Elements of Mathematical Analysis: An Informal Introduction for Physics and Engineering Students [1 ed.] 3031458532, 9783031458538, 9783031458545

Table of contents :
Preface
Contents
1 Functions
1.1 Real Numbers
1.2 Functions
1.3 Domain of Definition of a Function
1.4 Implicit and Multiple-Valued Functions
1.5 Exponential and Logarithmic Functions
1.6 Linear Function
1.7 Quadratic Function
1.8 Even and Odd Functions
1.9 Periodic Functions
1.10 Inverse Function
1.11 Monotonicity of a Function
References
2 Derivative and Differential
2.1 Definition
2.2 Differentiation Rules
2.3 Derivatives of Trigonometric Functions
2.4 Table of Derivatives of Elementary Functions
2.5 Derivatives of Composite Functions
2.6 Derivatives of Functions of the Form y=[f(x)](x)
2.7 Differential of a Function
2.8 Differential Operators
2.9 Derivative of a Composite Function by Using the Differential
2.10 Geometrical Significance of the Derivative and the Differential
2.11 Higher-Order Derivatives
2.12 Derivatives of Implicit Functions
References
3 Some Applications of Derivatives
3.1 Tangent and Normal Lines on Curves
3.2 Angle of Intersection of Two Curves
3.3 Maximum and Minimum Values of a Function
3.4 Indeterminate Forms and L’Hospital’s Rule
References
4 Indefinite Integral
4.1 Antiderivatives of a Function
4.2 The Indefinite Integral
4.3 Basic Integration Rules
4.4 Integration by Substitution (Change of Variable)
4.5 Integration by Parts (Partial Integration)
4.6 Integration of Rational Functions
5 Definite Integral
5.1 Definition and Properties
5.2 Integration by Substitution
5.3 Integration of Even, Odd and Periodic Functions
5.4 Integrals with Variable Limits
5.5 Improper Integrals: Infinite Limits
5.6 Improper Integrals: Unbounded Integrand
5.7 The Definite Integral as a Plane Area
Reference
6 Series
6.1 Series of Constants
6.2 Positive Series
6.3 Absolutely Convergent Series
6.4 Functional Series
6.5 Expansion of Functions into Power Series
Reference
7 An Elementary Introduction to Differential Equations
7.1 Two Basic Theorems
7.2 First-Order Differential Equations
7.3 Some Special Cases
7.4 Examples
8 Introduction to Differentiation in Higher Dimensions
8.1 Partial Derivatives and Total Differential
8.2 Exact Differential Equations
8.3 Integrating Factor
8.4 Line Integrals on the Plane
References
9 Complex Numbers
9.1 The Notion of a Complex Number
9.2 Polar Form of a Complex Number
9.3 Exponential Form of a Complex Number
9.4 Powers and Roots of Complex Numbers
Reference
10 Introduction to Complex Analysis
10.1 Analytic Functions and the Cauchy-Riemann Relations
10.2 Integrals of Complex Functions
10.3 The Cauchy-Goursat Theorem
10.4 Indefinite Integral of an Analytic Function
References
Appendix
Trigonometric Formulas
Answers to Selected Exercises
Selected Bibliography
Index