heat of capacity cp/J kg−1 K−1 2300

960

840

1000

460

1700

2210

1960

Cubic expansivity γ/K−1 16.3

12.7

12.2

11.9

11.3

12.2

14.3

10.7

× 10−4

Thermal conductivity λ/W m−1 K−1 0.127

0.121

0.103

0.144

0.140

0.161

0.180

Surface tension σ/N m−1 17

27.1

26.8

32.3

41.5

28.9

23.7

27.6

× 10−3

Viscosity η/N s m−2 0.242

0.569

0.972

0.375

0.993

0.647

0.324

1.219

× 10−3

Refractive index n 1.3538

1.4467

1.4607

1.6276

1.66

1.5011

1.3620 (288 K)

1.3718

2.49

Bulk modulus of rigidity K/GPa 0.69

1.1

1.12

1.16

1.58

1.10

~ 0.8

8

7

6

5

4

3

2

1

186 • Mathematical Formulas and Scientific Data

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Ethyl alcohol (C2H6O)

Glycerol (C3H8O3)

Mercury (Hg)

Methyl alcohol (CH4O)

Nitrobenzene (C6H5NO2)

Olive oil

Paraffin oil

Phenol (C6H6O)

Toluene (C7H8)

Turpentine

Water (H2O)

Water, sea

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

789

MFSDH.Part II_2PP.indd 187

1025

998

870

867

1073

800

920

1175

791

13546

1262

264

273

263

178

314

279

179

234

293

156

~ 377

373

429

384

455

570

484

337

630

563

352

226

29

35

53

33

112

29

83

85

3900

4190

1760

1670

2350

2130

1970

1400

2500

140

2400

2500

2.1

9.7

10.7

7.9

900

7.0

8.6

11.9

1.82

4.7

10.8

0.591

0.136

0.134

0.150

0.170

0.160

0.201

7.96

0.270

0.177

72.7

27

28.4

40.9

26

32

43.9

22.6

472

63

22.3

1.000

1.49

0.585

12.74

~ 1000

84

2.03

0.594

1.552

1495

1.197

1.343

1.333

1.48

1.4969

1.5425 (313 K)

1.43

1.48

1.5530

1.3276

1.73

1.4730

1.3610

2.05

1.28

1.09

1.62

1.60

2.2

0.97

26.2

4.03

1.32

9

20

19

18

17

16

15

14

13

12

11

10

Properties of Liquids (at 293 K) • 187

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MFSDH.Part II_2PP.indd 188

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CHAPTER

15

Properties of Gases at STP

MFSDH.Part II_2PP.indd 189

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MFSDH.Part II_2PP.indd 190

Acetylene (C2H2)

Air

Ammonia (NH3)

Argon (Ar)

Carbon dioxide (CO2)

Carbon monoxide (CO)

Chlorine (Cl2)

Cyanogen (C2N2)

1.

2.

3.

4.

5.

6.

7.

8.

Substance

Density ρ/kg m−3 2.337

3.214

1.250

1.977

1.784

0.771

1.293

1.173

Boiling point T/K 252

238

81

195

87

240

83

189

Specific latent heat of vaporization l/J kg−1 43.2

28.1

21.1

36.4

15.8

137.1

21.4

× 10−4

Specific Heat Capacity cp/J kg−1 K−1 1720

478

1050

834

524

2190

993

1590

Ratio of specific heats γ = cP/cv 1.26

1.36

1.404

1.304

1.667

1.310

1.402

1.26

Thermal conductivity (λ/W m−1 K−1) 72

232

145

162

218

241

184

× 10−4

Viscosity η/Ns m−2 9.28

12.9

16.6

14

21

9.18

18.325 (300 K)

9.35

× 10−6

Refractivity (n − 1) 835

773

338

451

281

376

292

606

× 10−6

Critical temperature Tc /K 401

417

134

304

151

405

132

309

Critical pressure (Pc /MPa) 6.0

7.71

3.50

7.38

4.86

11.3

3.77

6.14

Critical volume (Vc/m3 mol−1) 124

93.1

94.0

75.2

72.5

113∗

× 10−6

8

7

6

5

4

3

2

1

190 • Mathematical Formulas and Scientific Data

2/15/2023 12:45:35 PM

∗

MFSDH.Part II_2PP.indd 191

Helium (He)

Hydrogen (H2)

Hydrogen chloride (HCl)

Hydrogen sulfide (H2S)

Methane (CH4)

Nitric oxide (NO)

Nitrogen (N2)

Nitrous oxide (N2O)

Oxygen (O2)

Sulfur dioxide (SO2)

Water Vapor (273 K) (H2O)

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

0.800

2.927

1.429

1.978

1.250

1.340

0.717

1.538

1.640

0.090

0.179

1.260

170

263

90

183

77

121

109

211

189

20.35

4.25

226.1

40.3

24.3

37.6

20.9

46.2

51.1

55.3

41.4

45.3

2.5

48.4

2020 (373 K)

645

913

892

1040

972

2200

1020

796

14300

5240

1500

1.26

1.40

1.303

1.404

1.394

1.313

1.32

1.40

1.41

1.66

1.26

164

158

77

244

151

243

238

302

120

1684

1415

9.7

8.7

11.7

19.2

13.5

16.7

17.8

10.3

11.7

13.8

8.35

18.6

254

686

272

516

297

297

444

634

447

132

36

696

The critical volume is here defined as the volume of one mole of the gas at its critical temperature and pressure.

Ethylene (C2H4)

9.

10.

647

430

154

310

126

179

191

374

325

33.3

5.3

283

22.12

7.88

5.08

7.24

3.39

6.5

4.62

9.01

8.26

12.94

0.23

5.12

56.8

122

78

96.7

90.1

98.7

97.9

87

65.5

58

127.4

9

20

19

18

17

16

15

14

13

12

11

10

Properties of Gases at STP • 191

2/15/2023 12:45:35 PM

MFSDH.Part II_2PP.indd 192

2/15/2023 12:45:35 PM

CHAPTER

16

Mechanical Data

The Mohs Scale of Hardness Substance

Hardness

Substance

Hardness

Substance

Hardness

Talc

1

Felspar

6

Fused zirconia

11

Gypsum

2

Vitreous silica

7

Fused alumina

12

Calcite

3

Quartz

8

Silicon carbide

13

Fluorite

4

Topaz

9

Boron carbide

14

Apatite

5

Garnet

10

Diamond

15

Approximate Hardness of Some Common Materials Substance Agate Aluminum Amber Asbestos Brass

MFSDH.Part II_2PP.indd 193

Hardness

Substance

6–7

Calcium

2–3

Carborundum

2–2.5 5 3–4

Chromium Copper Finger nail

Hardness 1.5 9–10 9 2.5–3 2.5

Substance Glass

Hardness 4.5–6.5

Marble

3–4

Penknife blade

6.5

Silver Steel (mild)

2.5–2.7 4–5

2/15/2023 12:45:35 PM

194 • Mathematical Formulas and Scientific Data

Viscosities of Liquids and Their Temperature Dependence, η, Ns m−2 Substance

0°C

10°C

20°C

30°C

40°C

50°C

Water

0.001787

0.001304

0.001002

0.00080

0.000653

0.000547

Aniline

0.0102

0.0065

0.0044

0.00316

0.00237

0.00185

Benzene

0.000912

0.0000758

0.000652

0.000564

0.000503

0.000442

Ethanol

0.00177

0.00147

0.0012

0.00100

0.000834

0.00070

Glycerol (propane 1, 2, 3-triol)

10.59

3.44

1.34

0.629

0.289

0.141

Rape oil

2.53

0.385

0.163

0.096

—

—

EMF of Standard Cells Weston (Cadmium) cell (20°C) Clark cell (l5°C)

=

l.0l86 volts (absolute)

=

l.0l83 volts (international)

=

l.4333 volts (absolute)

=

l.4328 volts (international)

Temperature dependence Weston cell Et = 1.0l86 − 0.0000406 (t − 20) − 9.5 × 10−7(t − 20)2 absolute volts Clark cell Et = 1.4333 − 0.00ll9 (t − 15) − 7 × 10−6(t − l5)2 absolute volts Approximate EMFs of Cells Bichromate

2 volts

Accumulator 2.0 volts (Ranges l.85−2.2 volts)

Bunsen

l.9 volts

Dry cell

l.5 volts

Daniell

l.08 volts

Nickel-Cadmium

l.3 volts

Grove

l.8 volts

Nickel-Iron

l.4 volts

Leclanché

l.46 volts

Zinc-Silver oxide

l.8 volts

MFSDH.Part II_2PP.indd 194

2/15/2023 12:45:35 PM

Mechanical Data • 195

Relative Permittivities (εr) of Various Substances at Room Temperature (293 K) Solid

εr

Liquid

εr

Gas

εr

Amber

2.8

Acetone

21.3

Air

1.000536

Ebonite

2.7−2.9

Benzene

2.28

Argon

1.000545

Glass

5−10

Carbon tetrachloride

2.17

Carbon dioxide

1.000986

Ice (268 K)

75

Castor oil

4.5

Carbon monoxide

1.00070

Marble

8.5

Ether

4.34

Deuterium

1.000270

Mica

5.7−6.7

Ethyl alcohol

25.7

Helium

1.00007

Paraffin wax

2−2.3

Glycerine

43

Hydrogen

1.00027

Perspex

3.5

Medicinal paraffin

2.2

Neon

1.000127

Polystyrene

2.55

Nitrobenzene

35.7

Nitrogen

1.000580

P.V.C.

4.5

Pentane

1.83

Oxygen

1.00053

Shellac

3−3.7

Silicon oil

2.2

Sulfur dioxide

1.00082

Sulfur

3.6−4.3

Turpentine

2.23

Water vapor

1.00060

Teflon

2.1

Water

80.37

(393 K)

Values given in the table above refer to low frequencies, gases at l atmosphere pressure. Temperature—EMF Data for Thermocouples The table gives the emf in millivolts for “hot junction” temperature from 0° to l00°C. The “cold junction” is maintained at 0°C. Thermocouple

0°

10°

20°

30°

40°

50°

60°

70°

80°

90°

100°

Platinum− Platinum

0

0.06

0.11

0.17

0.23

0.30

0.36

0.43

0.50

0.57

0.64

Copper− Constantan

0

0.39

0.79

1.19

1.61

2.03

2.47

2.91

3.36

3.81

4.28

Iron−Constantan

0

0.52

1.05

1.58

2.12

2.66

3.20

3.75

4.30

4.85

5.40

(90%), Rhodium (10%)

Magnetic Mass Susceptibility, χm (at 293 K) The mass susceptibility is given by the expression, χm = (μr − l)/ρ; where μ, is the relative permeability and ρ the density of the specimen.

MFSDH.Part II_2PP.indd 195

2/15/2023 12:45:35 PM

196 • Mathematical Formulas and Scientific Data

χm/m3

χm/m3

χm/m3

× 10−8

× 10−8

× 10−8

Aluminum

+ 0.82

Glass

− 1.3

Oxygen

+ 133.6

Araldite

− 0.63

Helium

− 0.59

Perspex

− 0.5

Carbon (graphite)

− 4.4

Hydrogen

− 2.49

Polyethylene

+ 0.2

Copper

− 0.108

Lead chloride

− 0.40

P.V.C.

− 0.75

Copper sulfate

+ 7.7

Manganese chloride

+ 134

Sodium chloride

− 0.63

Ebonite

+0.75

Manganese dioxide

+ 48.3

Sulfur

− 0.62

Iron ammonium alum

+ 38.2

Manganese sulfate

+ 111

Sulfuric acid

− 0.50

Ferric hydroxide

+ 197

Mercury

− 0.21

Water

− 0.90

Ferrous sulfate

+ 52.2

Nitrogen

− 0.54

Maximum Relative permeability (μi max)

Coercive force (Hs/A m−2)

Energy loss per cycle (E/J m−3)

Resistivity, ρ (ohm m)

Saturation induction (BM/T)

Magnetic Properties of Some “Soft” Magnetic Materials

200 000

4.0

30

10

2.15

Mild steel

2 000

143

− 500

10

Silicon iron (1.25% Si)

6 100

67.6

220

Silicon iron (4.25% Si)

9 000

23.9

70

60

2.0

Isotropic

Silicon iron (3% Si)

40 000

12

30

47

2.0

Anisotropic, (110).100

Silicon iron (3.8% Si)

1 400 000

Alloy

Iron, pure (total impurities < 0.005%)

MFSDH.Part II_2PP.indd 196

80. Three “families” are known in which one substance decays to another which in turn continues the process until a stable material (lead) is attained. The decay process involves the emission of an electron (β-particle) or an α-particle from the nucleus. In the former case, the mass number, A, remains unchanged while Z increases by unity, while the latter emission involves a decrease in A of four and a decrease in Z of two as the α-particle is the helium nucleus. In any one “family” the mass numbers alter in steps of four only. In the Thorium family, each value of A can be described by the number (4n), the Uranium family by (4n + 2), and the Actinium family by (4n + 3). The apparently missing family (4n + 1) has been found as a result of the artificial production of heavy isotopes. It does not appear naturally because the longest half-life is short compared with the age of the Earth. The Law of Radioactive Decay All radioactive substances transform at a rate that is proportional to the number of atoms present. If there are N0 atoms present at the zero of time, then at time, t, there are Nt, where

= N t N 0 exp − (λ t)

MFSDH.Part II_2PP.indd 218

2/15/2023 12:45:38 PM

Radioactivity • 219

Here, λ, is a constant for the particular type of atom considered and is known as the transformation constant. The rate at which an atom decay is often measured in terms of the mean lifetime of the atom, τ, or the half-value period, Tt, which is often abbreviated to the half-life. The relation between these constants is:

τ = 1/λ = Tt / log e 2

MFSDH.Part II_2PP.indd 219

2/15/2023 12:45:38 PM

MFSDH.Part II_2PP.indd 220

2/15/2023 12:45:38 PM

CHAPTER

22

Properties of Inorganic Compounds In the following table, properties refer to room temperature, 293 K. Enthalpies of Formation refer to the substance in the crystalline (c), liquid (lq), or gaseous (g) states at 293 K. A negative value indicates that heat is evolved in the formation of the compound, while a positive value indicates absorption of heat. The following abbreviations are used:

MFSDH.Part II_2PP.indd 221

bl.

black

effi.

efflorescent

s.

sublimes

col.

colorless

ex.

explodes

tetr.

tetragonal

crys.

crystals

gn.

green

trig.

trigonal

cub.

cubic

hex.

hexagonal

visc.

viscous

d.

dissociates

mono.

monoclinic

w.

white

delq.

deliquescent

rh.

rhombic

yel.

yellow

2/15/2023 12:45:38 PM

Refractive index n

Enthalpy of formation, ΔHtθ kJ/mol1

Density ρ, kg/m3

Boiling point, Ta K

Melting point, TM K

Molecular weight, M g/mol−1

222 • Mathematical Formulas and Scientific Data

Al

Al2O3

101.96

2290

3250

3965

1.768

− 1670 c

Corundum. w. trig.

Ag

AgBr

187.78

705

1600 (d)

6473

2.252

− 99.5 c

pale yel. cub

AgCl

143.32

728

1820

5560

2.071

− 127 c

w. cub.

AgNO3

169.87

485

717 (d)

4352

1.744

− 123 c

col. rh.

AsBr3

314.65

306

494

3540

− 195.0 c

col. prisms

AsCl3

181.28

265

403

2163

− 335 lq

Oily liquid

As2O3

197.84

588

3738

− 1310 c

col. cub. (As4O6)

Au

AuCl3

303.33

527 (d)

3900

− 118 c

red delq.

Ba

BaCl2

208.25

1240

1820

3856

1.736

− 860.1 c

col. mono.

BaO

153.34

2196

2300

5720

1.98

− 558.1 c

col. cub.

BeCl2

79.92

678

790

1899

1.719

− 511.7 c

w. delq. needles

BeO

25.01

2800

4170

3010

1.719

− 610.9 c

w. hex.

CO

28.01

74

84

1.25

− 110.5 g

col. gas

CO2

44.01

162

195

1.98

− 393.5 g

col. gas

CaCO3

100.09

1612

d

2930

1.6809

− 1206.9

Aragonite. col. rh.

CaCl2

110.99

1045

1900

2150

1.52

− 795.0 c

w. delq. cub.

CaO

56.08

2850

3120

3300

1.837

− 635.5 c

col. cub.

CdBr2

272.22

840

1136

5192

− 314.4c

w. effi. needles

CdCl2

183.32

841

1233

4047

− 389.1 c

w. cub.

CdO

128.40

1200 (d)

8150

− 254.6 c

brown cub.

CoCl2

129.84

997∗

2940

− 326 c

blue crys. ∗ in HCl gas

CoO

74.93

2208

6450

− 239 c

brown cub.

Formula

As

Be

C Ca

Cd

Co

MFSDH.Part II_2PP.indd 222

1322

1.755

Description

2/15/2023 12:45:39 PM

CsCl

168.36

919

Cu

CuO

79.54

1599

CuSO4

223.14

Formula

1560

3988

1.534

6400

Boiling point, Ta K

249.68

3605

1.733

− 548.9 c

rose-red rh.

− 433.0 c

col. delq. cub.

− 155.2 c

bl. cub. or trig.

− 769.9 c

gn/w. rh.

Enthalpy of formation, ΔHtθ, kJ/ mol1

Cs

CuSO4

3597

Refractive index, n

d

Density ρ, kg/m3

92.95

Melting point, TM K

Co(OH)2

Molecular weight, M g/mol−1

Properties of Inorganic Compounds • 223

Description

2284

1.537

− 2278c

blue trig.

6000

2.705

− 166.7c

red cub.

− 95.1c

blue hex.

5H2O Fe

H

Hg

K

MFSDH.Part II_2PP.indd 223

Cu2O

143.08

1508

FeS

87.91

1470

Fe2O3

159.69

1838

5240

3.042

− 822.2c

red or bl. trig.

Fe3O4

231.54

1810 (d)

5180

2.42

− 1117c

bl. cub.

HBr

80.92

185

206

3.5

− 36.2 g

col. gas

HCl

36.46

158

188

1.0

− 92.3 g

col. gas

HF

20.01

190

293

0.99

− 268.6 g

col. gas

HI

127.91

222

238

5.66

+ 25.9 g

col. gas

HNO3

63.01

231

356

1503

− 173.2 lq

col. liquid

H2O

18.02

273

373

1000

− 285.9 lq

col. liquid

H2SO4

98.08

284

610

1841

− 814.0 lq

col. visc. liquid

HgCl

236.05

670 (s)

HgCl2

271.50

549

HgO

216.59

800 (d)

KCl

74.56

1049

d

575

1770 (s)

4740

1.333

7150

1.973

− 265∗ c

w. tetr. (∗Hg2Cl2)

5440

1.859

− 230 c

w. rh.

11100

2.5

− 90.4 c

yel. or red rh.

1984

1.490

− 435.9 c

col. cub

2/15/2023 12:45:39 PM

224 • Mathematical Formulas and Scientific Data

KHCO3

100.12

400 (d)

K2CO3

138.21

1164

K2O

94.20

620 (d)

Li

LiCl

42.39

887

Mg

MgBr2

184.13

970

MgCO3

84.32

620

1200

2958

1.700

MgCl2

95.22

981

1685

2320

MgF2

62.31

1539

2512

MgH3

26.33

550 (d)

MgI

278.12

1000 (d)

MgO

40.31

3100

Mg(OH)2

58.33

MgSO4

120.37

MnO

70.94

MnO3

86.94

MnO3

102.94

Mn2O3

Mn

N

d

2170

1.482

− 959.4 c

mono.

2428

1.531

− 1146.1 c

w. delq.

− 361.5 c

w. cub.

− 408.8 c

w. delq. cub.

− 517.6 c

w. delq.

− 1112 c

w. trig.

1.675

− 641.8 c

col. hex

1.378

− 1102 c

col. tetr.

2320 1600

2068

1.662

3720

w. tetr. 4430 3900

− 359 c

w. delq.

3580

1.736

− 601.8 c

col. cub.

620

2360

1.562

− 924.7 c

w. trig.

1397

2660

1.56

− 1278 c

col. rh.

5440

2.16

− 385 c

gray/gn cub.

808 (d)

5026

− 520.9 c

bl. rh.

157.87

1350 (d)

4500

− 971.1 c

Mn2O7

221.87

279

Mn3O4

228.81

1978

NH3

17.03

195

NH4Cl

53.49

613 (s)

MFSDH.Part II_2PP.indd 224

red delq.

328 (d)

2396 4856

240

red oil 2.46

0.77 1527

brown/bl. cub.

1.64

− 1386 c

brown/bl. tetr

− 46.2 g

col. gas

− 315.4 c

w. cub.

2/15/2023 12:45:39 PM

Na

Ni P

MFSDH.Part II_2PP.indd 225

Description

Enthalpy of formation, ΔHtθ, kJ/mol1

Refractive index, n

Density ρ, kg/m3

Boiling point, Ta K

Melting point, TM K

Formula

Molecular weight, M g/mol−1

Properties of Inorganic Compounds • 225

NO

30.01

110

121

1.34

+ 90.4 g

col. gas

NO2

44.01

182

185

1.98

+ 33.8 g

red/brown gas (N2O4)

N2O3

76.01

171

277 (d)

1447

+ 83.8 g

red/brown gas

NaBr

102.90

1028

1660

3203

1.641

− 359.9 c

col.cub.

NaCl

58.44

1074

1686

2165

1.544

− 411.0 c

col.cub.

NaF

41.99

1261

1968

2558

1.326

− 569 c

col.tetr.

NaH

24.00

1100 (d)

920

1.470

− 57.3 c

silver needles

NaHCO3

84.00

540 (d)

2159

1.500

− 947.7 c

w. mono. powder

NaHSO4

120.06

590

d

2435

− 1126 c

col. tricl.

Nal

149.89

924

1577

3667

− 288.0 c

col.cub.

NaOH

40.00

592

1660

2130

− 426.7 c

w. delq.

Na2CO3

105.99

1124

d

2532

− 1131 c

w. powder

Na2O

61.98

1548 (s)

− 416 c

w/gray delq.

Na2SO4

142.04

− 1384 c

mono (→hex at 510 K)

NiCl2

129.62

1274

3550

− 316 c

yel. delq.

NiO

74.71

2260

6670

2.37

− 244 c

gm/bl. cub.

PCl3

137.33

161

349

1574

1.503

− 320 c

col. fuming liquid

PCl5

208.24

435 (s)

4.65

− 463.2g

delq. tetr.

PH3

34.00

140

+ 5.2 g

col. gas

P2O3

109.95

297

− 820 lq

w. delq. mono ∗ in N2

1.774 1.535

2270 2680

185 447

∗

2135

1.477

2/15/2023 12:45:39 PM

226 • Mathematical Formulas and Scientific Data

P2O4

125.95

370

450∗

2540

P2O5

141.94

850

875

2390

PbCl2

278.10

774

1220

5850

PbCl4

349.00

258

378 (ex)

3180

PbO

223.19

1161

9530

− 219.2 c

red amor.

PbO2

239.19

560 (d)

9375

2.229

− 276.6 c

brown tetr.

PbS

239.25

1387

7500

3.912

− 100.4 c

lead gray cub.

Pb3O4

685.57

770 (d)

9100

− 718.4 c

red amor.

Rb

RbCl

120.92

988

1660

2800

− 430.5 c

cub.

S

SO2

64.06

200

263

2.93

− 296.9 g

col. gas

SO3

80.06

306

318

1927∗

− 395.2 g

col. gas (∗liquid)

SbBr3

361.48

370

550

4148

− 260 c

col. rh.

SbCl3

228.11

347

556

3140

− 382 c

col. rh. delq.

SbCl3

299.02

276

352

2336

SiC

40.10

3000

SiCl4

169.90

203

SiH4

32.12

SiO

44.09

SiO2

60.08

Pb

Sb

Si

MFSDH.Part II_2PP.indd 226

w. delq. rh. ∗in vacuo

2.217

− 3012∗ c

w. delq. amor. ∗ P4O10

− 359.2 c

w. rh. yel. liquid

1.494

1.74

− 438 lq

pale yel. liquid

3217

2.654

− 111.7 c

blue/bl. trig.

331

1483

1.412

− 640.2 lq

col. fuming liquid

88

161

1.44

+ 34 g

col. gas

1975

2150

2130

1880

2500

1.544

w. cub. − 911 c

Quartz. hex.

2/15/2023 12:45:39 PM

Ti U W Zn

− 511.3 lq

cot. fuming liquid

6446

− 286 c

bl. cub.

1.997

− 581 c

w. tetr.

1.536

− 828 c

w. rh.

1.870

− 590 c

col. cub.

− 750 lq

col. liquid

Refractive index, n

2226

Density ρ, kg/m3

Boiling point, Ta K

Description

Sr

SnCl4

260.05

240

SnO

134.69

1350 (d)

SnO2

150.69

1400

2100 (s)

6950

SrCl2

158.53

1146

1520

3052

SrO

103.62

2700

3300

4700

TiCl4

189.71

248

409

1726

TiO2

79.90

2098

UC2

262.05

2650

UO2

270.03

2800

WC

195.86

3140

WO3

231.85

1746

7160

ZnCO3

125.39

570 (d)

4398

1.818

ZnCl3

136.28

556

2910

ZnO

81.37

2100

5606

MFSDH.Part II_2PP.indd 227

387

Enthalpy of formation, ΔHtθ, kJ/mol1

Sn

Melting point, TM K

Formula

Molecular weight, M g/mol−1

Properties of Inorganic Compounds • 227

4170 4640 6300

1005

− 912 c

bl.rh.

11280

2.586

− 176 c

metallic crystals

10960

− 1130 c

bl. rh.

15630

− 38.0 c

gray. cub. powder

− 840 c

yel. rh.

− 813 c

w. trig.

1.687

− 416 c

w. delq.

2.004

− 348 c

w. hex.

2/15/2023 12:45:39 PM

MFSDH.Part II_2PP.indd 228

2/15/2023 12:45:39 PM

CHAPTER

23

Properties of Organic Compounds (at 293 K) Enthalpies of Formation refer to the substance in the crystalline (c), liquid (lq), or gaseous (g) states at 293 K. A negative value indicates the evolution of heat during the formation of the compound, while a positive value indicates absorption of heat. Enthalpy changes in combustion refer to combustion at a pressure of l atmosphere and temperature 293 K, the final products being liquid water, and gaseous carbon dioxide and nitrogen.

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Enthalpy of formation, ΔHtθ kJ/mol1

Heat of combustion, ΔHc, kJ/mol1

109

− 74.85 g

890.4 g

Ethane, C2H6

30.07

90

185

− 84.7 g

1560 g

Propane, C3H8

44.11

83

231

− 103.8 g

2220 g

n-Butane, n-C4H10

58.13

135

273

579

−146.2 lq

2877 g

2-Methyl propane iso-C4H10

58.13

114

261

557

− 134.6 g

2869 g

n-Pentane, n-C5H12

72.15

143

309

626

1.3575

− 173 lq

3509 lq

n-Hexane, n-C6H14

86.18

178

342

660

1.3751

− 198.8 lq

4163 lq

n-Heptane, n-C7H16

100.21

183

372

638

1.3878

− 224.4 lq

4853 lq

n-Octane, n-C8H18

114.23

216

399

702

1.3974

− 250 lq

5512 lq

Enthalpy of formation, ΔHtθ kJ/mol1

Heat of combustion, ΔHc, kJ/mol1

Boiling point, TB K

Molecular weight

Refractive index, n

91

Density, ρ kg/m3

16.04

Name and formula

Melting point, TM K

Methane, CH4

Refractive index, n

230 • Mathematical Formulas and Scientific Data

Alternative name

Hydrocarbons

1.3543

Density, ρ kg/m3

Boiling point, TB K

Melting point, TM K

Name and formula

Molecular weight

Isobutane

Ethene, n-C3H4

28.05

104 169

1.26

Propene, C3H6

42.08

88

226

519

Ethyne, C2H2

26.04

192 189

618

Benzene, C6H6

78.12

279 353

879

Cyclohexane, C6H12 84.16

280 354

779

Alternative Name

+ 52.3 g

1411 g

Ethylene

+ 20.4 g

2059 g

Propylene

+ 226.7 g

1300 g

Acetylene

1.5011

+ 48.7 lq

3273 lq

1.4266

− 156.2 lq 3924 lq

1.3567

Halogen derivatives of

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Properties of Organic Compounds (at 293 K) • 231

Hydrocarbons Monochloro-methane, 50.49 CH3Cl

175 249

916

− 81.9 g

687 g

Methyl chloride

Dichloromethane, CH2Cl2

84.93

178 313

1327

1.4242

− 117 lq

447 g

Methylene dichloride

Trichloromethane, CHCl3

119.38 210 335

1483

1.4459

− 132 lq

373 lq

Chloroform

Tetrachloromethane, CCl4

153.82 250 350

1594

1.4601

−139.5 lq 156 lq

Cargon tetrachloride

Bromomethane, CH3Br

94.94

180 277

1676

1.4218

− 35.6 g

770 g

Methyl bromide

Iodomethane, CH3I

141.94 207 316

2279

1.5380

− 8.4 lq

815 lq

Methyl iodine

Methanol, CH3OH

32.04

179 338

791

1.3288

− 238.7 lq 726.5 lq

Ethanol, C2H5OH

46.07

156 352

789

1.3611

− 277.7 lq 1371 lq

60.11

147 371

803

1.3850

− 300 lq

293 d

1261

1.4746

− 103.9 lq 1661 lq

Glyerol

Alcohols

n-Propanol, n-C3H7OH

Propane-1,2,3-triol, 92.11 C3H8O3

2017 lq

Acids Ethanoic acid, CH3COOH

60.05

290 391

1049

1.3716

− 488.3 lq 876 lq

Acetic acid

Propanoic acid, C2H5COOH

74.08

252 414

993

1.3869

− 509 lq

Propionic acid

88.12

269 437

958

1.3980

− 538.9 lq 2194 lq

122.13 396 522

1266

1.504

− 390 c

3227 c

Ethanal, CH3CHO

44.05

152 294

783

1.3316

− 166.4 g

1192 lq

Acetaldehyde

2-Propanone, CH3 • CO • CH3

58.08

178 329

790

1.3588

− 216.7 g

1821 g

Acetone

Methoxymethane, CH3 •O • CH3

46.07

135 250

− 185 g

1454 g

Dimethylether

Ethoxyethane, C2H3 •O • C2H5

74.12

157 308

714

1.3526

− 279.6 lq 2761 lq

Urea, CO(NH2)2

60.06

408 d

1323

1.484

− 333.2 c

634 c

d

828

− 528.6 c

981 c

n-Butanoic acid, n-C3H7COOH Benzoic acid, C6H5COOH

1527 lq

Miscellaneous

Glycine, NH2 • CH2 • 75.07 COOH

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Diethylether

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CHAPTER

24

The Fundamental Constants Certain physical constants have special importance on account of their universality or place in fundamental theory. These are given below, first in SI and then in cgs units. The figure in brackets which follows the final digit, is the estimated uncertainty in the last digit. Thus c = 2.997 924 590(8) × l08m s−1 could be written

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c = (2.997 924 590 ± 0.000 000 008) × 108 m s−1.

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234 • Mathematical Formulas and Scientific Data

General contents

Symbol

Quantity Speed of light in vacuo

2.997 924 590 (8)

108ms−1

μ0

Permeability of free space

4π

10−7 H m−1

ε0

Permittivity of free space

8.854 19 (1)

10−12 F m−1

e

Elementary charge

1.602 192 (7)

10−19 C

or 4.803 25(2)

—

6.626 20(5)

10−34 J s

1.054 592(8)

10−34 J s

h

Planck’s constant

h/2π h/e

Quantum charge ratio

4.135 708(14) or 1.379 523(5)

10−15 J s C−1 —

α

Fine structure

7.297 351(11)

10−3

1.370 360(2)

102

e2 2hε 0c

1/α G

Gravitational constant

6.673(3)

10−11 N m2 kg−2

Z0

Impedance of free space

3.767 304(1)

102 Ohm

Electron rest mass

9.109 56(5)

10−31 kg

Electron rest energy

8.187 26(6)

10−14 J

or 5.110 041(16)

10−1 Me V

1.758 803(5)

1011 C kg−1

me mec Electron

2

Proton

Multiplier and units SI

C

constant =

Neutron

Value

e/mc

Electron change mass ratio

or 5.272 759(16)

—

λc

Compton wave length of electron

2.426 310(7)

10−12 m

rc

Classical radius of electron

2.817 939(13)

10−15 m

mp

Proton rest mass

1.672 614(11)

10−27 kg

mpC2

Proton rest energy

1.503 271(15)

10−10 J

or 9.382 59(5)

102 MeV

9.578 97(11)

107 C kg−1

or 2.871 70(3)

—

e/mp

Proton charge-mass ratio

λcp

Proton Compton wavelength

1.321 441(9)

10−15 m

γ

Gyromagnetic ratio

2.675 197(8)

108 S−1 T−1

eγ

Gyromagnetic ratio (uncorrected for diamagnetism

2.675 127(8)

108 S−1 T−1

mn mnc2

Neutron rest mas Neutron rest energy

1.674 920(11) 1.505 353(15) or 9.395 53(5)

10−27 kg 10−10 J 102 MeV

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Matter in bulk

Atomic constants

The Fundamental Constants • 235

MFSDH.Part II_2PP.indd 235

R a0 μ μ0 μ/hc

Rydberg constant Bohr radius Bohr magneton Nuclear magneton Zeeman splitting constant

1.097 3731(1) 5.291 772(8) 9.274 10(6) 5.050 95(5) 4.668 60(7)

107m−1 10−11 m 10−24 J T−1 10−27 J T−1 101 m−1 T−1

N

Avogadro constant

F

Faraday

V0 R k σ

Normal volume of perfect gas Gas constant Boltzmann constant Stefan’s constant

6.022 17(4) or 6.022 17(4) 9.648 67(5) or 2.892 599(16) 2.241 36(30) 8.314 3(3) 1.380 62(6) 5.669 9(9)

1023 mol−1 1026 kg mol−1 104 C mol−1 — 10−2m3/mol 100 J/K mol 10−23 J/K 10−8 W/m2 K4

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