Maple в инженерных расчетах 5-8265-0211-8

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Ɇɢɧɢɫɬɟɪɫɬɜɨ ɨɛɪɚɡɨɜɚɧɢɹ Ɋɨɫɫɢɣɫɤɨɣ Ɏɟɞɟɪɚɰɢɢ Ɍɚɦɛɨɜɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ

Ⱥ. Ⱥ. Ʉɨɩɬɟɜ, Ⱥ. Ⱥ. ɉɚɫɶɤɨ, Ⱥ. Ⱥ. Ȼɚɪɚɧɨɜ Maple ɜ ɢɧɠɟɧɟɪɧɵɯ ɪɚɫɱɟɬɚɯ

ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ

ɍɬɜɟɪɠɞɟɧɨ ɍɱɟɧɵɦ ɫɨɜɟɬɨɦ ɭɧɢɜɟɪɫɢɬɟɬɚ ɜ ɤɚɱɟɫɬɜɟ ɭɱɟɛɧɨɝɨ ɩɨɫɨɛɢɹ

ɍȾɄ 519.67 ȻȻɄ ɡ973-018.2 Ʉ55

Ɍɚɦɛɨɜ ɂɡɞɚɬɟɥɶɫɬɜɨ ɌȽɌɍ 2003

Ɋɟɰɟɧɡɟɧɬɵ: Ʉɚɧɞɢɞɚɬ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ . . Ⱦɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ . И.

Ʉ55

Ʉɨɩɬɟɜ Ⱥ. Ⱥ., ɉɚɫɶɤɨ Ⱥ. Ⱥ., Ȼɚɪɚɧɨɜ Ⱥ. Ⱥ. Maple ɜ ɢɧɠɟɧɟɪɧɵɯ ɪɚɫɱɟɬɚɯ: ɍɱɟɛ. ɩɨɫɨɛɢɟ. Ɍɚɦɛɨɜ: ɂɡɞ-ɜɨ Ɍɚɦɛ. ɝɨɫ. ɬɟɯɧ. ɭɧ-ɬɚ, 2003. 80 ɫ. ISBN 5-8265-0211-8 ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɩɨɫɜɹɳɟɧɨ ɫɢɫɬɟɦɟ ɫɢɦɜɨɥɶɧɵɯ ɜɵɱɢɫɥɟɧɢɣ Maple. ɉɪɟɞɫɬɚɜɥɟɧɵ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɹɡɵɤɚ Maple ɢ ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɦɵɟ ɮɭɧɤɰɢɢ. ɉɨɞɪɨɛɧɨ ɪɚɫɫɦɨɬɪɟɧɵ ɜɨɩɪɨɫɵ ɝɪɚɮɢɱɟɫɤɨɝɨ ɨɬɨɛɪɚɠɟɧɢɹ ɩɨɥɭɱɟɧɧɵɯ ɫ ɩɨɦɨɳɶɸ Maple ɪɟɲɟɧɢɣ. Ⱦɚɧɨ ɜɜɟɞɟɧɢɟ ɜ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ ɧɚ ɹɡɵɤɟ Maple. Ɉɫɨɛɨɟ ɜɧɢɦɚɧɢɟ ɭɞɟɥɟɧɨ ɩɪɢɦɟɧɟɧɢɸ Maple ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɪɚɫɱɟɬɚ ɦɚɲɢɧ ɢ ɚɩɩɚɪɚɬɨɜ ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɢɡɜɨɞɫɬɜ. ɉɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɢ ɚɫɩɢɪɚɧɬɨɜ, ɢɫɩɨɥɶɡɭɸɳɢɯ ɩɟɪɫɨɧɚɥɶɧɵɣ ɤɨɦɩɶɸɬɟɪ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɢ ɩɪɢɤɥɚɞɧɨɝɨ ɯɚɪɚɤɬɟɪɚ.

ɍȾɄ 519.67 ȻȻɄ ɡ973-018.2

ISBN 5-8265-0211-8

 Ɍɚɦɛɨɜɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ (ɌȽɌɍ), 2003  Ʉɨɩɬɟɜ Ⱥ. Ⱥ., ɉɚɫɶɤɨ Ⱥ. Ⱥ., Ȼɚɪɚɧɨɜ Ⱥ. Ⱥ., 2003

ɍɱɟɛɧɨɟ ɢɡɞɚɧɢɟ ɄɈɉɌȿȼ Ⱥɧɞɪɟɣ Ⱥɥɟɤɫɟɟɜɢɱ, ɉȺɋɖɄɈ Ⱥɥɟɤɫɚɧɞɪ Ⱥɧɚɬɨɥɶɟɜɢɱ, ȻȺɊȺɇɈȼ Ⱥɧɞɪɟɣ Ⱥɥɟɤɫɟɟɜɢɱ Maple ɜ ɢɧɠɟɧɟɪɧɵɯ ɪɚɫɱɟɬɚɯ ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ Ɋɟɞɚɤɬɨɪ ȼ. ɇ. Ɇɢɬɪɨɮɚɧɨɜɚ Ʉɨɦɩɶɸɬɟɪɧɨɟ ɦɚɤɟɬɢɪɨɜɚɧɢɟ ȿ. ȼ. Ʉɨɪɚɛɥɟɜɨɣ ɉɨɞɩɢɫɚɧɨ ɤ ɩɟɱɚɬɢ 24.01.2003 Ɏɨɪɦɚɬ 60 × 84/16. Ȼɭɦɚɝɚ ɨɮɫɟɬɧɚɹ. ɉɟɱɚɬɶ ɨɮɫɟɬɧɚɹ Ɉɛɴɟɦ: 4,65 ɭɫɥ. ɩɟɱ. ɥ.; 4,56 ɭɱ. ɢɡɞ. ɥ. Ɍɢɪɚɠ 200 ɷɤɡ. ɋ. 40 ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ɌȽɌɍ 392000, ɌȺɆȻɈȼ, ɋɈȼȿɌɋɄȺə, 106, Ʉ. 14

ȼȼȿȾȿɇɂȿ ȼɫɥɟɞɫɬɜɢɟ ɧɟɩɪɟɪɵɜɧɨ ɜɨɡɪɚɫɬɚɸɳɢɯ ɬɪɟɛɨɜɚɧɢɣ ɤ ɤɚɱɟɫɬɜɭ, ɷɤɨɧɨɦɢɱɧɨɫɬɢ, ɧɚɞɟɠɧɨɫɬɢ, ɛɵɫɬɪɨɞɟɣɫɬɜɢɸ, ɫɧɢɠɟɧɢɸ ɦɚɬɟɪɢɚɥɨɟɦɤɨɫɬɢ ɨɛɨɪɭɞɨɜɚɧɢɹ, ɫɨɜɪɟɦɟɧɧɵɟ ɢɧɠɟɧɟɪɧɵɟ ɪɚɫɱɟɬɵ ɜɫɟ ɛɨɥɟɟ ɭɫɥɨɠɧɹɸɬɫɹ. Ɉɧɢ ɞɨɥɠɧɵ ɭɱɢɬɵɜɚɬɶ ɜɫɟ ɮɚɤɬɨɪɵ, ɨɤɚɡɵɜɚɸɳɢɟ ɫɨɜɨɤɭɩɧɨɟ ɜɥɢɹɧɢɟ ɧɚ ɪɚɛɨɬɭ ɫɨɜɪɟɦɟɧɧɵɯ ɦɚɲɢɧ ɢ ɚɩɩɚɪɚɬɨɜ: ɪɟɠɢɦɵ ɷɤɫɩɥɭɚɬɚɰɢɢ, ɫɜɨɣɫɬɜɚ ɦɚɬɟɪɢɚɥɨɜ, ɭɫɥɨɜɢɹ ɧɚɝɪɭɠɟɧɢɹ ɢ ɬ.ɩ. ɉɪɢ ɷɬɨɦ ɧɟɨɛɯɨɞɢɦɨ ɫɨɛɥɸɞɟɧɢɟ ɨɫɧɨɜɧɵɯ ɩɨɤɚɡɚɬɟɥɟɣ ɧɚɞɟɠɧɨɫɬɢ, ɩɪɨɱɧɨɫɬɢ ɢ ɞɨɥɝɨɜɟɱɧɨɫɬɢ. ɉɨɷɬɨɦɭ ɜ ɪɚɫɱɟɬɚɯ ɜɫɟ ɲɢɪɟ ɩɪɢɦɟɧɹɸɬɫɹ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɪɟɡɭɥɶɬɚɬɵ, ɫɬɪɨɝɨ ɦɚɬɟɦɚɬɢɱɟɫɤɢ ɨɩɢɫɵɜɚɸɳɢɟ ɡɚɞɚɱɭ, ɢ ɜɫɟ ɦɟɧɶɲɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɨɪɢɟɧɬɢɪɨɜɨɱɧɵɟ, ɩɪɢɛɥɢɠɟɧɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ. Ɋɟɲɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɡɚɞɚɱ, ɜɨɡɧɢɤɚɸɳɢɯ ɩɟɪɟɞ ɢɧɠɟɧɟɪɨɦ, ɧɟɜɨɡɦɨɠɧɨ ɛɟɡ ɭɦɟɥɨɝɨ ɩɪɢɦɟɧɟɧɢɹ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɢ ɢ ɟɟ ɩɪɨɝɪɚɦɦɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ. ɋɭɳɟɫɬɜɟɧɧɨ ɨɛɥɟɝɱɢɬɶ ɪɚɫɱɟɬɵ ɜ ɢɧɠɟɧɟɪɧɵɯ ɡɚɞɚɱɚɯ, ɩɨɜɵɫɢɬɶ ɢɯ ɤɚɱɟɫɬɜɨ ɢ ɛɵɫɬɪɨɬɭ ɦɨɠɟɬ ɭɧɢɜɟɪɫɚɥɶɧɵɣ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɩɚɤɟɬ Maple ɤɨɦɩɚɧɢɢ Waterloo Maple, ɤɨɬɨɪɵɣ ɩɨ ɩɪɚɜɭ ɫɱɢɬɚɟɬɫɹ ɨɞɧɨɣ ɢɡ ɥɭɱɲɢɯ ɩɪɨɝɪɚɦɦ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ, ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ, ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ. ɐɟɧɧɨɫɬɶ ɱɢɫɬɨ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɤɨɧɫɬɪɭɤɰɢɣ, ɫ ɤɨɬɨɪɵɦɢ ɦɚɧɢɩɭɥɢɪɭɟɬ Maple, ɩɨɡɜɨɥɹɟɬ ɢɡɛɟɠɚɬɶ ɩɨɝɪɟɲɧɨɫɬɟɣ ɧɟɢɡɛɟɠɧɵɯ ɩɪɢ ɱɢɫɥɟɧɧɵɯ ɪɟɲɟɧɢɹɯ, ɩɨɥɭɱɢɬɶ ɭɞɨɛɧɵɟ ɪɚɫɱɟɬɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ, ɭɜɟɥɢɱɢɬɶ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɱɢɫɥɟɧɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ. Ɉɞɧɚɤɨ ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɛɟɡ ɩɪɢɜɥɟɱɟɧɢɹ ɱɢɫɥɟɧɧɵɯ ɦɟɬɨɞɨɜ ɧɟɜɨɡɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɪɟɲɟɧɢɟ ɦɧɨɝɢɯ ɫɨɜɪɟɦɟɧɧɵɯ ɢɧɠɟɧɟɪɧɵɯ ɡɚɞɚɱ. ȼ ɞɚɧɧɨɣ ɫɢɬɭɚɰɢɢ ɫ ɭɫɩɟɯɨɦ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɭɧɢɜɟɪɫɚɥɶɧɵɟ ɱɢɫɥɟɧɧɵɟ ɚɥɝɨɪɢɬɦɵ, ɜɯɨɞɹɳɢɟ ɜ ɫɢɫɬɟɦɭ Maple ɫ ɜɨɡɦɨɠɧɨɫɬɶɸ ɪɟɝɭɥɢɪɨɜɚɬɶ ɬɨɱɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ ɞɨ ɩɨɪɹɞɤɨɜ ɧɟ ɞɨɫɬɭɩɧɵɯ ɦɚɤɫɢɦɚɥɶɧɵɦ ɚɩɩɚɪɚɬɧɵɦ ɡɧɚɱɟɧɢɹɦ ɫɨɜɪɟɦɟɧɧɵɯ ɤɨɦɩɶɸɬɟɪɨɜ. ȼ ɞɚɧɧɨɦ ɩɨɫɨɛɢɢ ɨɩɢɫɚɧɵ ɧɟɤɨɬɨɪɵɟ ɜɨɡɦɨɠɧɨɫɬɢ ɩɚɤɟɬɚ Maple 8, ɧɨ ɦɧɨɝɢɟ ɢɡ ɷɬɢɯ ɜɨɡɦɨɠɧɨɫɬɟɣ ɢɦɟɸɬɫɹ ɢ ɜ ɛɨɥɟɟ ɪɚɧɧɢɯ ɜɟɪɫɢɹɯ, ɩɪɢ ɷɬɨɦ ɩɪɨɛɧɭɸ ɜɟɪɫɢɸ ɩɚɤɟɬɚ Maple 8 ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɧɚ ɫɟɪɜɟɪɟ ɤɨɦɩɚɧɢɢ Waterloo Maple http://www.maplesoft.com.

1 ɂɇɌȿɊɎȿɃɋ MAPLE Ʉɚɤ ɥɸɛɨɟ Windows – ɩɪɢɥɨɠɟɧɢɟ Maple ɢɦɟɟɬ ɨɤɨɧɧɵɣ ɢɧɬɟɪɮɟɣɫ, ɫɬɪɨɤɢ ɤɨɦɚɧɞ ɤɨɬɨɪɨɝɨ ɛɭɞɭɬ ɧɟɫɤɨɥɶɤɨ ɨɬɥɢɱɚɬɶɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɥɟɞɭɸɳɢɯ ɞɟɣɫɬɜɢɣ: • ɪɟɞɚɤɬɢɪɨɜɚɧɢɟ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ – ɫɬɚɧɞɚɪɬɧɵɣ ɢɧɬɟɪɮɟɣɫ; • ɪɚɛɨɱɟɝɨ ɥɢɫɬɚ; • ɩɪɨɫɦɨɬɪ ɫɩɪɚɜɤɢ – ɢɧɬɟɪɮɟɣɫ ɫɩɪɚɜɨɱɧɨɣ ɫɢɫɬɟɦɵ; • ɞɜɭɯɦɟɪɧɵɟ ɩɨɫɬɪɨɟɧɢɹ – ɢɧɬɟɪɮɟɣɫ ɝɪɚɮɢɱɟɫɤɨɣ ɞɜɭɯɦɟɪɧɨɣ ɫɢɫɬɟɦɵ; • ɬɪɟɯɦɟɪɧɵɟ ɩɨɫɬɪɨɟɧɢɹ – ɢɧɬɟɪɮɟɣɫ ɬɪɟɯɦɟɪɧɨɣ ɝɪɚɮɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ.

1.1 ɂɇɌȿɊɎȿɃɋ ɊȺȻɈɑȿȽɈ ȾɈɄɍɆȿɇɌȺ ɂɧɬɟɪɮɟɣɫ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ ɛɭɞɟɬ ɩɨɤɚɡɚɧ ɧɚ ɷɤɪɚɧɟ, ɟɫɥɢ ɩɨɥɶɡɨɜɚɬɟɥɶ ɪɚɛɨɬɚɟɬ ɜ ɪɚɛɨɱɟɦ ɞɨɤɭɦɟɧɬɟ ɢ ɬɚɦ ɠɟ ɪɚɫɩɨɥɨɠɟɧ ɤɭɪɫɨɪ ɜɜɨɞɚ. ȼɢɞ ɞɚɧɧɨɝɨ ɢɧɬɟɪɮɟɣɫɚ ɛɵɥ ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫɭɧɤɟ. Ʉɚɤ ɜɢɞɧɨ ɢɡ ɪɢɫɭɧɤɚ, ɜ ɫɬɪɨɤɟ ɤɨɦɚɧɞ ɫɨɞɟɪɠɢɬɫɹ ɜɨɫɟɦɶ ɩɭɧɤɬɨɜ ɦɟɧɸ: • File – ɤɨɦɚɧɞɵ ɞɥɹ ɪɚɛɨɬɵ ɫ ɮɚɣɥɚɦɢ ɫɟɫɫɢɢ Maple; • Edit – ɤɨɦɚɧɞɵ ɞɥɹ ɪɚɛɨɬɵ ɫ ɨɬɞɟɥɶɧɵɦ ɪɟɝɢɨɧɨɦ ɢɥɢ ɟɝɨ ɱɚɫɬɶɸ; • View – ɢɡɦɟɧɟɧɢɟ ɜɢɞɚ ɫɨɞɟɪɠɢɦɨɝɨ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ ɢ ɩɚɧɟɥɟɣ ɭɩɪɚɜɥɟɧɢɹ; • Insert – ɜɫɬɚɜɤɚ ɪɚɡɥɢɱɧɵɯ ɨɛɴɟɤɬɨɜ ɢ ɬɟɤɫɬɚ ɜ ɨɬɤɪɵɬɵɣ ɞɨɤɭɦɟɧɬ; • Format – ɤɨɦɚɧɞɵ ɮɨɪɦɚɬɢɪɨɜɚɧɢɹ ɬɟɤɫɬɚ; • Spreadsheet – ɤɨɦɚɧɞɵ ɞɥɹ ɪɚɛɨɬɵ ɫ ɷɥɟɤɬɪɨɧɧɵɦɢ ɬɚɛɥɢɰɚɦɢ; • Window – ɤɨɦɚɧɞɵ ɞɥɹ ɡɚɤɪɵɬɢɹ, ɭɩɨɪɹɞɨɱɢɜɚɧɢɹ ɢ ɜɵɜɨɞɚ ɫɩɢɫɤɚ ɨɬɤɪɵɬɵɯ ɪɚɛɨɱɢɯ ɞɨɤɭɦɟɧɬɨɜ; • Help – ɤɨɦɚɧɞɵ ɞɥɹ ɪɚɛɨɬɵ ɫɨ ɫɩɪɚɜɨɱɧɨɣ ɫɢɫɬɟɦɨɣ ɢ ɢɡɦɟɧɟɧɢɹ ɛɚɡɵ ɞɚɧɧɵɯ ɩɨɦɨɳɢ. ɇɚɢɛɨɥɟɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɦɵɟ ɤɨɦɚɧɞɵ ɭɩɪɚɜɥɟɧɢɹ ɪɚɛɨɱɢɦ ɞɨɤɭɦɟɧɬɨɦ ɜɵɧɟɫɟɧɵ ɜ ɩɢɤɬɨɝɪɚɮɢɱɟɫɤɨɟ ɦɟɧɸ, ɨɩɢɫɚɧɢɟ ɤɨɬɨɪɵɯ ɩɪɢɜɟɞɟɧɨ ɧɢɠɟ.

– ɨɬɤɪɵɬɢɟ ɧɨɜɨɝɨ ɞɨɤɭɦɟɧɬɚ;

– ɨɬɤɪɵɬɢɟ ɫɭɳɟɫɬɜɭɸɳɟɝɨ ɞɨɤɭɦɟɧɬɚ; – ɨɬɤɪɵɬɢɟ URL; – ɫɨɯɪɚɧɹɟɬ ɬɟɤɭɳɢɣ ɞɨɤɭɦɟɧɬ ɜ ɮɚɣɥɟ ɧɚ ɞɢɫɤɟ; – ɩɟɱɚɬɶ ɚɤɬɢɜɧɨɝɨ ɞɨɤɭɦɟɧɬɚ; – ɜɵɪɟɡɚɬɶ ɜɵɞɟɥɟɧɧɭɸ ɱɚɫɬɶ ɞɨɤɭɦɟɧɬɚ ɢ ɨɬɩɪɚɜɢɬɶ ɟɝɨ ɜ ɛɭɮɟɪ ɨɛɦɟɧɚ; – ɤɨɩɢɪɨɜɚɬɶ ɜɵɞɟɥɟɧɧɭɸ ɱɚɫɬɶ ɞɨɤɭɦɟɧɬɚ ɜ ɛɭɮɟɪ ɨɛɦɟɧɚ; – ɜɫɬɚɜɢɬɶ ɫɨɞɟɪɠɢɦɨɟ ɛɭɮɟɪɚ ɨɛɦɟɧɚ ɜ ɚɤɬɢɜɧɵɣ ɞɨɤɭɦɟɧɬ; – ɨɬɦɟɧɢɬɶ ɩɨɫɥɟɞɧɸɸ ɨɩɟɪɚɰɢɸ ɪɟɞɚɤɬɢɪɨɜɚɧɢɹ; – ɜɟɪɧɭɬɶ ɨɬɦɟɧɟɧɧɭɸ ɨɩɟɪɚɰɢɸ; – ɜɫɬɚɜɤɚ ɤɨɦɚɧɞɵ Maple ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜ ɬɭ ɱɚɫɬɶ ɞɨɤɭɦɟɧɬɚ, ɝɞɟ ɧɚɯɨɞɢɬɫɹ ɤɭɪɫɨɪ; – ɜɫɬɚɜɤɚ ɢ ɮɨɪɦɚɬɢɪɨɜɚɧɢɟ ɬɟɫɬɨɜɨɝɨ ɤɨɦɦɟɧɬɚɪɢɹ; – ɜɫɬɚɜɤɚ ɝɪɭɩɩɵ ɜɵɩɨɥɧɹɟɦɵɯ ɤɨɦɚɧɞ; – ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɜɵɞɟɥɟɧɢɟ ɜ ɩɨɞɫɟɤɰɢɸ; – ɞɟɣɫɬɜɢɟ, ɨɛɪɚɬɧɨɟ ɩɪɟɞɵɞɭɳɟɦɭ; – ɲɚɝ ɧɚɡɚɞ ɩɪɢ ɪɚɛɨɬɟ ɫ ɝɢɩɟɪɫɫɵɥɤɚɦɢ; – ɲɚɝ ɜɩɟɪɟɞ ɩɪɢ ɪɚɛɨɬɟ ɫ ɝɢɩɟɪɫɫɵɥɤɚɦɢ; – ɩɪɟɪɜɚɬɶ ɜɵɱɢɫɥɟɧɢɹ; ɦɚɫɲɬɚɛ ɨɬɨɛɪɚɠɟɧɢɹ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ (100 %, 150 % ɢ 200 % ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ); – ɩɨɤɚɡɚɬɶ /ɫɤɪɵɬɶ ɫɩɟɰɢɚɥɶɧɵɟ ɫɢɦɜɨɥɵ; – ɭɜɟɥɢɱɢɬɶ ɪɚɡɦɟɪ ɚɤɬɢɜɧɨɝɨ ɨɤɧɚ; – ɨɱɢɫɬɢɬɶ ɜɧɭɬɪɟɧɧɸɸ ɩɚɦɹɬɶ (restart); – ɩɟɪɟɤɥɸɱɚɟɬ ɨɬɨɛɪɚɠɟɧɢɟ ɫɬɪɨɤɢ ɤɨɦɚɧɞ ɢɡ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɜ Maple-ɧɨɬɚɰɢɸ ɢ ɨɛɪɚɬɧɨ; – ɜɵɩɨɥɧɹɬɶ/ɧɟ ɜɵɩɨɥɧɹɬɶ ɜɵɪɚɠɟɧɢɟ; – ɚɜɬɨɦɚɬɢɱɟɫɤɚɹ ɤɨɪɪɟɤɰɢɹ ɫɢɧɬɚɤɫɢɫɚ ɜɵɪɚɠɟɧɢɹ; – ɜɵɩɨɥɧɢɬɶ ɬɟɤɭɳɟɟ ɜɵɪɚɠɟɧɢɟ; – ɜɵɩɨɥɧɢɬɶ ɪɚɛɨɱɢɣ ɞɨɤɭɦɟɧɬ. 1.2 ɂɇɌȿɊɎȿɃɋ ɋɉɊȺȼɈɑɇɈɃ ɋɂɋɌȿɆɕ Maple ɫɧɚɛɠɟɧ ɦɨɳɧɨɣ ɞɢɚɥɨɝɨɜɨɣ ɫɢɫɬɟɦɨɣ ɤɨɧɬɟɤɫɬɧɨɣ ɧɚɫɬɪɚɢɜɚɟɦɨɣ ɩɨɦɨɳɢ. ɉɪɢ ɪɚɛɨɬɟ ɫɨ ɫɩɪɚɜɨɱɧɨɣ ɫɢɫɬɟɦɨɣ ɦɨɠɧɨ ɭɜɢɞɟɬɶ ɬɚɤɭɸ ɤɚɪɬɢɧɭ.

ɋɩɪɚɜɨɱɧɭɸ ɢɧɮɨɪɦɚɰɢɸ ɦɨɠɧɨ ɢɫɤɚɬɶ ɩɨ ɨɩɪɟɞɟɥɟɧɧɨɣ ɬɟɦɟ ɢɥɢ ɤɨɦɚɧɞɟ, ɚ ɬɚɤɠɟ ɩɨ ɲɢɪɨɤɨɦɭ ɞɢɚɩɚɡɨɧɭ ɞɨɫɬɭɩɧɵɯ ɤɨɦɚɧɞ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɫɩɪɚɜɤɢ ɩɨ ɤɨɧɤɪɟɬɧɨɣ ɤɨɦɚɧɞɟ ɫɥɟɞɭɟɬ ɜ ɪɚɛɨɱɟɦ ɞɨɤɭɦɟɧɬɟ ɜɜɟɫɬɢ "?" ɢ ɢɦɹ ɤɨɦɚɧɞɵ, ɥɢɛɨ ɭɫɬɚɧɨɜɢɬɶ ɤɭɪɫɨɪ ɧɚ ɢɧɬɟɪɟɫɭɸɳɭɸ ɤɨɦɚɧɞɭ ɢ ɧɚɠɚɬɶ ɤɥɚɜɢɲɭ F1. 1.3 ɂɇɌȿɊɎȿɃɋ ȾȼɍɏɆȿɊɇɈɃ ȽɊȺɎɂɑȿɋɄɈɃ ɋɂɋɌȿɆɕ ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɝɪɚɮɢɱɟɫɤɢɯ ɩɨɫɬɪɨɟɧɢɣ ɧɚ ɩɥɨɫɤɨɫɬɢ ɩɟɪɟɞ ɩɨɥɶɡɨɜɚɬɟɥɟɦ ɩɨɹɜɥɹɟɬɫɹ ɢɧɬɟɪɮɟɣɫ ɞɜɭɯɦɟɪɧɨɣ ɝɪɚɮɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ.

ɉɪɢ ɷɬɨɦ ɤɨɦɚɧɞɧɚɹ ɫɬɪɨɤɚ ɫɨɞɟɪɠɢɬ ɫɥɟɞɭɸɳɢɟ ɩɭɧɤɬɵ ɦɟɧɸ: • File –ɫɬɚɧɞɚɪɬɧɨɟ ɦɟɧɸ ɢɧɬɟɪɮɟɣɫɚ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ;

• Edit – ɫɬɚɧɞɚɪɬɧɨɟ ɦɟɧɸ ɢɧɬɟɪɮɟɣɫɚ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ; • View – ɫɬɚɧɞɚɪɬɧɨɟ ɦɟɧɸ ɢɧɬɟɪɮɟɣɫɚ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ; • Format – ɤɨɦɚɧɞɵ ɮɨɪɦɚɬɢɪɨɜɚɧɢɹ; • Style – ɨɩɪɟɞɟɥɹɟɬ ɫɬɢɥɶ ɩɨɫɬɪɨɟɧɢɹ; • Legend – ɪɟɞɚɤɬɢɪɨɜɚɧɢɟ ɢ ɩɨɤɚɡ ɥɟɝɟɧɞɵ; • Axes – ɭɩɪɚɜɥɹɟɬ ɫɬɢɥɟɦ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ; • Projection – ɨɩɪɟɞɟɥɹɟɬ ɦɚɫɲɬɚɛ ɢɡɨɛɪɚɠɟɧɢɹ; • Animation – ɚɧɢɦɚɰɢɹ ɝɪɚɮɢɤɨɜ; • Export – ɫɨɯɪɚɧɟɧɢɟ ɝɪɚɮɢɤɢ ɜ ɮɚɣɥɵ ɪɚɡɥɢɱɧɵɯ ɮɨɪɦɚɬɨɜ; • Window – ɫɬɚɧɞɚɪɬɧɨɟ ɦɟɧɸ ɢɧɬɟɪɮɟɣɫɚ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ; • Help – ɫɬɚɧɞɚɪɬɧɨɟ ɦɟɧɸ ɢɧɬɟɪɮɟɣɫɚ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ. ɇɚɢɛɨɥɟɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɦɵɟ ɤɨɦɚɧɞɵ ɭɩɪɚɜɥɟɧɢɹ ɞɜɭɯɦɟɪɧɨɣ ɝɪɚɮɢɱɟɫɤɨɣ ɫɢɫɬɟɦɨɣ ɜɵɧɟɫɟɧɵ ɜ ɩɢɤɬɨɝɪɚɮɢɱɟɫɤɨɟ ɦɟɧɸ, ɨɩɢɫɚɧɢɟ ɤɨɬɨɪɵɯ ɩɪɢɜɟɞɟɧɨ ɧɢɠɟ. – ɤɨɨɪɞɢɧɚɬɵ ɤɭɪɫɨɪɚ; – ɫɬɢɥɢ ɩɨɫɬɪɨɟɧɢɹ; – ɫɬɢɥɢ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ; – ɦɚɫɲɬɚɛɢɪɨɜɚɧɢɟ ɢɡɨɛɪɚɠɟɧɢɹ. ɍɩɪɚɜɥɹɬɶ ɞɜɭɯɦɟɪɧɨɣ ɝɪɚɮɢɱɟɫɤɨɣ ɫɢɫɬɟɦɨɣ ɦɨɠɧɨ ɢɫɩɨɥɶɡɭɹ ɤɨɧɬɟɤɫɬɧɨɟ ɦɟɧɸ. Ɉɧɨ ɜɵɡɵɜɚɟɬɫɹ ɧɚɠɚɬɢɟɦ ɩɪɚɜɨɣ ɤɥɚɜɢɲɢ ɦɵɲɢ ɧɚ ɩɨɥɟ ɢɡɨɛɪɚɠɟɧɢɹ.

1.4 ɂɇɌȿɊɎȿɃɋ ɌɊȿɏɆȿɊɇɈɃ ȽɊȺɎɂɑȿɋɄɈɃ ɋɂɋɌȿɆɕ ɉɪɢ ɥɸɛɨɦ ɜɢɞɟ ɬɪɟɯɦɟɪɧɨɝɨ ɩɨɫɬɪɨɟɧɢɹ ɩɟɪɟɞ ɩɨɥɶɡɨɜɚɬɟɥɟɦ ɜɨɡɧɢɤɚɟɬ ɢɧɬɟɪɮɟɣɫ ɬɪɟɯɦɟɪɧɨɣ ɝɪɚɮɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ.

ɉɟɪɟɱɢɫɥɢɦ ɩɭɧɤɬɵ ɦɟɧɸ, ɧɟ ɨɩɢɫɚɧɧɵɟ ɜ ɫɬɚɧɞɚɪɬɧɨɦ ɢɧɬɟɪɮɟɣɫɟ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ: • Style – ɨɩɪɟɞɟɥɹɟɬ ɫɬɢɥɶ ɩɨɫɬɪɨɟɧɢɹ; • Color – ɨɩɪɟɞɟɥɹɟɬ ɰɜɟɬ ɩɨɫɬɪɨɟɧɢɹ; • Axes – ɭɩɪɚɜɥɹɟɬ ɫɬɢɥɟɦ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ; • Projection – ɨɩɪɟɞɟɥɹɟɬ ɦɚɫɲɬɚɛ ɢɡɨɛɪɚɠɟɧɢɹ; • Animation – ɚɧɢɦɚɰɢɹ ɝɪɚɮɢɤɨɜ; • Export – ɫɨɯɪɚɧɟɧɢɟ ɝɪɚɮɢɤɢ ɜ ɮɚɣɥɵ ɪɚɡɥɢɱɧɵɯ ɮɨɪɦɚɬɨɜ. ȼ ɩɢɤɬɨɝɪɚɮɢɱɟɫɤɨɟ ɦɟɧɸ ɜɵɧɟɫɟɧɵ ɫɥɟɞɭɸɳɢɟ ɤɨɦɚɧɞɵ:

– ɧɚɩɪɚɜɥɟɧɢɟ ɜɡɝɥɹɞɚ ɧɚ ɨɛɴɟɤɬ; – ɫɬɢɥɢ ɩɨɫɬɪɨɟɧɢɹ; – ɫɬɢɥɢ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ; – ɦɚɫɲɬɚɛɢɪɨɜɚɧɢɟ ɢɡɨɛɪɚɠɟɧɢɹ. 2 ɋɂɇɌȺɄɋɂɋ əɁɕɄȺ MAPLE 2.1 ɉɊɈɋɌɕȿ ȼɕɑɂɋɅȿɇɂə ȼ Maple ɜɵɩɨɥɧɹɟɦɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɜɵɪɚɠɟɧɢɹ ɜɜɨɞɹɬɫɹ ɜɫɟɝɞɚ ɩɨɫɥɟ ɫɢɦɜɨɥɚ >, ɚ ɡɚɤɚɧɱɢɜɚɸɬɫɹ ɬɨɱɤɨɣ ɫ ɡɚɩɹɬɨɣ ɢɥɢ ɞɜɨɟɬɨɱɢɟɦ, ɟɫɥɢ ɪɟɡɭɥɶɬɚɬ ɧɟ ɧɚɞɨ ɜɵɜɨɞɢɬɶ ɧɚ ɷɤɪɚɧ. ɑɬɨɛɵ ɩɪɨɞɨɥɠɢɬɶ ɡɚɩɢɫɶ ɩɪɟɞɥɨɠɟɧɢɹ ɧɚ ɫɥɟɞɭɸɳɟɣ ɫɬɪɨɤɟ ɢɫɩɨɥɶɡɭɸɬ ɤɨɦɛɢɧɚɰɢɸ "Shift + Enter". ɉɪɢ ɧɚɠɚɬɢɢ ɤɥɚɜɢɲɢ "Enter" ɩɪɟɞɥɨɠɟɧɢɟ ɜɵɩɨɥɧɹɟɬɫɹ. Ɉɛɧɚɪɭɠɢɜ ɨɲɢɛɤɭ, Maple ɜɵɜɨɞɢɬ ɫɨɨɛɳɟɧɢɟ ɨ ɧɟɣ ɜ ɫɥɟɞɭɸɳɟɣ ɫɬɪɨɤɟ. > 1+2; 3

> 12*4/3; 16

> 1+3/2; 5 2

> 1.125/2;

0.5625000000

> 1/0; Error, numeric exception: division by zero ɐɟɥɵɟ ɱɢɫɥɚ ɜ Maple ɢɦɟɸɬ ɧɚɢɜɵɫɲɢɣ ɩɪɢɨɪɢɬɟɬ, ɩɨɷɬɨɦɭ 1 + 3 / 2 = , ɚ ɧɟ 2.5. ȼɫɟɝɞɚ ɦɨɠɧɨ ɩɨ5 2

ɥɭɱɢɬɶ ɪɟɡɭɥɶɬɚɬ ɜ ɜɢɞɟ ɞɟɫɹɬɢɱɧɨɣ ɞɪɨɛɢ ɩɪɢ ɩɨɦɨɳɢ ɮɭɧɤɰɢɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ evalf ( [ ]).

,

> evalf(1+3/2); 2.500000000

> evalf(113/112); 1.008928571

> evalf(113/112,5); 1.0089

> evalf(113/112,20); 1.0089285714285714286

ɉɨ ɭɦɨɥɱɚɧɢɸ Maple ɩɪɨɜɨɞɢɬ ɜɵɱɢɫɥɟɧɢɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɞɟɫɹɬɨɝɨ ɡɧɚɤɚ ɩɨɫɥɟ ɡɚɩɹɬɨɣ, ɨɞɧɚɤɨ, ɡɚɞɚɜɚɹ ɧɟɨɛɹɡɚɬɟɥɶɧɵɣ ɩɚɪɚɦɟɬɪ ɜ ɮɭɧɤɰɢɢ evalf, ɦɨɠɧɨ ɤɚɤ ɭɦɟɧɶɲɢɬɶ, ɬɚɤ ɢ ɭɜɟɥɢɱɢɬɶ ɷɬɨ ɡɧɚɱɟɧɢɟ. ȿɫɥɢ ɧɟɨɛɯɨɞɢɦɨ ɢɡɦɟɧɢɬɶ ɬɨɱɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ ɞɥɹ ɜɫɟɯ ɜɵɪɚɠɟɧɢɣ, ɬɨ ɞɥɹ ɷɬɨɝɨ ɫɥɟɞɭɟɬ ɩɪɢɫɜɨɢɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɩɟɪɟɦɟɧɧɨɣ Digits. > Digits:=25: evalf(13/7); 1.857142857142857142857143

Ʉɪɨɦɟ ɨɛɵɱɧɵɯ ɡɧɚɤɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ ɢɫɩɨɥɶɡɭɸɬ: ** ɢɥɢ ^ – ɜɨɡɜɟɞɟɧɢɟ ɜ ɫɬɟɩɟɧɶ; ! – ɮɚɤɬɨɪɢɚɥ; := – ɡɧɚɤ ɩɪɢɫɜɨɟɧɢɹ; , >=, 100!; 9332621544394415268169923885626670049071596826438162146859296\3895217599993229915608941 463976156518286253697920827223758251\185210916864000000000000000000000000 ȼ Maple ɢɫɩɨɥɶɡɭɸɬɫɹ ɨɛɳɟɩɪɢɧɹɬɵɟ ɧɚɡɜɚɧɢɹ ɨɫɧɨɜɧɵɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ. Ɏɭɧɤɰɢɹ abs sqrt log log10 ln exp round trunc Re Im argument

Ɉɩɢɫɚɧɢɟ ɦɨɞɭɥɶ ɤɜɚɞɪɚɬɧɵɣ ɤɨɪɟɧɶ ɨɛɵɤɧɨɜɟɧɧɵɣ ɥɨɝɚɪɢɮɦ ɞɟɫɹɬɢɱɧɵɣ ɥɨɝɚɪɢɮɦ ɧɚɬɭɪɚɥɶɧɵɣ ɥɨɝɚɪɢɮɦ ɷɤɫɩɨɧɟɧɬɚ ɨɤɪɭɝɥɟɧɢɟ ɨɬɫɟɱɟɧɢɟ ɞɪɨɛɧɨɣ ɱɚɫɬɢ ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ ɱɚɫɬɶ ɤɨɦɩɥɟɤɫɧɚɹ ɱɚɫɬɶ ɚɪɝɭɦɟɧɬ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ

Ɍɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɧɟ ɬɪɟɛɭɸɬ ɞɟɬɚɥɶɧɨɝɨ ɨɩɢɫɚɧɢɹ: sin(x), cos(x), tan(x), sec(x), csc(x), cot(x), sinh(x), cosh(x), tanh(x), sech(x), csch(x), coth(x), arcsin(x), arccos(x), arctan(x), arcsec(x), arccsc(x), arccot(x), arcsinh(x), arccosh(x), arctanh(x), arcsech(x), arccsch(x), arccoth(x), arctan(y, x). ȼɚɠɧɟɣɲɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɤɨɧɫɬɚɧɬɵ π ɢ i = − 1 ɧɚɱɢɧɚɸɬɫɹ ɫ ɛɨɥɶɲɢɯ ɛɭɤɜ. Ɉɫɧɨɜɚɧɢɟ ɧɚɬɭɪɚɥɶɧɨɝɨ ɥɨɝɚɪɢɮɦɚ – e, ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ exp. > Pi; evalf(%);

π 3.141592653589793238462643

> exp(1); evalf(%); e 2.718281828459045235360287

> I; I ∞

> infinity;

ȼɵɪɚɠɟɧɢɹ ɦɨɠɧɨ ɩɪɢɫɜɚɢɜɚɬɶ ɩɟɪɟɦɟɧɧɵɦ, ɩɪɢ ɷɬɨɦ ɤɚɠɞɚɹ ɩɟɪɟɦɟɧɧɚɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɬɢɩɨɦ ɢ ɢɦɟɧɟɦ – ɧɚɛɨɪɨɦ ɫɢɦɜɨɥɨɜ, ɜ ɤɨɬɨɪɵɯ ɫɬɪɨɱɧɵɟ ɢ ɩɪɨɩɢɫɧɵɟ ɛɭɤɜɵ ɪɚɡɥɢɱɚɸɬɫɹ. Ʉɚɤ ɨɛɵɱɧɨ, ɢɦɹ ɧɟ ɞɨɥɠɧɨ ɫɨɜɩɚɞɚɬɶ ɫ ɫɭɳɟɫɬɜɭɸɳɢɦɢ ɭɠɟ ɢɦɟɧɚɦɢ. > a:=5; A:=12; b:=15; B:=6; c:=a/b; C:=A/B; a := 5 A := 12 b := 15 B := 6 1 c := 3 C := 2

2.2 ȼɕɑɂɋɅȿɇɂȿ ɋɍɆɆɕ ɊəȾȺ, ɉɊɈɂɁȼȿȾȿɇɂə ɂ ɉɊȿȾȿɅȺ ȼɵɱɢɫɥɟɧɢɟ ɫɭɦɦɵ ɱɥɟɧɨɜ ɧɟɤɨɬɨɪɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ f (k) ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɢɧɞɟɤɫɚ k ɨɬ ɡɧɚɱɟɧɢɹ m ɞɨ ɡɧɚɱɟɧɢɹ n, ɬ.ɟ.

∑ f (k ) = f (m) + f (m + 1) + ... + f (n − 1) + f (n) , n

k =m

ɹɜɥɹɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɨɣ ɨɩɟɪɚɰɢɟɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ. Ⱦɥɹ ɜɵɱɢɫɥɹɟɦɨɣ ɢ ɢɧɟɪɬɧɨɣ ɮɨɪɦ ɜɵɱɢɫɥɟɧɢɹ ɫɭɦɦ ɫɥɭɠɚɬ ɮɭɧɤɰɢɢ sum ɢ Sum. ȼɵɱɢɫɥɹɟɦɚɹ ɮɨɪɦɚ ɫɭɦɦɵ. > sum(k^2,k=1..10); 385

ɂɧɟɪɬɧɚɹ ɮɨɪɦɚ ɫɭɦɦɵ.

∑ k2

> Sum(k^2,k=1..10);

10

k=1

Ɉɮɨɪɦɥɟɧɢɟ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɫɱɟɬɚ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɢɧɟɪɬɧɨɣ ɢ ɜɵɱɢɫɥɹɟɦɨɣ ɮɨɪɦ. > Sum(k^2,k=1..10)=sum(k^2,k=1..10);

∑ k2 = 385 10

k=1

Ɉɬɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ ɩɟɪɟɦɟɧɧɨɣ-ɢɧɞɟɤɫɭ (k) ɤ ɦɨɦɟɧɬɭ ɜɵɱɢɫɥɟɧɢɹ ɫɭɦɦɵ ɭɠɟ ɩɪɢɫɜɨɟɧɨ ɤɚɤɨɟɥɢɛɨ ɡɧɚɱɟɧɢɟ, ɬɨ ɮɭɧɤɰɢɹ sum ɩɪɢɜɟɞɟɬ ɤ ɨɲɢɛɤɟ. > k:=125;

k := 125

> sum(k^2,k = 1 .. 10); Error, (in sum) summation variable previously assigned, second argument evaluates to 125 = 1 .. 10 Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɢɡɛɟɠɚɬɶ ɨɲɢɛɤɢ ɫɥɟɞɭɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɨɞɢɧɚɪɧɵɟ ɤɚɜɵɱɤɢ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɢɠɟ. > sum('k^2','k'=1..10); 385

Ɏɭɧɤɰɢɹ value ɫɥɭɠɢɬ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɢɧɟɪɬɧɵɯ ɮɨɪɦ. > S:=Sum('k^2','k'=1..10);

∑ k2 10

S :=

k=1

> value(S); 385

Ɇɧɨɝɢɟ ɛɟɫɤɨɧɟɱɧɵɟ ɫɭɦɦɵ ɫɯɨɞɹɬɫɹ ɤ ɨɩɪɟɞɟɥɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ ɢ Maple ɫɩɨɫɨɛɟɧ ɢɯ ɜɵɱɢɫɥɢɬɶ.

∑

> Sum(1/k!, k=0..infinity)=sum(1/k!, k=0..infinity); ∞

1 =e k!

k=0

> Sum(1/k^2, k=1..infinity)=sum(1/k^2, k=1..infinity);

∑

1 π2 = 6 k2

∞

k=1

Ⱦɥɹ ɜɵɱɢɫɥɹɟɦɨɣ ɢ ɢɧɟɪɬɧɨɣ ɮɨɪɦ ɧɚɯɨɠɞɟɧɢɹ ɩɪɨɢɡɜɟɞɟɧɢɣ ɫɥɭɠɚɬ ɮɭɧɤɰɢɢ product ɢ Product.

∏ k2 = 14400

> Product(k^2,k = 1 .. 5)=product(k^2,k = 1 .. 5); 5

k=1

Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɪɟɞɟɥɨɜ ɫɥɭɠɚɬ ɮɭɧɤɰɢɢ limit ɢ Limit. ȼɵɱɢɫɥɢɦ ɩɪɟɞɟɥ ɮɭɧɤɰɢɢ y = 12 ⋅ sin( x) ɜ ɬɨɱɤɟ π/4. > Limit(12*sin(x),x=Pi/4)=limit(12*sin(x),x=Pi/4); lim

ɉɪɟɞɟɥ ɮɭɧɤɰɢɢ 1/x ɜ ɬɨɱɤɟ x = 0.

π x →    4

12 sin ( x ) = 6 2

> Limit(1/x,x=0)=limit(1/x,x=0);

1 = undefined x

lim

ɋɩɪɚɜɚ ɨɬ ɧɭɥɹ.

x→0

> Limit(1/x,x=0,right)=limit(1/x,x=0,right); ɋɥɟɜɚ ɨɬ ɧɭɥɹ.

lim

1 =∞ x

lim

1 = −∞ x

x → 0+

> Limit(1/x,x=0,left)=limit(1/x,x=0,left); ɉɟɪɜɵɣ ɡɚɦɟɱɚɬɟɥɶɧɵɣ ɩɪɟɞɟɥ.

x → 0-

> Limit(sin(x)/x,x=0)=limit(sin(x)/x,x=0); lim

x→0

sin ( x ) =1 x

2.3 ɈɋɇɈȼɇɕȿ Ɍɂɉɕ ȾȺɇɇɕɏ Ɋɚɫɫɦɨɬɪɢɦ ɨɫɧɨɜɧɵɟ ɬɢɩɵ ɞɚɧɧɵɯ, ɫ ɤɨɬɨɪɵɦɢ ɩɪɢɯɨɞɢɬɫɹ ɜɫɬɪɟɱɚɬɶɫɹ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɪɚɡɥɢɱɧɵɯ ɜɵɱɢɫɥɟɧɢɣ. Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɩɪɢɧɚɞɥɟɠɧɨɫɬɢ ɜɵɪɚɠɟɧɢɹ ɤ ɨɩɪɟɞɟɥɟɧɧɨɦɭ ɬɢɩɭ ɫɥɭɠɚɬ ɞɜɟ ɮɭɧɤɰɢɢ:

• whattype( ) ɜɨɡɜɪɚɳɚɟɬ ɬɢɩ ɜɵɪɚɠɟɧɢɹ; • type( , ) ɜɨɡɜɪɚɳɚɟɬ true ( ), ɟɫɥɢ false ( ) ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ.

ɩɪɢɧɚɞɥɟɠɢɬ ɤ ɭɤɚɡɚɧɧɨɦɭ

,ɢ

ɐɟɥɵɟ ȼɵɪɚɠɟɧɢɟ ɩɪɢɧɚɞɥɟɠɢɬ ɤ ɰɟɥɨɦɭ ɬɢɩɭ (ɬɢɩ integer), ɟɫɥɢ ɨɧɨ ɫɨɫɬɨɢɬ ɢɡ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɰɢɮɪ, ɧɟ ɪɚɡɞɟɥɟɧɧɵɯ ɦɟɠɞɭ ɫɨɛɨɣ ɧɢɤɚɤɢɦɢ ɡɧɚɤɚɦɢ. Maple ɦɨɠɟɬ ɪɚɛɨɬɚɬɶ ɫ ɰɟɥɵɦɢ ɱɢɫɥɚɦɢ ɩɪɚɤɬɢɱɟɫɤɢ ɛɟɫɤɨɧɟɱɧɨɣ ɞɥɢɧɵ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɜ Maple 8 ɨɝɪɚɧɢɱɟɧɢɟ ɧɚ ɞɥɢɧɭ ɰɟɥɵɯ ɱɢɫɟɥ – 228 ɰɢɮɪ. ɑɢɫɥɚ ɬɢɩɚ integer ɦɨɝɭɬ ɛɵɬɶ ɤɚɤ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ, ɬɚɤ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ. > whattype(-125); integer

> type(-125,integer); true

Ⱦɪɨɛɧɵɟ Ⱦɪɨɛɢ (ɬɢɩ fraction) ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɜ ɜɢɞɟ:

a b

, ɝɞɟ a – ɰɟɥɨɟ ɱɢɫɥɨ ɫɨ ɡɧɚɤɨɦ, b – ɰɟɥɨɟ ɱɢɫɥɨ ɛɟɡ

ɡɧɚɤɚ. ȼ ɜɵɪɚɠɟɧɢɢ ɬɢɩɚ fraction ɨɛɹɡɚɬɟɥɶɧɨ ɩɪɢɫɭɬɫɬɜɭɸɬ ɞɜɚ ɩɨɥɹ: ɱɢɫɥɢɬɟɥɶ ɢ ɡɧɚɦɟɧɚɬɟɥɶ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɭɱɟɧɵ ɮɭɧɤɰɢɟɣ op. > type(-3/7,integer); false

> whattype(-3/7); fraction

> op(-3/7); -3, 7

ɑɢɫɥɚ ɫ ɩɥɚɜɚɸɳɟɣ ɬɨɱɤɨɣ ɑɢɫɥɚ ɫ ɩɥɚɜɚɸɳɟɣ ɬɨɱɤɨɣ (ɬɢɩ float) ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. 1 ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɢɫɟɥ, ɪɚɡɞɟɥɟɧɧɵɯ ɬɨɱɤɨɣ: ɚ) . ɛ) . ɜ) . 2 ɜ ɜɢɞɟ: Float(M, E), ɬ.ɟ. M*10E. > whattype(0.123); float

> Float(2,3); 2000.

ɋɬɪɨɤɨɜɵɟ ɬɢɩɵ ȼɵɪɚɠɟɧɢɟ ɫɬɪɨɤɨɜɨɝɨ ɬɢɩɚ (ɬɢɩ string) – ɷɬɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɢɦɜɨɥɨɜ, ɡɚɤɥɸɱɟɧɧɵɯ ɜ ɞɜɨɣɧɵɟ ɤɚɜɵɱɤɢ. > str:="ɗɬɨ ɫɬɪɨɤɚ !"; str := "ɗɬɨ ɫɬɪɨɤɚ !"

> whattype(str); string

Ɇɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɞɥɢɧɭ ɫɬɪɨɤɢ: > length(str); 12

ɂɡ ɫɬɪɨɤɢ ɦɨɠɧɨ ɢɡɜɥɟɱɶ ɩɨɞɫɬɪɨɤɭ: > substring(str,5..10); "ɫɬɪɨɤɚ"

Ȼɭɥɟɜɵ ɜɵɪɚɠɟɧɢɹ Ȼɭɥɟɜɵ ɜɵɪɚɠɟɧɢɹ (ɬɢɩ boolean) ɦɨɝɭɬ ɩɪɢɧɢɦɚɬɶ ɨɞɧɨ ɢɡ ɞɜɭɯ ɡɧɚɱɟɧɢɣ: true ( ) ɢɥɢ false ( ). ȼ ɛɭɥɟɜɵɯ ɜɵɪɚɠɟɧɢɹɯ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɥɟɞɭɸɳɢɟ ɨɩɟɪɚɬɨɪɵ and, or, xor, implies, not, ɚ ɬɚɤɠɟ ɨɩɟɪɚɬɨɪɵ ɨɬɧɨɲɟɧɢɣ =, =, . Ɏɭɧɤɰɢɹ evalb ɜɵɱɢɫɥɹɟɬ ɫɥɨɠɧɨɟ ɥɨɝɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ. > 5>3;

3 evalb(5>3); true

ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ (ɬɢɩ exprseq) – ɧɚɛɨɪ ɷɥɟɦɟɧɬɨɜ, ɪɚɡɞɟɥɟɧɧɵɯ ɡɚɩɹɬɵɦɢ, ɛɟɡ ɫɤɨɛɨɤ. > S:=1,2,3,4,5,6,7,8,9,10; S := 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

> whattype(S); exprseq

Ⱦɥɹ ɝɟɧɟɪɚɰɢɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ ɫɥɭɠɢɬ ɮɭɧɤɰɢɹ seq: > S:=seq(i,i=1..10); S := 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

> Q:=seq(i^2,i=1..10); Q := 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɦɨɠɧɨ ɬɚɤɠɟ ɩɨɥɭɱɢɬɶ ɩɪɢ ɩɨɦɨɳɢ ɨɩɟɪɚɬɨɪɚ ɮɨɪɦɢɪɨɜɚɧɢɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ – $. > $ 2..5; 2, 3, 4, 5

> i^2 $ i = 2/3 .. 8/3; 4 25 64 , , 9 9 9

> a[i] $ i = 1..3; a 1, a 2, a 3

> x$4; x, x, x, x

ɉɭɫɬɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɛɨɡɧɚɱɚɟɬɫɹ NULL. Ɇɧɨɠɟɫɬɜɚ Ɇɧɨɠɟɫɬɜɨ (ɬɢɩ set) – ɧɚɛɨɪ ɷɥɟɦɟɧɬɨɜ, ɪɚɡɞɟɥɟɧɧɵɯ ɡɚɩɹɬɵɦɢ ɢ ɡɚɤɥɸɱɟɧɧɵɣ ɜ ɮɢɝɭɪɧɵɟ ɫɤɨɛɤɢ.

Ⱦɥɹ ɦɧɨɠɟɫɬɜ ɞɟɣɫɬɜɢɬɟɥɶɧɵ ɜɫɟ ɩɪɚɜɢɥɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɩɪɢɧɹɬɵɟ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɚɬɟɦɚɬɢɤɟ. Ɂɚɞɚɧɢɟ ɦɧɨɠɟɫɬɜɚ ɢ ɨɩɪɟɞɟɥɟɧɢɟ ɱɢɫɥɚ ɷɥɟɦɟɧɬɨɜ. > {1, 2, 3, 4, 5, 1}; { 1, 2, 3, 4, 5 }

> whattype(%); set

Ʉɨɥɢɱɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɜɨ ɦɧɨɠɟɫɬɜɟ. > nops(%%); 5

Ʉɚɤ ɜɢɞɧɨ ɢɡ ɩɪɟɞɵɞɭɳɟɝɨ ɩɪɢɦɟɪɚ, ɦɧɨɠɟɫɬɜɨ ɧɟ ɦɨɠɟɬ ɫɨɞɟɪɠɚɬɶ ɞɜɚ ɨɞɢɧɚɤɨɜɵɯ ɷɥɟɦɟɧɬɚ. Ɉɛɴɟɞɢɧɟɧɢɟ ɦɧɨɠɟɫɬɜ > {1, 2, 3, 4} union {3, 4, 5, 6}; ɉɟɪɟɫɟɱɟɧɢɟ ɦɧɨɠɟɫɬɜ

{ 1, 2, 3, 4, 5, 6 } .

> {1, 2, 3, 4} intersect {3, 4, 5, 6}; { 3, 4 } .

ȼɵɱɢɬɚɧɢɟ ɦɧɨɠɟɫɬɜ > {1, 2, 3, 4} minus {3, 4, 5, 6};

{ 1, 2 } .

Ɇɧɨɠɟɫɬɜɚ ɦɨɝɭɬ ɫɨɫɬɨɹɬɶ ɧɟ ɬɨɥɶɤɨ ɢɡ ɱɢɫɟɥ > U:={a, b, c, d, e, f};

U := { a, b, c, d, e, f } .

ɂɡɜɥɟɱɟɧɢɟ ɫ 2 ɩɨ 4 ɷɥɟɦɟɧɬ ɢɡ ɦɧɨɠɟɫɬɜɚ U. > op(2..4,U);

b , c, d

ɉɪɨɜɟɪɤɚ ɩɪɢɧɚɞɥɟɠɧɨɫɬɢ ɷɥɟɦɟɧɬɚ ɦɧɨɠɟɫɬɜɭ U. > member(a,U); true

> member(g,U); false

Ɂɚɞɚɧɢɟ ɦɧɨɠɟɫɬɜɚ ɜ ɜɢɞɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ > P:={seq(a[i],i=1..5)}; P := { a 1, a 2, a3, a 4, a 5 }

Ⱦɨɛɚɜɥɟɧɢɟ ɷɥɟɦɟɧɬɚ ɤ ɦɧɨɠɟɫɬɜɭ P > P:={op(P),a[6]}; P := { a 1, a 2, a 3, a 4, a 5, a 6 }

ɍɞɚɥɟɧɢɟ ɬɪɟɬɶɟɝɨ ɷɥɟɦɟɧɬɚ ɢɡ ɦɧɨɠɟɫɬɜɚ > P:=subsop(3=NULL,P);

P := { a 1, a 2, a4, a 5, a 6 }

ɋɩɢɫɤɢ ɋɩɢɫɨɤ (ɬɢɩ list) – ɧɚɛɨɪ ɷɥɟɦɟɧɬɨɜ, ɪɚɡɞɟɥɟɧɧɵɯ ɡɚɩɹɬɵɦɢ ɢ ɡɚɤɥɸɱɟɧɧɵɣ ɜ ɤɜɚɞɪɚɬɧɵɟ ɫɤɨɛɤɢ.

> [a, b, c, d]; [ a, b, c, d ]

> whattype(%); list

ɋɨ ɫɩɢɫɤɚɦɢ ɦɨɠɧɨ ɩɪɨɜɨɞɢɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɨɩɟɪɚɰɢɢ, ɧɚɩɪɢɦɟɪ, ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ: > L:=[sin, cos, tan]; L := [ sin, cos, tan ]

> D(L);

[ cos , −sin , 1 + tan 2 ]

Ɂɚɞɚɧɢɟ ɫɩɢɫɤɚ ɜ ɜɢɞɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ. > [seq(x[i],i=1..5)];

[ x1, x2, x3, x4, x5 ]

ɋɩɢɫɨɤ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɦɧɨɠɟɫɬɜɚ, ɦɨɠɟɬ ɫɨɞɟɪɠɚɬɶ ɨɞɢɧɚɤɨɜɵɟ ɷɥɟɦɟɧɬɵ. > [1, 2, 3, 4, 5, 2]; [ 1, 2, 3, 4, 5, 2 ]

ȼɨ ɦɧɨɠɟɫɬɜɟ ɩɨɪɹɞɨɤ ɫɥɟɞɨɜɚɧɢɹ ɷɥɟɦɟɧɬɨɜ ɧɟ ɢɦɟɟɬ ɡɧɚɱɟɧɢɹ, ɚ ɜ ɫɩɢɫɤɟ ɨɧ ɫɭɳɟɫɬɜɟɧɟɧ: > evalb({a,b,c}={c,b,a}); true

> evalb([a,b,c]=[c,b,a]); false

Ɇɚɫɫɢɜɵ Ɇɚɫɫɢɜ (ɬɢɩ array) – ɤɨɧɟɱɧɵɣ ɫɩɢɫɨɤ ɫ ɰɟɥɨɱɢɫɥɟɧɧɵɦɢ ɢɧɞɟɤɫɚɦɢ. Ⱦɥɹ ɫɨɡɞɚɧɢɹ ɦɚɫɫɢɜɚ ɫɥɭɠɢɬ ɮɭɧɤɰɢɹ array. ɋɨɡɞɚɟɦ ɩɭɫɬɨɣ ɦɚɫɫɢɜ ɢɡ ɩɹɬɢ ɷɥɟɦɟɧɬɨɜ, ɡɚɩɨɥɧɹɟɦ ɟɝɨ ɜ ɰɢɤɥɟ for ɤɜɚɞɪɚɬɚɦɢ ɢɧɞɟɤɫɨɜ ɢ ɜɵɜɨɞɢɦ ɧɚ ɩɟɱɚɬɶ ɮɭɧɤɰɢɟɣ print: > A:=array(1..5); A := array( 1 .. 5, [ ] )

> whattype(%); array

> for i from 1 to 5 do A[i]:=i^2 end do; A1 := 1 A2 := 4 A3 := 9 A4 := 16 A5 := 25

> print(A); ɋɨɡɞɚɟɦ ɞɜɭɯɦɟɪɧɵɣ ɦɚɫɫɢɜ 2 × 2 ɢ ɫɪɚɡɭ ɩɪɢɫɜɚɢɜɚɟɦ ɡɧɚɱɟɧɢɹ [ 1, 4, 9, 16, 25 ]

> B:=array(1..2, 1..2, [[1, 3], [1/2, 5]]);

 1  B :=  1  2 

3   5 

3  x C :=  sin( x ) 

3   2.33

Ɇɚɫɫɢɜ ɜ Maple ɦɨɠɟɬ ɫɨɞɟɪɠɚɬɶ ɷɥɟɦɟɧɬɵ ɪɚɡɧɵɯ ɬɢɩɨɜ. > C:=array(1..2, 1..2, [[x^3, 3], [sin(x), 2.33]]);

Ɏɭɧɤɰɢɹ map ɩɨɡɜɨɥɹɟɬ ɜɵɩɨɥɧɹɬɶ ɤɚɤɭɸ-ɥɢɛɨ ɨɩɟɪɚɰɢɸ ɧɚɞ ɜɫɟɦɢ ɷɥɟɦɟɧɬɚɦɢ ɦɚɫɫɢɜɚ. Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɩɨ x ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɦɚɫɫɢɜɚ > map(diff,C,x);

2  3x  cos ( x ) 

0  0

ɂɡɜɥɟɱɟɧɢɟ ɤɜɚɞɪɚɬɧɨɝɨ ɤɨɪɧɹ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɦɚɫɫɢɜɚ > map(sqrt,C);

x3     sin( x )

   1.526433752 3

Ɍɚɛɥɢɰɵ

ȼ ɨɬɥɢɱɢɟ ɨɬ ɦɚɫɫɢɜɚ, ɝɞɟ ɢɧɞɟɤɫɵ – ɰɟɥɨɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɪɚɫɩɨɥɨɠɟɧɧɵɟ ɩɨ ɩɨɪɹɞɤɭ ɧɨɦɟɪɨɜ, ɢɧɞɟɤɫɵ ɭ ɬɚɛɥɢɰɵ (ɬɢɩ table) – ɥɸɛɵɟ ɡɧɚɱɟɧɢɹ. ȿɫɥɢ ɢɧɞɟɤɫɵ ɧɟ ɨɩɪɟɞɟɥɟɧɵ, ɬɨ Maple ɩɪɢɫɜɚɢɜɚɟɬ ɩɨ ɩɨɪɹɞɤɭ ɰɟɥɨɱɢɫɥɟɧɧɵɟ ɢɧɞɟɤɫɵ > A:=table([ɂɜɚɧɨɜ,ɉɟɬɪɨɜ,ɋɢɞɨɪɨɜ]);

A := table( [ 1 = И

,2=

,3=

])

> whattype(%); table

> A[2]; ɂɧɞɟɤɫɵ ɬɚɛɥɢɰɵ ɦɨɠɧɨ ɩɪɢɫɜɚɢɜɚɬɶ ɩɪɨɢɡɜɨɥɶɧɨ. > B:=table([(ɩɟɪɜɵɣ)=ɂɜɚɧɨɜ, (ɜɬɨɪɨɣ)=ɉɟɬɪɨɜ, (ɬɪɟɬɢɣ)=ɋɢɞɨɪɨɜ]); B := table( [

=И

,

=

,

=

])

> B[ɜɬɨɪɨɣ]; 2.4 ɈɉȿɊȺɐɂɂ ɋ ɎɈɊɆɍɅȺɆɂ ɉɪɢ ɪɚɛɨɬɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɜɵɪɚɠɟɧɢɹɦɢ ɩɪɢɯɨɞɢɬɫɹ ɜɵɩɨɥɧɹɬɶ ɬɚɤɢɟ ɨɩɟɪɚɰɢɢ, ɤɚɤ ɩɪɢɜɟɞɟɧɢɟ ɩɨɞɨɛɧɵɯ ɱɥɟɧɨɜ, ɪɚɫɤɪɵɬɢɟ ɫɤɨɛɨɤ, ɪɚɡɥɨɠɟɧɢɟ ɧɚ ɦɧɨɠɢɬɟɥɢ. ȼ ɩɚɤɟɬɟ Maple ɷɬɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɩɪɢ ɩɨɦɨɳɢ ɫɩɟɰɢɚɥɶɧɵɯ ɮɭɧɤɰɢɣ. Ɏɭɧɤɰɢɹ simplify factor expand normal convert coeff

Ɉɩɢɫɚɧɢɟ ɭɩɪɨɫɬɢɬɶ ɜɵɪɚɠɟɧɢɟ ɮɚɤɬɨɪɢɡɨɜɚɬɶ ɪɚɡɥɨɠɢɬɶ (ɪɚɫɤɪɵɬɶ ɜɫɟ ɫɤɨɛɤɢ) ɩɪɢɜɟɫɬɢ ɜɵɪɚɠɟɧɢɟ ɤ "ɧɨɪɦɚɥɶɧɨɦɭ" ɜɢɞɭ ɩɟɪɟɩɢɫɚɬɶ ɜ ɡɚɞɚɧɧɨɦ ɜɢɞɟ ɜɵɞɟɥɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɨɥɢɧɨɦɚ

collect

ɫɨɛɪɚɬɶ ɜɦɟɫɬɟ ɱɚɫɬɢ ɜɵɪɚɠɟɧɢɹ

ɍɩɪɨɳɟɧɢɟ ɜɵɪɚɠɟɧɢɣ. > sin(x)^2+cos(x)^2;

sin ( x ) 2 + cos ( x ) 2

> simplify(%); 1

> cos(x)^5+sin(x)^4+2*cos(x)^2-2*sin(x)^2-cos(2*x);

cos ( x ) 5 + sin ( x ) 4 + 2 cos ( x ) 2 − 2 sin ( x ) 2 − cos ( 2 x )

> simplify(%);

cos ( x ) 4 ( cos ( x ) + 1 )

> sqrt(x^2); x2

> simplify(%); csgn( x ) x

ȼ ɩɨɫɥɟɞɧɟɦ ɩɪɢɦɟɪɟ ɮɭɧɤɰɢɹ csgn ɜɨɡɜɪɚɳɚɟɬ ɡɧɚɤ ɞɟɣɫɬɜɢɬɟɥɶɧɨɝɨ ɢɥɢ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ  1 c sgn( x) =  − 1

ɟɫɥɢ Re( x) > 0 ɢɥɢ Re( x) = 0 ɢ Im( x) > 0

ɟɫɥɢ Re( x) < 0 ɢɥɢ Re( x) = 0 ɢ Im( x) < 0

.

Ɏɚɤɬɨɪɢɡɢɪɨɜɚɬɶ – ɡɧɚɱɢɬ ɪɚɡɥɨɠɢɬɶ ɜɵɪɚɠɟɧɢɟ ɧɚ ɦɧɨɠɢɬɟɥɢ. > 6*x^2+18*x-24; > factor(%);

6 x 2 + 18 x − 24

6 (x + 4) (x − 1)

Ⱦɥɹ ɪɚɡɥɨɠɟɧɢɹ ɧɚ ɦɧɨɠɢɬɟɥɢ ɰɟɥɵɯ ɱɢɫɟɥ ɫɥɭɠɢɬ ɮɭɧɤɰɢɹ ifactor. > ifactor(132); ( 2 ) 2 ( 3 ) ( 11 )

Ɋɚɫɤɪɵɬɢɟ ɫɤɨɛɨɤ. > (x+1)*(x+2);

( x + 1) (x + 2)

> expand(%);

x2 + 3 x + 2

> (x+1)/(x+2);

x+1 x+2

> expand(%);

x 1 + x+2 x+2

Ɏɭɧɤɰɢɹ normal ɨɛɵɱɧɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɩɨɥɢɧɨɦɨɜ ɢ ɪɚɰɢɨɧɚɥɶɧɵɯ ɮɭɧɤɰɢɣ, ɯɨɬɹ ɢɧɨɝɞɚ ɩɪɢɦɟɧɢɦɚ ɢ ɞɥɹ ɛɨɥɟɟ ɨɛɳɢɯ ɜɵɪɚɠɟɧɢɣ. > 1/x+x/(x+1);

1 x + x x+1

> normal(%);

x + 1 + x2 x (x + 1)

> normal( %%, expanded );

x + 1 + x2 x2 + x

ȼ ɩɨɫɥɟɞɧɟɦ ɩɪɢɦɟɪɟ ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɩɪɨɢɫɯɨɞɢɬ ɪɚɫɤɪɵɬɢɟ

ɫɤɨɛɨɤ.

> sin(x)+1/sin(x)^2;

sin( x ) +

1 sin( x ) 2

sin ( x ) 3 + 1 sin ( x ) 2

> normal(%);

Ɏɭɧɤɰɢɹ convert ɩɪɟɨɛɪɚɡɭɟɬ ɜɵɪɚɠɟɧɢɟ ɜ ɪɚɡɥɢɱɧɵɟ ɮɨɪɦɵ. ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɱɢɫɟɥ ɜ ɪɚɡɥɢɱɧɵɟ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ: > convert(1019, binary); 1111111011

> convert(1019, hex); 3FB

ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɞɟɫɹɬɢɱɧɨɣ ɞɪɨɛɢ ɜ ɧɚɬɭɪɚɥɶɧɭɸ. > convert( 1.23456, fraction ); 3858 3125

ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɜɵɪɚɠɟɧɢɹ ɜ ɷɥɟɦɟɧɬɚɪɧɵɟ ɞɪɨɛɢ. x3 + x x2 − 1

> (x^3+x)/(x^2-1); > convert(%, parfrac, x);

x+

1 1 + x−1 x+1

ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɤ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɦɭ ɜɢɞɭ. > 1/4*exp(x)^2-1/4/exp(x)^2;

> convert(%,trig);

1 x 2 1 1 (e ) − 4 4 ( e x )2

1 1 1 ( cosh ( x ) + sinh ( x ) ) 2 − 4 4 ( cosh ( x ) + sinh ( x ) ) 2

ɂ ɨɛɪɚɬɧɨ ɜ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɣ. > convert(%,exp);

1 x 2 1 1 (e ) − 4 4 ( e x )2

ȼɵɞɟɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɢɧɨɦɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ coeff ( ). > p := 2*x^2 + 3*y^3 - 5; > coeff(p,x,0);

,

,

p := 2 x 2 + 3 y 3 − 5

3 y3 − 5

> coeff(p,x,1); 0

> coeff(p,x,2); 2

Ɏɭɧɤɰɢɹ collect ɩɨɡɜɨɥɹɟɬ ɫɨɛɢɪɚɬɶ ɜɦɟɫɬɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ. ȼ ɫɥɟɞɭɸɳɢɯ ɩɪɢɦɟɪɚɯ ɫɨɛɢɪɚɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ln(x) ɢ x.

> a*ln(x)-ln(x)*x-x;

a ln( x ) − ln( x ) x − x

> collect(%,ln(x));

( a − x ) ln( x ) − x

> y/x+2*z/x+x^(1/3)-y*x^(1/3); > collect(%,x);

y 2z ( 1/3 ) ( 1/3 ) + +x −yx x x (1 − y) x

( 1/3 )

+

y+2z x

Ɏɭɧɤɰɢɹ trigsubs ɜɵɞɚɟɬ ɜɫɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɷɤɜɢɜɚɥɟɧɬɵ ɜɵɪɚɠɟɧɢɹ ɜ ɜɢɞɟ ɫɩɢɫɤɚ. > trigsubs(sin(alpha+beta));

   1  sin( α + β ), −sin( −α − β ), 2 sin α + β  cos α + β ,   2 2  csc( α + β ) , 2 2         α β  2 tan +    1 2 2 -1 ( ( α + β ) I ) ( −I ( α + β ) )  ,  , − I (e −e )  2 csc( −α − β ) 2  α β  1 + tan +   2 2   

ɇɚɩɨɦɧɢɦ, ɱɬɨ ɢɡɜɥɟɤɚɬɶ ɷɥɟɦɟɧɬɵ ɢɡ ɫɩɢɫɤɚ ɦɨɠɧɨ ɩɪɢ ɩɨɦɨɳɢ ɮɭɧɤɰɢɢ op, ɭɤɚɡɚɜ ɩɟɪɜɵɦ ɩɚɪɚɦɟɬɪɨɦ ɧɨɦɟɪ ɷɥɟɦɟɧɬɚ. ɂɡɜɥɟɱɟɦ ɬɪɟɬɢɣ ɷɥɟɦɟɧɬ ɢɡ ɩɨɫɥɟɞɧɟɝɨ ɫɩɢɫɤɚ. > op(3,%);

α β α β 2 sin  +  cos  +  2 2 2   2

2.5 ɉɊɈɂɁȼɈȾɇɕȿ ɂ ɂɇɌȿȽɊȺɅɕ ȼɵɱɢɫɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɵɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɣ diff ɢ Diff. ɉɟɪɜɵɦ ɩɚɪɚɦɟɬɪɨɦ ɷɬɢɯ ɮɭɧɤɰɢɣ ɹɜɥɹɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɟ ɜɵɪɚɠɟɧɢɟ, ɞɚɥɟɟ – ɢɦɹ ɩɟɪɟɦɟɧɧɨɣ ɢɥɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɢɦɟɧ ɩɟɪɟɦɟɧɧɵɯ. Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦ ɮɭɧɤɰɢɸ sin(x) ɩɨ x. > diff(sin(x),x); cos ( x )

ɂ ɜɬɨɪɨɣ ɪɚɡ. > diff(%,x);

−sin( x )

ɇɨ ɞɜɨɣɧɨɟ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɦɨɠɧɨ ɜɵɩɨɥɧɢɬɶ ɬɚɤ > diff(sin(x),x,x); ɂɥɢ ɬɚɤ > diff(sin(x),x$2);

−sin( x ) .

−sin( x ) .

ȼ ɩɨɫɥɟɞɧɟɦ ɩɪɢɦɟɪɟ ɞɥɹ ɡɚɞɚɧɢɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɢɡ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ x ɢɫɩɨɥɶɡɭɟɬɫɹ ɨɩɟɪɚɬɨɪ ɝɟɧɟɪɚɰɢɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ – $. Ɉɞɧɚ ɮɭɧɤɰɢɹ diff ɦɨɠɟɬ ɜɵɩɨɥɧɢɬɶ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɩɨ ɧɟɫɤɨɥɶɤɢɦ ɩɟɪɟɦɟɧɧɵɦ.

> Diff(y*sin(x)/cos(y),x,y) = diff(y*sin(x)/cos(y),x,y);

∂2  y sin( x )  cos ( x ) y cos ( x ) sin( y ) +  = ∂y ∂x  cos ( y )  cos ( y ) cos ( y ) 2

Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɢɧɬɟɝɪɚɥɨɜ ɢɫɩɨɥɶɡɭɸɬɫɹ ɮɭɧɤɰɢɢ int ɢ Int. ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɧɟɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɩɟɪɜɵɣ ɩɚɪɚɦɟɬɪ – ɢɧɬɟɝɪɢɪɭɟɦɨɟ ɜɵɪɚɠɟɧɢɟ, ɜɬɨɪɨɣ – ɢɦɹ ɩɟɪɟɦɟɧɧɨɣ. > Int(sin(x),x)=int(sin(x),x);

⌠ sin ( x ) d x = − cos ( x )  ⌡

> Int(x/(x^3-1),x)=int(x/(x^3-1),x);

⌠ x 1 1  (2 x + 1) 3  1   + ln( x − 1 ) 3 arctan  d x = − ln( x 2 + x + 1 ) +  3  6 3 3 x − 1   3  ⌡

> eq:=exp(-x^2)*ln(x); eq := e

> int(eq,x);

2 ( −x )

ln( x )

⌠ ( −x 2 ) e ln( x ) dx  ⌡

ȿɫɥɢ ɢɧɬɟɝɪɚɥ ɧɟ ɛɟɪɟɬɫɹ, ɤɚɤ ɜ ɩɨɫɥɟɞɧɟɦ ɩɪɢɦɟɪɟ, ɬɨ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɨ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɮɭɧɤɰɢɟɣ series. Ɋɚɡɥɨɠɢɦ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɜ ɪɹɞ ɞɨ 8-ɝɨ ɩɨɪɹɞɤɚ. > series(eq,x,8); ln( x ) − ln( x ) x 2 +

1 1 ln( x ) x 4 − ln( x ) x 6 + O ( x 8 ) 2 6

Ɍɟɩɟɪɶ ɦɨɠɧɨ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɪɹɞ. > int(%,x); x ln( x ) − x −

1 x3 1 x5 1 x7 + − + O( x 9 ) ln( x ) x3 + ln( x ) x5 − ln( x ) x7 + 9 10 50 42 294 3

ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɩɪɟɞɟɥɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ. > Int(sin(x),x=0..Pi/4)=int(sin(x),x=0..Pi/4); π 4

⌠ sin( x ) dx = − 2 + 1  2 ⌡0

ɑɢɫɥɟɧɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ. > evalf(Int(sin(x),x=0..Pi/4)); 0.2928932188

ȼɵɱɢɫɥɟɧɢɟ ɢɧɬɟɝɪɚɥɚ ɫ ɛɟɫɤɨɧɟɱɧɵɦ ɜɟɪɯɧɢɦ ɩɪɟɞɟɥɨɦ. > Int(exp(-x),x=0..infinity)=int(exp(-x), x=0..infinity);

∞

⌠ e ( −x ) d x = 1   ⌡0

2.6 ɉȺɄȿɌɕ ɊȺɋɒɂɊȿɇɂɃ ɂ ɊȺȻɈɌȺ ɋ ɇɂɆɂ ɇɟɤɨɬɨɪɵɟ ɮɭɧɤɰɢɢ Maple ɩɨɦɢɦɨ ɹɞɪɚ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɚɤɟɬɚɯ ɪɚɫɲɢɪɟɧɢɣ, ɜɯɨɞɹɳɢɯ ɜ ɛɚɡɨɜɭɸ ɩɨɫɬɚɜɤɭ ɫɢɫɬɟɦɵ. ɉɟɪɟɞ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɬɚɤɢɯ ɮɭɧɤɰɢɣ ɢɯ ɧɚɞɨ ɡɚɝɪɭɡɢɬɶ. Ⱦɥɹ ɡɚɝɪɭɡɤɢ ɜɫɟɯ ɮɭɧɤɰɢɣ ɤɚɤɨɝɨ-ɥɢɛɨ ɩɚɤɟɬɚ ɢɫɩɨɥɶɡɭɟɬɫɹ ɮɭɧɤɰɢɹ with ( _ ). Ⱦɥɹ ɡɚɝɪɭɡɤɢ ɢɡɛɪɚɧɧɵɯ ɮɭɧɤɰɢɣ ɩɚɤɟɬɚ – with ( _ , _1, _2, …). Ɉɬɦɟɬɢɦ, ɱɬɨ ɧɟɤɨɬɨɪɵɟ ɮɭɧɤɰɢɢ ɩɚɤɟɬɨɜ ɪɚɫɲɢɪɟɧɢɣ ɦɨɝɭɬ ɩɟɪɟɨɩɪɟɞɟɥɹɬɶ ɨɞɧɨɢɦɟɧɧɵɟ ɮɭɧɤɰɢɢ ɹɞɪɚ. ɉɚɤɟɬ linalg ɫɨɞɟɪɠɢɬ ɛɨɥɟɟ ɫɬɚ ɮɭɧɤɰɢɣ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɥɢɧɟɣɧɨɣ ɚɥɝɟɛɪɵ. Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɢɡ ɧɢɯ ɧɚ ɩɪɢɦɟɪɟ ɞɜɭɯ ɦɚɬɪɢɰ 3 × 3 ɫɨɡɞɚɧɧɵɯ ɩɪɢ ɩɨɦɨɳɢ ɮɭɧɤɰɢɢ matrix, ɚɧɚɥɨɝɢɱɧɨɣ ɮɭɧɤɰɢɢ array. > with(linalg): > A := matrix(3,3,[[1,2,3],[4,5,6],[7,8,9]]) ; 1  A :=  4  7

3  6  9

2 5 8

> B := matrix(3,3,[[7,4,3],[1,2,5],[8,9,6]]); 7  B :=  1  8

3  5  6

4 2 9

ȼɵɱɢɫɥɢɦ ɞɟɬɟɪɦɢɧɚɧɬ (ɨɩɪɟɞɟɥɢɬɟɥɶ) ɦɚɬɪɢɰ A ɢ B. > det(A); 0

> det(B); -116

ɋɭɦɦɚ ɦɚɬɪɢɰ > matadd(A,B);

 8   5   15

6 7 17

6  11  15

ɉɪɨɢɡɜɟɞɟɧɢɟ ɦɚɬɪɢɰ > multiply(A, B);

 33   81   129

35 80 125

31  73  115

Ɍɪɚɧɫɩɨɧɢɪɨɜɚɧɢɟ ɦɚɬɪɢɰɵ A > transpose(A);

1  2   3

4 5 6

7  8  9

Ɏɭɧɤɰɢɹ multiply ɭɦɧɨɠɚɟɬ ɦɚɬɪɢɰɵ, ɚ inverse ɧɚɯɨɞɢɬ ɨɛɪɚɬɧɭɸ ɦɚɬɪɢɰɭ.

> multiply(B, inverse(B)); 1  0   0

0 1 0

0  0  1

Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɮɭɧɤɰɢɢ ɩɚɤɟɬɚ student, ɤɨɬɨɪɵɣ ɩɨɡɜɨɥɹɟɬ ɩɪɨɜɨɞɢɬɶ ɜɵɱɢɫɥɟɧɢɹ ɩɨɷɬɚɩɧɨ, ɱɬɨ ɦɨɠɟɬ ɛɵɬɶ ɨɫɨɛɟɧɧɨ ɩɨɥɟɡɧɨ ɫɬɭɞɟɧɬɚɦ. Ɏɭɧɤɰɢɹ intparts – ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ. > with(student): > Int(x*cos(x),x);

⌠ x cos ( x ) d x  ⌡

> intparts(Int(x*cos(x),x),x); x sin ( x ) − ⌠  sin ( x ) d x ⌡

> value(%);

x sin ( x ) + cos ( x )

ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨɞɫɬɚɧɨɜɤɨɣ – ɮɭɧɤɰɢɹ changevar. > Int((cos(x)+1)^3*sin(x), x); ⌠( cos ( x ) + 1 ) 3 sin( x ) dx  ⌡

> changevar(cos(x)+1=u, Int((cos(x)+1)^3*sin(x), x), u); ⌠−u 3 d u  ⌡

> Int(sqrt(1-x^2), x=a...b);

⌠   ⌡a

b

1 − x2 dx

> changevar(x=sin(u), Int(sqrt(1-x^2), x=a...b), u); ⌠ 1 − sin( u ) 2 cos( u ) du   ⌡arcsin( a ) arcsin( b )

Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɩɨɫɥɟɞɧɟɦ ɩɪɢɦɟɪɟ ɩɪɟɞɟɥɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢɡɦɟɧɢɥɢɫɶ ɚɜɬɨɦɚɬɢɱɟɫɤɢ. Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɞɜɨɣɧɵɯ ɢ ɬɪɨɣɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɫɥɭɠɚɬ ɮɭɧɤɰɢɢ Doubleint ɢ Tripleint. > Tripleint((r^2*sin(f),r=0..4*cos(f),f=0..Pi/4, t=0..Pi/2)); π 2

π 4

⌠ ⌠ ⌠    ⌡0 ⌡0 ⌡0

> value(%);

4 cos ( f )

r 2 sin ( f ) d r d f d t

2π

Ɏɭɧɤɰɢɢ simpson ɢ trapezoid ɪɟɚɥɢɡɭɸɬ ɱɢɫɥɨɜɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ɤ ɢɧɬɟɝɪɚɥɭ ɦɟɬɨɞɚɦɢ ɋɢɦɩɫɨɧɚ ɢ ɬɪɚɩɟɰɢɣ. > simpson(x^k*ln(x), x=1..3);

 1 1 1 k 2 2 1 1  3 ln( 3 ) +  ∑  + i  ln + i   +  ∑ ( 1 + i ) k ln( 1 + i )  6 3  i = 1  2   2   3  i = 1  k

> trapezoid(x^k*ln(x), x=1..3);

1  3  i i  1 1 +  ln 1 +   + 3k ln( 3 )  ∑   2  i = 1  2  2   4 k

Ɏɭɧɤɰɢɢ leftbox, middlebox ɢ rightbox ɢɥɥɸɫɬɪɢɪɭɸɬ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɦɟɬɨɞɚɦɢ ɥɟɜɵɯ, ɰɟɧɬɪɚɥɶɧɵɯ ɢ ɩɪɚɜɵɯ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ. > middlebox(x^4*ln(x), x=2..4);

Ɏɭɧɤɰɢɹ showtangent ɫɬɪɨɢɬ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɢ ɤɚɫɚɬɟɥɶɧɭɸ ɜ ɭɤɚɡɚɧɧɨɣ ɬɨɱɤɟ. > showtangent(x^2+5, x = 5);

3 ȾȼɍɏɆȿɊɇȺə ȽɊȺɎɂɄȺ Ɉɫɧɨɜɧɨɣ ɮɭɧɤɰɢɟɣ ɩɨɫɬɪɨɟɧɢɹ ɝɪɚɮɢɤɨɜ ɹɜɥɹɟɬɫɹ plot. > plot(sin(x),x=0..2*Pi);

Ʉɪɨɦɟ ɫɚɦɨɣ ɮɭɧɤɰɢɢ, ɝɪɚɮɢɤ ɤɨɬɨɪɨɣ ɧɭɠɧɨ ɩɨɫɬɪɨɢɬɶ, ɨɛɹɡɚɬɟɥɶɧɵɦ ɩɚɪɚɦɟɬɪɨɦ ɹɜɥɹɟɬɫɹ ɨɛɥɚɫɬɶ. – ɷɬɨ ɨɤɧɨ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɜ ɤɨɬɨɪɨɦ ɫɬɪɨɢɬɫɹ ɝɪɚɮɢɤ. ȿɫɥɢ ɜ ɨɛɥɚɫɬɢ ɡɚɞɚɧ ɬɨɥɶɤɨ ɞɢɚɩɚɡɨɧ ɩɨ x (ɤɚɤ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ), ɬɨ ɞɢɚɩɚɡɨɧ ɩɨ y ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɚɜɬɨɦɚɬɢɱɟɫɤɢ. > plot(sin(x),x=0..2*Pi,y=-0.5..0.5); plot(sin(x),x=0..2*Pi,y=-2..2);

Ɉɛɥɚɫɬɢ ɦɨɠɧɨ ɡɚɞɚɜɚɬɶ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɤɨɧɫɬɚɧɬ, ɜ ɬɨɦ ɱɢɫɥɟ infinity. > plot(exp(-x),x=0..infinity);

Ɏɭɧɤɰɢɹ plot ɦɨɠɟɬ ɢɦɟɬɶ 27 ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ; ɧɟɤɨɬɨɪɵɟ ɢɡ ɧɢɯ ɨɩɢɫɚɧɵ ɧɢɠɟ. ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ ɝɪɚɮɢɤɨɜ ɦɨɠɧɨ ɜɵɛɢɪɚɬɶ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ. ɋɬɢɥɶ ɡɚɞɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɤɥɸɱɟɜɨɝɨ ɫɥɨɜɚ style. ɋɭɳɟɫɬɜɭɸɬ ɬɪɢ ɫɬɢɥɹ: • POINT – ɝɪɚɮɢɤ ɫɬɪɨɢɬɫɹ ɩɨ ɬɨɱɤɚɦ; • LINE – ɬɨɱɤɢ ɫɨɟɞɢɧɹɸɬɫɹ ɩɪɹɦɵɦɢ. ɂɫɩɨɥɶɡɭɟɬɫɹ ɩɨ ɭɦɨɥɱɚɧɢɸ; • PATCH – ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɪɚɫɤɪɚɲɟɧɧɵɯ ɦɧɨɝɨɭɝɨɥɶɧɢɤɨɜ. > plot(sin(x),x=0..2*Pi,style=POINT);

Ɍɢɩ ɥɢɧɢɢ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧ ɩɚɪɚɦɟɬɪɨɦ linestyle. Ⱦɨɫɬɭɩɧɵ ɫɥɟɞɭɸɳɢɟ ɫɬɢɥɢ: • SOLID – ɫɩɥɨɲɧɚɹ ɥɢɧɢɹ; • DOT – ɥɢɧɢɹ ɢɡ ɬɨɱɟɤ; • DASH – ɲɬɪɢɯɨɜɚɹ ɥɢɧɢɹ; • DASHDOT – ɲɬɪɢɯɩɭɧɤɬɢɪɧɚɹ ɥɢɧɢɹ. ɐɜɟɬ ɥɢɧɢɢ ɡɚɞɚɟɬɫɹ ɩɚɪɚɦɟɬɪɨɦ color, ɬɨɥɳɢɧɚ ɥɢɧɢɢ ɩɚɪɚɦɟɬɪɨɦ thickness. > plot(sin(x),x=0..2*Pi,linestyle=DASH, color=blue,thickness=3); ɉɪɢ ɜɵɜɨɞɟ ɝɪɚɮɢɤɨɜ Maple ɜɵɛɢɪɚɟɬ ɦɚɫɲɬɚɛɵ ɩɨ ɨɫɹɦ ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɬɚɤ, ɱɬɨɛɵ ɝɪɚɮɢɤ ɛɵɥ ɧɚɢɛɨɥɟɟ ɢɧɮɨɪɦɚɬɢɜɟɧ, ɧɨ, ɢɫɩɨɥɶɡɭɹ ɩɚɪɚɦɟɬɪ scaling (ɦɚɫɲɬɚɛɢɪɨɜɚɧɢɟ), ɦɨɠɧɨ ɡɚɩɪɟɬɢɬɶ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɪɚɡɥɢɱɧɵɯ ɦɚɫɲɬɚɛɨɜ ɩɨ ɨɫɹɦ, ɤɚɤ ɷɬɨ ɫɞɟɥɚɧɨ ɜ ɫɥɟɞɭɸɳɟɦ ɩɪɢɦɟɪɟ.

> plot(sin(x),x=0..2*Pi,scaling=CONSTRAINED);

ɉɚɪɚɦɟɬɪ coords ɩɨɡɜɨɥɹɟɬ ɜɵɛɪɚɬɶ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ. ɉɨ ɭɦɨɥɱɚɧɢɸ Maple ɫɬɪɨɢɬ ɝɪɚɮɢɤɢ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ. ȼ ɫɥɟɞɭɸɳɟɦ ɩɪɢɦɟɪɟ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ y = x ɩɨɫɬɪɨɟɧ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ. > plot(x,x=0..4*Pi,coords=polar, scaling=CONSTRAINED);

3.1 ɋɈȼɆȿɓȿɇɂȿ ȽɊȺɎɂɄɈȼ Maple ɦɨɠɟɬ ɩɨɫɬɪɨɢɬɶ ɧɟɫɤɨɥɶɤɨ ɝɪɚɮɢɤɨɜ ɧɚ ɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ⱦɥɹ ɷɬɨɝɨ ɞɨɫɬɚɬɨɱɧɨ ɭɤɚɡɚɬɶ ɜ ɮɭɧɤɰɢɢ plot ɦɧɨɠɟɫɬɜɨ ɢɥɢ ɫɩɢɫɨɤ ɮɭɧɤɰɢɣ, ɩɪɢ ɷɬɨɦ ɞɥɹ ɪɚɡɧɵɯ ɝɪɚɮɢɤɨɜ ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɜɵɛɢɪɚɸɬɫɹ ɪɚɡɥɢɱɧɵɟ ɰɜɟɬɚ. ɉɪɢ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɞɥɹ ɤɚɠɞɨɣ ɮɭɧɤɰɢɢ ɦɨɠɧɨ ɭɤɚɡɚɬɶ ɠɟɥɚɟɦɵɣ ɰɜɟɬ ɢ ɫɬɢɥɶ ɩɨɫɬɪɨɟɧɢɹ. > plot([x^2,exp(-x)],x=0..1,color=[blue,violet], linestyle=[DASH,DASHDOT]);

3.2 ȺɇɂɆȺɐɂə ȽɊȺɎɂɄɈȼ

Ɏɭɧɤɰɢɹ animate ɩɨɡɜɨɥɹɟɬ ɫɨɡɞɚɜɚɬɶ ɚɧɢɦɢɪɨɜɚɧɧɵɟ ɢɡɨɛɪɚɠɟɧɢɹ. ɗɬɚ ɮɭɧɤɰɢɹ ɧɚɯɨɞɢɬɫɹ ɜ ɩɚɤɟɬɟ plots, ɤɨɬɨɪɵɣ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɞɨɥɠɟɧ ɛɵɬɶ ɩɨɞɤɥɸɱɟɧ. ɋɭɬɶ ɚɧɢɦɚɰɢɢ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɩɨɫɬɪɨɟɧɢɢ ɫɟɪɢɢ ɢɡɨɛɪɚɠɟɧɢɣ, ɩɪɢɱɟɦ ɤɚɠɞɨɟ ɢɡɨɛɪɚɠɟɧɢɟ (ɮɪɟɣɦ) ɫɜɹɡɚɧɨ ɫ ɢɡɦɟɧɹɟɦɨɣ ɜɨ ɜɪɟɦɟɧɢ ɩɟɪɟɦɟɧɧɨɣ t. > with(plots): animate( sin(x*t),x=-10..10,t=1..2); ɉɪɢ ɜɵɞɟɥɟɧɢɢ ɩɨɥɭɱɟɧɧɨɝɨ ɢɡɨɛɪɚɠɟɧɢɹ ɩɨɹɜɥɹɟɬɫɹ ɩɚɧɟɥɶ ɩɪɨɢɝɪɵɜɚɧɢɹ ɚɧɢɦɚɰɢɨɧɧɵɯ ɤɥɢɩɨɜ. Ɉɧɚ ɢɦɟɟɬ ɤɧɨɩɤɢ ɭɩɪɚɜɥɟɧɢɹ ɫ ɨɛɨɡɧɚɱɟɧɢɹɦɢ, ɩɪɢɧɹɬɵɦɢ ɭ ɦɚɝɧɢɬɨɮɨɧɨɜ.

ɇɚɠɚɜ ɤɧɨɩɤɭ

, ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ ɚɧɢɦɢɪɨɜɚɧɧɨɟ ɢɡɨɛɪɚɠɟɧɢɟ. 3.3 ɉɈɋɌɊɈȿɇɂȿ ȽɊȺɎɂɄȺ ɇȿəȼɇɈɃ ɎɍɇɄɐɂɂ

Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ ɡɚɞɚɧɧɨɣ ɧɟɹɜɧɨ ɫɥɭɠɢɬ ɮɭɧɤɰɢɹ implicitplot ɢɡ ɩɚɤɟɬɚ plots.

> with(plots): implicitplot(2*x^2 + 3*y^2 = 1,x=-1..1,y=-1..1);

ɋ ɩɨɦɨɳɶɸ implicitplot ɦɨɠɧɨ ɬɚɤɠɟ ɫɬɪɨɢɬɶ ɝɪɚɮɢɤɢ ɹɜɧɨ ɡɚɞɚɧɧɵɯ ɮɭɧɤɰɢɣ.

> implicitplot(y = exp(x),x=-1..1,y=-1..3);

ɍɤɚɡɚɜ ɜ ɤɚɱɟɫɬɜɟ ɩɟɪɜɨɝɨ ɩɚɪɚɦɟɬɪɚ ɮɭɧɤɰɢɢ implicitplot ɦɧɨɠɟɫɬɜɨ ɮɭɧɤɰɢɣ, ɦɨɠɧɨ ɫɨɜɦɟɫɬɢɬɶ ɝɪɚɮɢɤɢ ɜ ɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ. > implicitplot({2*x^2 + 3*y^2 = 1,y = exp(x)},

x=-1..1,y=-1..3);

3.4 ɉɈɋɌɊɈȿɇɂȿ ȽɊȺɎɂɄɈȼ ɅɂɇɂəɆɂ ɊȺȼɇɈȽɈ ɍɊɈȼɇə Ʌɢɧɢɢ ɪɚɜɧɨɝɨ ɭɪɨɜɧɹ ɨɛɪɚɡɭɸɬɫɹ, ɟɫɥɢ ɦɵɫɥɟɧɧɨ ɪɚɫɫɟɱɶ ɬɪɟɯɦɟɪɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɩɥɨɫɤɨɫɬɹɦɢ, ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɩɥɨɫɤɨɫɬɢ XY. Ʌɢɧɢɢ ɩɟɪɟɫɟɱɟɧɢɟ ɷɬɢɯ ɩɥɨɫɤɨɫɬɟɣ ɫ ɬɪɟɯɦɟɪɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢ ɹɜɥɹɸɬɫɹ ɥɢɧɢɹɦɢ ɪɚɜɧɨɝɨ ɭɪɨɜɧɹ. Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɬɚɤɢɯ ɝɪɚɮɢɤɨɜ ɢɫɩɨɥɶɡɭɟɬɫɹ ɮɭɧɤɰɢɹ contourplot ɢɡ ɩɚɤɟɬɚ plots. > with(plots): contourplot(sin(x*y),x=-3..3,y=-3..3);

ȼ ɤɚɱɟɫɬɜɟ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧ ɪɚɡɦɟɪ ɫɟɬɤɢ (grid), ɤɨɬɨɪɚɹ ɛɭɞɟɬ ɢɫɩɨɥɶɡɨɜɚɧɚ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɝɪɚɮɢɤɨɜ ɢ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ, ɤɨɬɨɪɵɟ ɨɛɪɚɡɭɸɬ ɥɢɧɢɢ ɪɚɜɧɨɝɨ ɭɪɨɜɧɹ (contours). > contourplot(sin(x*y),x=-3..3,y=-3..3, grid=[50,50], contours=[-1/2,1/4,1/2,3/4]);

Ɉɩɰɢɹ filled = true ɨɛɟɫɩɟɱɢɜɚɟɬ ɨɤɪɚɲɢɜɚɧɢɟ ɡɚɦɤɧɭɬɵɯ ɨɛɥɚɫɬɟɣ, ɨɛɪɚɡɨɜɚɧɧɵɯ ɥɢɧɢɹɦɢ ɪɚɜɧɨɝɨ ɭɪɨɜɧɹ, ɱɬɨ ɩɪɢɞɚɟɬ ɝɪɚɮɢɤɭ ɛɨɥɶɲɭɸ ɜɵɪɚɡɢɬɟɥɶɧɨɫɬɶ. 3.5 ȽɊȺɎɂɄ ɉɅɈɌɇɈɋɌɂ ɂɧɨɝɞɚ ɬɪɟɯɦɟɪɧɵɟ ɩɨɜɟɪɯɧɨɫɬɢ ɧɟɨɛɯɨɞɢɦɨ ɨɬɨɛɪɚɡɢɬɶ ɧɚ ɩɥɨɫɤɨɫɬɢ ɤɚɤ ɝɪɚɮɢɤɢ ɩɥɨɬɧɨɫɬɢ ɨɤɪɚɫɤɢ – ɱɟɦ ɜɵɲɟ ɜɵɫɨɬɚ ɩɨɜɟɪɯɧɨɫɬɢ, ɬɟɦ ɩɥɨɬɧɟɟ ɨɤɪɚɫɤɚ. Ɍɚɤɨɣ ɜɢɞ ɝɪɚɮɢɤɨɜ ɫɨɡɞɚɟɬɫɹ ɮɭɧɤɰɢɟɣ densityplot, ɤɨɬɨɪɚɹ ɬɚɤɠɟ ɜɯɨɞɢɬ ɜ ɩɚɤɟɬ plots. > with(plots): densityplot(sin(x*y),x=-Pi..Pi,y=-Pi..Pi, grid=[25,25]);

ɋɨɡɞɚɜɚɟɦɨɟ ɢɡɨɛɪɚɠɟɧɢɟ ɪɚɡɛɢɬɨ ɧɚ ɩɪɹɦɨɭɝɨɥɶɧɢɤɢ, ɤɨɥɢɱɟɫɬɜɨ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɹɟɬ ɩɚɪɚɦɟɬɪ grid.

3.6 ȽɊȺɎɂɄ ȼȿɄɌɈɊɇɈȽɈ ɉɈɅə ȽɊȺȾɂȿɇɌȺ Ɏɭɧɤɰɢɹ ɩɚɤɟɬɚ plots – gradplot, ɫɥɭɠɢɬ ɞɥɹ ɨɬɨɛɪɚɠɟɧɢɹ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ ɝɪɚɞɢɟɧɬɚ. ɉɚɪɚɦɟɬɪ arrow ɩɨɡɜɨɥɹɟɬ ɢɡɦɟɧɹɬɶ ɮɨɪɦɭ ɫɬɪɟɥɨɤ. > with(plots): gradplot(sin(x*y),x=-1..1,y=-1..1,arrows=THICK);

3.7 ȽɊȺɎɂɄ ȼȿɄɌɈɊɇɈȽɈ ɉɈɅə Ɏɭɧɤɰɢɹ ɩɚɤɟɬɚ plots – fieldplot, ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɨɬɨɛɪɚɠɟɧɢɹ ɩɨɥɟɣ. Ɉɫɨɛɟɧɧɨɫɬɶ ɬɚɤɢɯ ɝɪɚɮɢɤɨɜ ɜ ɬɨɦ, ɱɬɨ ɞɥɹ ɢɯ ɩɨɫɬɪɨɟɧɢɹ ɢɫɩɨɥɶɡɭɸɬɫɹ ɫɬɪɟɥɤɢ, ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɵɯ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɩɪɚɜɥɟɧɢɸ ɢɡɦɟɧɟɧɢɹ ɝɪɚɞɢɟɧɬɚ ɩɨɥɹ, ɚ ɞɥɢɧɚ – ɡɧɚɱɟɧɢɸ ɝɪɚɞɢɟɧɬɚ. > with(plots): fieldplot([x/(x^2+y^2+4)^(1/2), -y/(x^2+y^2+4)^(1/2)], x=-1..1,y=-1..1, arrows=THICK);

3.8 ɋɈȼɆȿɓȿɇɂȿ ȽɊȺɎɂɄɈȼ ɉɈɋɌɊɈȿɇɇɕɏ ɊȺɁɅɂɑɇɕɆɂ ɎɍɇɄɐɂəɆɂ ɑɚɫɬɨ ɛɵɜɚɟɬ ɧɟɨɛɯɨɞɢɦɨ ɫɨɜɦɟɫɬɢɬɶ ɧɚ ɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ ɝɪɚɮɢɤɢ, ɩɨɫɬɪɨɟɧɧɵɟ ɪɚɡɧɵɦɢ ɮɭɧɤɰɢɹɦɢ. ɗɬɨ ɦɨɠɟɬ ɛɵɬɶ ɫɞɟɥɚɧɨ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ display. > with(plots): Ris_1:=gradplot(sin(x*y),x=-Pi..Pi,y=-Pi..Pi, arrows=THICK): Ris_2:=contourplot (sin(x*y),x=-Pi..Pi,y=-Pi..Pi, grid=[50,50],contours=4): display(Ris_1,Ris_2);

Ɏɭɧɤɰɢɹ display ɫ ɩɚɪɚɦɟɬɪɨɦ insequence = true (ɩɨ ɭɦɨɥɱɚɧɢɸ insequence = false) ɨɬɨɛɪɚɠɚɟɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ, ɦɧɨɠɟɫɬɜɚ ɢɥɢ ɫɩɢɫɤɢ ɫ ɝɪɚɮɢɱɟɫɤɢɦɢ ɞɚɧɧɵɦɢ, ɫɨɡɞɚɜɚɹ ɫɬɪɭɤɬɭɪɭ, ɚɧɚɥɨɝɢɱɧɭɸ animate. Ɏɪɟɣɦɵ ɩɨɤɚɡɵɜɚɸɬɫɹ ɞɪɭɝ ɡɚ ɞɪɭɝɨɦ, ɱɬɨ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɷɮɮɟɤɬ ɚɧɢɦɚɰɢɢ. ȼ ɫɥɟɞɭɸɳɟɦ ɩɪɢɦɟɪɟ ɫɨɡɞɚɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ, ɫɨɞɟɪɠɚɳɚɹ ɝɪɚɮɢɱɟɫɤɢɟ ɞɚɧɧɵɟ, ɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɢɯ ɩɨɤɚɡ ɩɨɫɪɟɞɫɬɜɨɦ ɮɭɧɤɰɢɢ display. Ʉɚɠɞɵɣ ɮɪɟɣɦ ɫɨɞɟɪɠɢɬ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ cos(x) ɢ ɪɚɡɥɨɠɟɧɢɟ ɷɬɨɣ ɮɭɧɤɰɢɢ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ n-ɨɣ ɫɬɟɩɟɧɢ ɜ ɬɨɱɤɟ x = 0. Ƚɪɚɮɢɤɢ ɫɬɪɨɹɬɫɹ ɧɚ ɢɧɬɟɪɜɚɥɟ ɨɬ –π ɞɨ π. > f:=cos(x); L:=seq(plot([f,convert(series(f,x=0,n),polynom)], x=-Pi..Pi, y=-1..1, style=[line, point], color=[blue, black],title=cat("n=",n) ), # end plot

n=1..10):

# end seq

plots[display](L, insequence=true); f := cos( x )

ɂɡ-ɡɚ ɧɟɜɨɡɦɨɠɧɨɫɬɢ ɩɟɪɟɞɚɬɶ ɷɮɮɟɤɬ ɚɧɢɦɚɰɢɢ ɧɚ ɛɭɦɚɝɟ, ɩɪɢɜɟɞɟɦ ɬɨɥɶɤɨ ɞɜɚ ɮɪɟɣɦɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɬɟɩɟɧɢ ɪɹɞɚ 3 ɢ 7. Ɉɬɦɟɬɢɦ, ɱɬɨ ɜ ɩɨɫɥɟɞɧɟɣ ɫɬɪɨɤɟ ɩɪɢɦɟɪɚ ɢɫɩɨɥɶɡɨɜɚɧ ɜɵɡɨɜ ɮɭɧɤɰɢɢ display ɢɡ ɩɚɤɟɬɚ plots ɛɟɡ ɡɚɝɪɭɡɤɢ ɜɫɟɯ ɮɭɧɤɰɢɣ ɩɚɤɟɬɚ. 4 ɌɊȿɏɆȿɊɇȺə ȽɊȺɎɂɄȺ Maple ɢɦɟɟɬ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɮɭɧɤɰɢɣ ɬɪɟɯɦɟɪɧɨɣ ɝɪɚɮɢɤɢ. Ɇɧɨɝɢɟ ɮɭɧɤɰɢɢ ɬɪɟɯɦɟɪɧɨɣ ɝɪɚɮɢɤɢ ɚɧɚɥɨɝɢɱɧɵ ɪɚɧɟɟ ɪɚɫɫɦɨɬɪɟɧɧɵɦ ɮɭɧɤɰɢɹ ɞɜɭɯɦɟɪɧɨɣ ɝɪɚɮɢɤɢ. Ɉɫɧɨɜɧɚɹ ɮɭɧɤɰɢɹ ɬɪɟɯɦɟɪɧɨɣ ɝɪɚɮɢɤɢ – plot3d. > plot3d(x*exp(-x^2-y^2),x=-2..2,y=-2..2, grid=[25,25]);

ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɩɚɪɚɦɟɬɪɚ style = CONTOUR ɜ ɮɭɧɤɰɢɢ plot3d ɩɨɡɜɨɥɹɟɬ ɩɨɫɬɪɨɢɬɶ ɥɢɧɢɢ ɪɚɜɧɨɝɨ ɭɪɨɜɧɹ. > plot3d(x*exp(-x^2-y^2),x=-2..2,y=-2..2, grid=[25,25], style=CONTOUR);

Ɏɭɧɤɰɢɹ plot3d ɦɨɠɟɬ ɩɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɡɚɞɚɧɧɨɣ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢ. > plot3d([x*sin(x)*cos(y),x*cos(x)*cos(y),x*sin(y)], x=0..2*Pi,y=0..Pi);

Ɉɫɧɨɜɧɵɟ ɮɭɧɤɰɢɢ ɬɪɟɯɦɟɪɧɨɣ ɝɪɚɮɢɤɢ ɩɚɤɟɬɚ plots

ɭɧɤɰɢɹ

ɇɚɡɧɚɱɟɧɢɟ

contourplot3d

ɋɬɪɨɢɬ ɥɢɧɢɢ ɪɚɜɧɨɝɨ ɭɪɨɜɧɹ. ɂɞɟɧɬɢɱɧɚ ɮɭɧɤɰɢɢ plot3d ɫ ɩɚɪɚɦɟɬɪɨɦ style = CONTOUR

gradplot3d

ɋɬɪɨɢɬ ɬɪɟɯɦɟɪɧɨɟ ɩɨɥɟ ɝɪɚɞɢɟɧɬɚ

fieldplot3d

ɋɬɪɨɢɬ ɬɪɟɯɦɟɪɧɨɟ ɜɟɤɬɨɪɧɨɟ ɩɨɥɟ

implicit-

ɋɬɪɨɢɬ ɧɟɹɜɧɨ ɡɚɞɚɧɧɭɸ ɮɭɧɤɰɢɸ ɬɪɟɯ ɩɟ-

plot3d

ɪɟɦɟɧɧɵɯ

matrixplot

ɋɬɪɨɢɬ ɩɨɜɟɪɯɧɨɫɬɶ, ɡɚɞɚɧɧɭɸ ɬɚɛɥɢɰɟɣ

cylinderplot

ɋɬɪɨɢɬ ɩɨɜɟɪɯɧɨɫɬɶ, ɡɚɞɚɧɧɭɸ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɢɯ ɤɨɨɪɞɢɧɚɬɚɯ

sphereplot

ɋɬɪɨɢɬ ɩɨɜɟɪɯɧɨɫɬɶ, ɡɚɞɚɧɧɭɸ ɜ ɫɮɟɪɢɱɟɫɤɢɯ ɤɨɨɪɞɢɧɚɬɚɯ

spacecurve

ɋɬɪɨɢɬ ɤɪɢɜɭɸ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ. Ʉɪɢɜɚɹ ɞɨɥɠɧɚ ɛɵɬɶ ɡɚɞɚɧɚ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢ

surfdata

ɋɬɪɨɢɬ ɩɨɜɟɪɯɧɨɫɬɶ, ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ ɡɚɞɚɧɧɵɟ ɬɨɱɤɢ

tuberplot

ɋɬɪɨɢɬ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɩɪɟɞɟɥɹɟɦɭɸ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢ ɡɚɞɚɧɧɨɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɪɢɜɨɣ ɢ ɪɚɞɢɭɫɨɦ

display3d

ȼɵɜɨɞɢɬ ɬɪɟɯɦɟɪɧɵɟ ɝɪɚɮɢɱɟɫɤɢɟ ɫɬɪɭɤɬɭɪɵ ɜ ɨɛɳɢɯ ɨɫɹɯ ɤɨɨɪɞɢɧɚɬ ɉɪɢɦɟɪɵ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɬɪɟɯɦɟɪɧɨɣ ɝɪɚɮɢɤɢ

Ɂɚɩɨɥɧɢɦ ɦɚɫɫɢɜ A ɪɚɡɦɟɪɨɦ 8 × 8 ɫɥɭɱɚɣɧɵɦɢ ɱɢɫɥɚɦɢ. ȼɵɜɟɞɟɦ ɟɝɨ ɧɚ ɩɟɱɚɬɶ, ɩɨɫɬɪɨɢɦ ɬɪɟɯɦɟɪɧɭɸ ɝɢɫɬɨɝɪɚɦɦɭ ɢ ɩɨɜɟɪɯɧɨɫɬɶ. > with(plots): rnd:=rand(1..100): A:=array(1..8,1..8,[]): for i from 1 to 8 do for j from 1 to 8 do A[i,j]:= rnd(): end do: end do: print(A); matrixplot(A,heights=histogram,axes=frame); 82  22   17   74    1  96   50   90 

71

98

64

77

39

86

10

56

64

58

61

75

62

8

50

87

99

67

82

75

67

74

43

92

12

39

14

21

45

66

75

10

61

83

93

14

36

62

49

4

24

96

38

58

100

95

29

16

69  86  10  94  92 78  74  56

> S:=[seq([seq([i,j,A[i,j]],j=1..8)],i=1..8)]: surfdata(S,axes=frame);

ȼ ɷɬɨɦ ɩɪɢɦɟɪɟ ɞɜɭɯɦɟɪɧɵɣ ɦɚɫɫɢɜ A ɡɚɩɨɥɧɹɟɬɫɹ ɩɨɫɪɟɞɫɬɜɨɦ ɞɜɭɯ ɜɥɨɠɟɧɧɵɯ ɰɢɤɥɨɜ, ɚ ɮɭɧɤɰɢɹ Maple rand (1 … 100) ɫɨɡɞɚɟɬ ɩɪɨɰɟɞɭɪɭ rnd ( ), ɝɟɧɟɪɢɪɭɸɳɢɟ ɫɥɭɱɚɣɧɵɟ ɱɢɫɥɚ ɜ ɞɢɚɩɚɡɨɧɟ ɨɬ 1 ɞɨ 100. Ɏɭɧɤɰɢɹ matrixplot ɫ ɩɚɪɚɦɟɬɪɨɦ heights = histogram ɫɬɪɨɢɬ ɬɪɟɯɦɟɪɧɭɸ ɝɢɫɬɨɝɪɚɦɦɭ, ɚ surfdata – ɩɨɜɟɪɯɧɨɫɬɶ ɩɨ ɬɨɱɤɚɦ, ɧɚɯɨɞɹɳɢɦɫɹ ɜ ɫɩɢɫɤɟ S. ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɮɭɧɤɰɢɢ cylinderplot. > with(plots): cylinderplot(1,theta=0..2*Pi,z=-1..1, axes=frame);

ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɮɭɧɤɰɢɢ sphereplot. > with(plots): sphereplot(1,theta=0..2*Pi,phi=0..Pi,axes=frame); sphereplot((1.3)^z*sin(theta),z=-1..2*Pi, theta=0..Pi, style=patch,color=z,axes=frame);

5 Ɋȿɒȿɇɂȿ ɍɊȺȼɇȿɇɂɃ, ɋɂɋɌȿɆ ɍɊȺȼɇȿɇɂɃ ɂ ɇȿɊȺȼȿɇɋɌȼ Ⱦɥɹ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ ɥɢɧɟɣɧɵɯ ɢ ɧɟɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢ ɫɢɫɬɟɦ ɫɥɭɠɢɬ ɮɭɧɤɰɢɹ solve, ɧɚɩɪɢɦɟɪ: > solve(a*x^2+b*x+c=0,x); −b +

b 2 − 4 a c −b − b 2 − 4 a c , 2a 2a

.

ȼ ɤɚɱɟɫɬɜɟ ɩɟɪɜɨɝɨ ɩɚɪɚɦɟɬɪɚ ɡɚɩɢɫɚɧɨ ɭɪɚɜɧɟɧɢɟ, ɚ ɜɬɨɪɨɝɨ – ɩɟɪɟɦɟɧɧɚɹ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɭɪɚɜɧɟɧɢɟ ɫɥɟɞɭɟɬ ɪɟɲɚɬɶ. ȿɫɥɢ ɩɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɚ ɧɭɥɸ, ɬɨ ɡɧɚɤ ɪɚɜɟɧɫɬɜɚ ɢ ɧɭɥɶ ɦɨɝɭɬ ɛɵɬɶ ɨɩɭɳɟɧɵ. > solve(a*x^2+b*x+c,x);

−b +

b 2 − 4 a c −b − b 2 − 4 a c , 2a 2a

ȿɫɥɢ ɧɚɣɞɟɧɨ ɧɟɫɤɨɥɶɤɨ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ, ɬɨ ɤɨɪɧɢ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɜɢɞɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ. Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɪɟɲɟɧɢɟ ɞɥɹ ɧɟɪɚɜɟɧɫɬɜɚ. > solve(x^2+x>5,x);  1  RealRange −∞, Open  − −   2

21 2

   1  , RealRange Open  − +    2

21 2

  , ∞   

Open – ɨɬɤɪɵɬɵɣ ɞɢɚɩɚɡɨɧ, ɬ.ɟ. ɭɤɚɡɚɧɧɨɟ ɜ ɫɤɨɛɤɚɯ ɡɧɚɱɟɧɢɟ ɜ ɧɟɝɨ ɧɟ ɜɯɨɞɢɬ. ȿɫɥɢ ɜ ɤɚɱɟɫɬɜɟ ɩɟɪɜɨɝɨ ɩɚɪɚɦɟɬɪɚ ɮɭɧɤɰɢɢ solve ɛɭɞɟɬ ɦɧɨɠɟɫɬɜɨ, ɫɨɫɬɨɹɳɟɟ ɢɡ ɭɪɚɜɧɟɧɢɣ, ɬɨ Maple ɛɭɞɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɷɬɨ ɦɧɨɠɟɫɬɜɨ ɤɚɤ ɫɢɫɬɟɦɭ. Ɋɟɲɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ > solve({x+5*y+z=1,2*x-y+4*z=4,x+2*y+2*z=12}, {x,y,z}); { z = 23, x = -42, y = 4 }

Ɋɚɫɫɦɨɬɪɢɦ ɟɳɟ ɨɞɢɧ ɩɪɢɦɟɪ. Ɋɟɲɢɦ ɫɢɫɬɟɦɭ ɧɟɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ  y = 2 x 2 ,  2  x + y 2 = 1

ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɜ ɪɟɲɟɧɢɟ ɝɪɚɮɢɱɟɫɤɢ. > plots[implicitplot]({y=x^2,x^2+y^2=1},x=-1..1,

y=-1..1);

> solve({y=x^2,x^2+y^2=1},{x,y});

{ x = RootOf ( −RootOf ( _Z + _Z 2 − 1, label = _L1 ) + _Z 2, label = _L2 ), y = RootOf ( _Z + _Z 2 − 1, label = _L1 ) }

ȼ ɪɟɲɟɧɢɢ ɩɪɢɫɭɬɫɬɜɭɟɬ ɜɵɪɚɠɟɧɢɟ RootOf, ɨɡɧɚɱɚɸɳɟɟ, ɱɬɨ ɪɟɲɟɧɢɟ ɩɨɥɭɱɟɧɨ ɜ ɧɟɹɜɧɨɣ ɮɨɪɦɟ. Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɪɟɲɟɧɢɹ ɜ ɹɜɧɨɣ ɮɨɪɦɟ ɫɥɟɞɭɟɬ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɮɭɧɤɰɢɟɣ allvalues. > allvalues(%); {y = −

1 5 ,x= + 2 2

{y = −

1 5 ,x= − 2 2

−2 + 2 2

−2 − 2 2

5

}, { y = −

5

}, { y = −

1 5 ,x= − + 2 2

1 5 − ,x=− 2 2

−2 + 2 5 }, 2

−2 − 2 5 } 2

> evalf(%);

{ y = 0.6180339880 , x = 0.7861513775 } , { y = 0.6180339880 , x = -0.7861513775 }, { y = -1.618033988 , x = 1.272019650 I },

{ y = -1.618033988, x = -1.272019650 I }

ɉɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɨɥɭɱɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɤ ɜɢɞɭ ɫ ɩɥɚɜɚɸɳɟɣ ɬɨɱɤɨɣ ɫɬɚɥɨ ɨɱɟɜɢɞɧɵɦ, ɱɬɨ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɞɜɚ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɹ (ɢɯ ɜɢɞɧɨ ɧɚ ɝɪɚɮɢɤɟ) ɢ ɞɜɚ ɤɨɦɩɥɟɤɫɧɵɯ. ȿɫɥɢ ɩɨ ɤɚɤɢɦ-ɥɢɛɨ ɩɪɢɱɢɧɚɦ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ solve ɧɟ ɭɞɚɥɨɫɶ ɧɚɣɬɢ ɪɟɲɟɧɢɟ, ɬɨ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɮɭɧɤɰɢɸ fsolve ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɪɟɲɟɧɢɹ ɱɢɫɥɟɧɧɵɦ ɦɟɬɨɞɨɦ. Ɋɟɲɢɦ ɭɪɚɜɧɟɧɢɟ cos( x) −

ɮɭɧɤɰɢɣ y = cos(x) ɢ y =

x+2 . x−2

x+2 = 0 . ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ ɞɥɹ ɜɵɹɫɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɤɨɪɧɟɣ ɩɨɫɬɪɨɢɦ ɝɪɚɮɢɤɢ x−2

> plot({cos(x),(x+2)/(x-2)}, x=-6*Pi..4*Pi, y=-2..2,color=[red, blue]);

Ʉɚɤ ɜɢɞɧɨ ɢɡ ɝɪɚɮɢɤɚ, ɝɢɩɟɪɛɨɥɚ y =

x+2 x−2

ɢɦɟɟɬ ɜɟɪɬɢɤɚɥɶɧɭɸ ɚɫɢɦɩɬɨɬɭ x = 2 ɢ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ

– y = 1. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɟɞɥɨɠɟɧɧɨɟ ɞɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɤɨɪɧɟɣ ɜ ɞɢɚɩɚɡɨɧɟ ɨɬ ɧɭɥɹ ɞɨ -∞. ɉɪɨɛɭɟɦ ɪɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ ɩɪɢ ɩɨɦɨɳɢ ɮɭɧɤɰɢɢ fsolve. > fsolve(cos(x)-(x+2)/(x-2),x); -1.662944360

ɇɚɣɞɟɧ ɛɥɢɠɚɣɲɢɣ ɤ ɧɭɥɸ ɤɨɪɟɧɶ. Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɧɚɣɬɢ ɫɥɟɞɭɸɳɢɣ ɤɨɪɟɧɶ ɮɭɧɤɰɢɢ fsolve ɧɟɨɛɯɨɞɢɦɨ ɭɤɚɡɚɬɶ ɢɧɬɟɪɜɚɥ ɞɥɹ ɩɨɢɫɤɚ, ɩɪɢ ɷɬɨɦ ɤɪɚɣɧɟ ɠɟɥɚɬɟɥɶɧɨ ɱɬɨɛɵ ɧɚ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ ɛɵɥ ɬɨɥɶɤɨ ɨɞɢɧ ɤɨɪɟɧɶ. ɂɬɚɤ, ɧɚɣɞɟɦ ɜɬɨɪɨɣ ɤɨɪɟɧɶ. > fsolve(cos(x)-(x+2)/(x-2),x=-6..-4); -5.170382990

6 ɍɉɊȺȼɅəɘɓɂȿ ɄɈɇɋɌɊɍɄɐɂɂ Ʌɸɛɨɣ ɚɥɝɨɪɢɬɦ ɦɨɠɧɨ ɪɟɚɥɢɡɨɜɚɬɶ, ɢɫɩɨɥɶɡɭɹ ɥɢɲɶ ɬɪɢ ɭɩɪɚɜɥɹɸɳɢɟ ɤɨɧɫɬɪɭɤɰɢɢ: ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɟ ɜɵɩɨɥɧɟɧɢɟ, ɜɟɬɜɥɟɧɢɟ ɢ ɰɢɤɥ. Ɉɩɟɪɚɬɨɪ ɜɟɬɜɥɟɧɢɟ ɦɨɠɧɨ ɢɡɨɛɪɚɡɢɬɶ ɫɥɟɞɭɸɳɟɣ ɫɢɧɬɚɤɫɢɱɟɫɤɨɣ ɞɢɚɝɪɚɦɦɨɣ: if then [ elif then < ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɩɟɪɚɬɨɪɨɜ > ] [ else < ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɩɟɪɚɬɨɪɨɜ > ] end if ȼɵɪɚɠɟɧɢɟ elif ɫɥɟɞɭɟɬ ɩɨɧɢɦɚɬɶ ɤɚɤ else-if, ɬ.ɟ. «ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɩɪɨɜɟɪɢɬɶ ɫɥɟɞɭɸɳɟɟ ɭɫɥɨɜɢɟ». ȼ ɤɜɚɞɪɚɬɧɵɯ ɫɤɨɛɤɚɯ – ɧɟɨɛɹɡɚɬɟɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ. ɉɨɤɚɠɟɦ, ɤɚɤ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɤɨɪɧɢ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɨɩɟɪɚɬɨɪɚ ɜɟɬɜɥɟɧɢɹ if: > a:=2;b:=6;c:=1; d:=b^2-4*a*c; if d>0 then (-b+sqrt(d))/2/a,(-b-sqrt(d))/2/a elif d=0 then -b/2/a else print(`Ⱦɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɧɟɬ !!!`) end if; a := 2 b := 6 c := 1 d := 28 3 7 3 7 − + ,− − 2 2 2 2

Ɉɩɟɪɚɬɨɪ ɜɟɬɜɥɟɧɢɹ ɦɨɠɧɨ ɡɚɞɚɜɚɬɶ ɜ ɮɨɪɦɟ ɮɭɧɤɰɢɢ, ɩɪɢ ɷɬɨɦ if ɞɨɥɠɧɨ ɛɵɬɶ ɡɚɤɥɸɱɟɧɨ ɜ ɨɛɪɚɬɧɵɟ ɤɚɜɵɱɤɢ: `if `(ɭɫɥɨɜɢɟ, ɢɫɬɢɧɧɨɟ ɜɵɪɚɠɟɧɢɟ, ɥɨɠɧɨɟ ɜɵɪɚɠɟɧɢɟ) ɇɚɩɪɢɦɟɪ, > d:=4; `IF`(D>=0, `ȿɋɌɖ ȾȿɃɋɌȼɂɌȿɅɖɇɕȿ ɄɈɊɇɂ.`, `ȾȿɃɋɌȼɂɌȿɅɖɇɕɏ ɄɈɊɇȿɃ ɇȿɌ !!!`); d := 4 .

> d:=-2; `if`(d>=0, `ȿɫɬɶ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɤɨɪɧɢ.`, `Ⱦɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɧɟɬ !!!`); d := -2 !!!

ɐɢɤɥɵ ɜ Maple ɦɨɠɧɨ ɡɚɞɚɜɚɬɶ ɞɜɭɯ ɬɢɩɨɜ: for-to ɢ while. [ for ] [ from ] [ by < ɜɵɪɚɠɟɧɢɟ > ] [ to < ɜɵɪɚɠɟɧɢɟ > ] [ while < ɜɵɪɚɠɟɧɢɟ > ] do < ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɩɟɪɚɬɨɪɨɜ > end do; ɢɥɢ [ for ] [ in < ɜɵɪɚɠɟɧɢɟ > ] [ while < ɜɵɪɚɠɟɧɢɟ > ] do < ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɩɟɪɚɬɨɪɨɜ > end do; ɉɨ ɭɦɨɥɱɚɧɢɸ ɡɧɚɱɟɧɢɹ ɜɵɪɚɠɟɧɢɣ from ɢ by ɪɚɜɧɵ ɟɞɢɧɢɰɟ. ɉɪɢɜɟɞɟɦ ɩɪɢɦɟɪɵ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɰɢɤɥɨɜ. ȼ ɫɥɟɞɭɸɳɟɦ ɩɪɢɦɟɪɟ ɨɪɝɚɧɢɡɨɜɚɧ ɰɢɤɥ ɩɨ ɩɟɪɟɦɟɧɧɨɣ k ɨɬ ɧɭɥɹ ɞɨ 3 ɫ ɲɚɝɨɦ 0,5. > for k from 0 to 3 by 0.5 do print(k) end do; 0 0.5 1.0 1.5 2.0 2.5 3.0

ȼɵɲɟ, ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɮɭɧɤɰɢɢ fsolve, ɛɵɥɢ ɧɚɣɞɟɧɵ ɞɜɚ ɛɥɢɠɚɣɲɢɯ ɤ ɧɭɥɸ ɤɨɪɧɹ ɭɪɚɜɧɟ-

ɧɢɹ cos( x) −

x+2 = 0. x−2

ɉɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ ɩɟɪɢɨɞ ɮɭɧɤɰɢɢ cos(x) ɧɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɜ ɤɚɠɞɵɣ ɢɧ-

ɬɟɪɜɚɥ [–i⋅π, –i⋅π + π], ɝɞɟ i = 1, 2, 3, 4, …, n, ɩɨɩɚɞɚɟɬ ɬɨɥɶɤɨ ɨɞɢɧ ɤɨɪɟɧɶ. ɋɥɟɞɭɸɳɢɣ ɩɪɢɦɟɪ ɞɟɦɨɧɫɬɪɢɪɭɟɬ ɧɚɯɨɠɞɟɧɢɟ n ɩɟɪɜɵɯ ɤɨɪɧɟɣ, ɭɤɚɡɚɧɧɨɝɨ ɜɵɲɟ ɭɪɚɜɧɟɧɢɹ. > n:=10; n := 10

> for i from 1 to n do fsolve(cos(x)-(x+2)/(x-2),x=-i*Pi..-i*Pi+Pi); end do; -1.662944360 -5.170382990 -7.250409918 -11.78482522 -13.30607789 -18.20951859 -19.46987598

-24.57696992 -25.67706817 -30.91781263

ɋɥɟɞɭɸɳɢɟ ɩɪɢɦɟɪɵ ɞɟɦɨɧɫɬɪɢɪɭɟɬ ɜɵɱɢɫɥɟɧɢɟ n! ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɰɢɤɥɚ for. > n:=7; z:=1: for x in seq(i,i=1..n) do z:=z*x; if x=n then print(z) end if end do: n := 7 5040

> n:=7; z:=1: for x while (x for i do if (i > 10) then break end if; if (i mod 2)=0 then next end if; print(i) end do; 1 3 5 7 9

7 ɉɊɈɐȿȾɍɊɕ ɂ ɎɍɇɄɐɂɂ ɉɪɨɰɟɞɭɪɵ ɹɜɥɹɸɬɫɹ ɜɚɠɧɵɦ ɷɥɟɦɟɧɬɨɦ ɫɬɪɭɤɬɭɪɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɢ ɫɥɭɠɚɬ ɫɪɟɞɫɬɜɨɦ ɪɚɫɲɢɪɟɧɢɹ ɜɨɡɦɨɠɧɨɫɬɟɣ Maple ɩɨɥɶɡɨɜɚɬɟɥɟɦ. ɉɪɨɰɟɞɭɪɵ ɢɦɟɸɬ ɢɦɹ ɢ ɫɩɢɫɨɤ ɩɚɪɚɦɟɬɪɨɜ, ɞɚɠɟ ɟɫɥɢ ɨɧ ɩɭɫɬɨɣ. ɉɪɨɰɟɞɭɪɵ ɜɵɡɵɜɚɸɬɫɹ, ɬɚɤɠɟ ɤɚɤ ɜɫɬɪɨɟɧɧɵɟ ɮɭɧɤɰɢɢ, ɭɤɚɡɚɧɢɟɦ ɢɯ ɢɦɟɧɢ ɫɨ ɫɩɢɫɤɨɦ ɮɚɤɬɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ. Ɉɛɳɚɹ ɮɨɪɦɚ ɡɚɞɚɧɢɹ ɩɪɨɰɟɞɭɪɵ: proc ( ɮɨɪɦɚɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ) local ɥɨɤɚɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ; global ɝɥɨɛɚɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ; options ɪɚɫɲɢɪɹɸɳɢɟ ɤɥɸɱɢ; description ɤɨɦɦɟɧɬɚɪɢɢ; ɬɟɥɨ ɩɪɨɰɟɞɭɪɵ end proc

ɋ ɩɨɦɨɳɶɸ ɡɧɚɤɚ :: ɩɨɫɥɟ ɢɦɟɧɢ ɩɟɪɟɦɟɧɧɨɣ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɟɟ ɬɢɩ. ɇɟɫɨɨɬɜɟɬɫɬɜɢɟ ɮɚɤɬɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɬɢɩɭ ɡɚɞɚɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɜɟɞɟɬ ɤ ɫɨɨɛɳɟɧɢɸ ɨɛ ɨɲɢɛɤɟ ɢ ɤ ɨɬɤɚɡɭ ɨɬ ɜɵɩɨɥɧɟɧɢɹ ɩɪɨɰɟɞɭɪɵ. ɉɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɜɟɬɜɥɟɧɢɹ, ɛɵɥɨ ɩɪɟɞɥɨɠɟɧɨ ɪɟɲɟɧɢɟ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. Ɉɮɨɪɦɢɦ ɷɬɨ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ ɩɪɨɰɟɞɭɪɵ. Ɉɛɴɹɜɢɦ ɮɚɤɬɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɚɥɝɟɛɪɚɢɱɟɫɤɢɦ ɬɢɩɨɦ, ɜ ɤɨɬɨɪɵɣ ɜɯɨɞɹɬ ɰɟɥɵɟ, ɞɪɨɛɧɵɟ ɢ ɱɢɫɥɚ ɫ ɩɥɚɜɚɸɳɟɣ ɬɨɱɤɨɣ. > A:=proc(a::algebraic,b::algebraic,c::algebraic) local d; description "Ɋɟɲɟɧɢɟ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ"; d:=b^2-4*a*c; if d>0 then (-b+sqrt(d))/2/a,(-b-sqrt(d))/2/a elif d=0 then -b/2/a else print(`Ⱦɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɧɟɬ !!!`) end if; end proc; A := proc(a::algebraic, b::algebraic, c::algebraic) elif d = 0 then −1/2×b/a local d; else print( ` description "Ɋɟɲɟɧɢɟ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ" ; d := b^2 − 4×a×c; end if if 0 < d then 1/2×( −b + sqrt( d ) )/a, 1/2×( −b − sqrt( d ) )/a end proc

> A(1,12,3);

!!!`

)

−6 + 33 , −6 − 33

> A(1,8.5,3); ,

-0.3689563260 , -8.131043675

> A(12,4,2); !!!

ȿɫɥɢ ɜ ɤɚɱɟɫɬɜɟ ɩɚɪɚɦɟɬɪɚ ɛɭɞɟɬ ɩɨɞɫɬɚɜɥɟɧɨ ɡɧɚɱɟɧɢɟ ɧɟɞɨɩɭɫɬɢɦɨɝɨ ɬɢɩɚ, ɷɬɨ ɩɪɢɜɟɞɟɬ ɤ ɨɲɢɛɤɟ ɢ ɫɨɨɛɳɟɧɢɸ ɨ ɧɟɞɨɩɭɫɬɢɦɨɦ ɬɢɩɟ ɩɚɪɚɦɟɬɪɚ. > A(12,"4",6); Error, invalid input: A expects its 2nd argument, b, to be of type algebraic, but received 4 ȿɫɥɢ ɜ ɬɟɥɟ ɩɪɨɰɟɞɭɪɵ ɢɦɟɸɬɫɹ ɨɩɟɪɚɰɢɢ ɩɪɢɫɜɨɟɧɢɹ ɞɥɹ ɪɚɧɟɟ ɨɩɪɟɞɟɥɟɧɧɵɯ (ɝɥɨɛɚɥɶɧɵɯ) ɩɟɪɟɦɟɧɧɵɯ, ɬɨ ɢɡɦɟɧɟɧɢɟ ɢɯ ɡɧɚɱɟɧɢɣ ɜ ɯɨɞɟ ɜɵɩɨɥɧɟɧɢɹ ɩɪɨɰɟɞɭɪɵ ɫɨɡɞɚɟɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ . Ɉɧ ɫɩɨɫɨɛɟɧ ɫɭɳɟɫɬɜɟɧɧɨ ɢɡɦɟɧɢɬɶ ɚɥɝɨɪɢɬɦ ɪɟɲɟɧɢɹ ɡɚɞɚɱ, ɢ ɩɨɷɬɨɦɭ, ɤɚɤ ɩɪɚɜɢɥɨ, ɧɟɞɨɩɭɫɬɢɦ. ȼɫɬɪɟɱɚɹ ɬɚɤɢɟ ɨɩɟɪɚɰɢɢ ɩɪɢɫɜɨɟɧɢɹ, Maple ɤɨɪɪɟɤɬɢɪɭɟɬ ɬɟɤɫɬ ɩɪɨɰɟɞɭɪɵ, ɞɨɛɚɜɥɹɹ ɜ ɧɟɟ ɨɛɴɹɜɥɟɧɢɟ ɩɟɪɟɦɟɧɧɵɯ ɫ ɩɨɦɨɳɶɸ local. ɉɪɢ ɷɬɨɦ ɜɵɞɚɟɬɫɹ ɩɪɟɞɭɩɪɟɠɞɟɧɢɟ ɜɢɞɚ: Warning, `d` is implicitly declared local to procedure `A` ȿɫɥɢ ɜɫɟ-ɬɚɤɢ ɪɚɛɨɬɚ ɫ ɝɥɨɛɚɥɶɧɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ɜɧɭɬɪɢ ɩɪɨɰɟɞɭɪɵ ɧɟɨɛɯɨɞɢɦɚ, ɬɨ ɷɬɢ ɩɟɪɟɦɟɧɧɵɟ ɞɨɥɠɧɵ ɛɵɬɶ ɨɛɴɹɜɥɟɧɵ ɜ ɩɪɨɰɟɞɭɪɟ ɫ ɩɨɦɨɳɶɸ global. ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɧɟɥɶɡɹ ɞɟɥɚɬɶ ɝɥɨɛɚɥɶɧɵɦɢ ɩɟɪɟɦɟɧɧɵɟ, ɭɤɚɡɚɧɧɵɟ ɜ ɫɩɢɫɤɟ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɞɭɪɵ, ɩɨɫɤɨɥɶɤɭ ɨɧɢ ɭɠɟ ɮɚɤɬɢɱɟɫɤɢ ɨɛɴɹɜɥɟɧɵ ɥɨɤɚɥɶɧɵɦɢ. Ɍɚɤɚɹ ɩɨɩɵɬɤɚ ɩɪɢɜɟɞɟɬ ɤ ɩɨɹɜɥɟɧɢɸ ɫɨɨɛɳɟɧɢɹ ɨɛ ɨɲɢɛɤɟ. Ɋɚɫɲɢɪɹɸɳɢɟ ɤɥɸɱɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɞɟɬɚɥɶɧɨɣ ɧɚɫɬɪɨɣɤɢ ɩɪɨɰɟɞɭɪɵ. Ʉɥɸɱ operator ɨɩɪɟɞɟɥɹɟɬ, ɱɬɨ c ɩɪɨɰɟɞɭɪɨɣ ɦɨɠɧɨ ɪɚɛɨɬɚɬɶ ɤɚɤ ɫ ɨɩɟɪɚɬɨɪɨɦ, ɚ ɤɥɸɱ arrow ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɨɩɟɪɚɬɨɪ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɫɬɪɟɥɨɱɧɨɣ ɧɨɬɚɰɢɢ. ɉɨɤɚɠɟɦ ɧɚ ɩɪɢɦɟɪɟ: > f:=proc(x) option operator, arrow; sin(x)+cos(x) end;

f := x → sin( x ) + cos( x )

ɉɨɫɥɟɞɧɹɹ ɩɪɨɰɟɞɭɪɚ ɷɤɜɢɜɚɥɟɧɬɧɚ ɡɚɩɢɫɢ:

> g:=x->sin(x)+cos(x);

g := x → sin( x ) + cos( x )

ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ f(x) ɢ g(x) ɜ ɬɨɱɤɟ x=0,5: > f(0.5); 1.357008100

> g(0.5); 1.357008100

ɉɪɨɜɟɪɢɦ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɶ f(x) ɢ g(x): > evalb(f(x)=g(x)); true

Ɍɚɤɭɸ ɮɨɪɦɭ ɩɪɨɰɟɞɭɪɵ ɢɧɨɝɞɚ ɧɚɡɵɜɚɸɬ ɩɪɨɰɟɞɭɪɭ-ɮɭɧɤɰɢɸ ɫ ɞɜɭɦɹ ɩɚɪɚɦɟɬɪɚɦɢ. > z:=(x,y)->x^2+y^2;

-

. ɋɥɟɞɭɸɳɢɣ ɩɪɢɦɟɪ ɞɟɦɨɧɫɬɪɢɪɭɟɬ

z := ( x, y ) → x 2 + y 2

> z(1.2,3.5); 13.69

ɉɪɨɰɟɞɭɪɚ ɮɭɧɤɰɢɹ ɦɨɠɟɬ ɧɟ ɢɦɟɬɶ ɩɚɪɚɦɟɬɪɨɜ, ɧɨ ɩɪɢ ɟɟ ɜɵɡɨɜɟ ɨɛɹɡɚɬɟɥɶɧɨ ɫɨɯɪɚɧɹɟɬɫɹ ɩɚɪɚ ɫɤɨɛɨɤ. > e:=()->evalf(exp(1));

e := ( ) → evalf( e )

> e; e

> e(); 2.718281828

> ln(e()); 0.9999999998

> ln(exp(1)); 1

Ɏɭɧɤɰɢɹ unapply ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɤɨɧɫɬɪɭɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɢɡ ɜɵɪɚɠɟɧɢɣ. > p := x^2 + sin(x) + 1;

p := x 2 + sin ( x ) + 1

> f := unapply(p,x);

f := x → x 2 + sin ( x ) + 1

π2 π + sin  + 1 144 12  

> f(Pi/12);

Ⱦɥɹ ɡɚɞɚɧɢɹ ɤɭɫɨɱɧɨɣ ɮɭɧɤɰɢɢ ɜ Maple ɢɦɟɟɬɫɹ ɫɩɟɰɢɚɥɶɧɨɟ ɫɪɟɞɫɬɜɨ – ɮɭɧɤɰɢɹ piecewise, ɤɨɬɨɪɚɹ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɟ ɩɚɪɚɦɟɬɪɵ: piecewise(ɢɧɬɟɪɜɚɥ_1, ɜɵɪɚɠɟɧɢɟ_1, ɢɧɬɟɪɜɚɥ_2, ɜɵɪɚɠɟɧɢɟ_2, ..., ɢɧɬɟɪɜɚɥ_n, ɜɵɪɚɠɟɧɢɟ_n [, ɜɵɪɚɠɟɧɢɟ]) ɉɨɫɥɟɞɧɟɟ ɧɟɨɛɹɡɚɬɟɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɡɚɞɚɟɬ ɮɭɧɤɰɢɸ ɞɥɹ ɧɟɨɯɜɚɱɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ. > p :=x-> piecewise(x0,x);

p := x → piecewise( x < 0, −x, 0 < x, x )

> plot(p(x),x=-1..1,scaling=constrained);

Ʉɭɫɨɱɧɵɟ ɮɭɧɤɰɢɢ ɢɧɬɟɝɪɢɪɭɸɬɫɹ ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɸɬɫɹ ɬɟɦɟ ɠɟ ɫɪɟɞɫɬɜɚɦɢ, ɱɬɨ ɢ ɨɛɵɱɧɵɟ ɮɭɧɤɰɢɢ. ȼ ɫɥɟɞɭɸɳɟɦ ɩɪɢɦɟɪɟ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɨɛɴɟɦɚ ɢ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɜɨɤɪɭɝ ɨɫɢ z ɤɭɫɨɱɧɨɣ ɮɭɧɤɰɢɢ f (z), ɢɫɩɨɥɶɡɨɜɚɥɢɫɶ ɢɡɜɟɫɬɧɵɟ ɜɵɪɚɠɟɧɢɹ: V = π∫ ( f ( z ) )2 dz b

a

ɢ

 df ( z )  S = 2π ∫ f ( z ) 1 +   dz .  dz  a 2

b

> f := z->piecewise( z>-2 and z=-1 and z1 and z plot(f(z),z=-2..2,scaling=constrained);

> plots[cylinderplot](f(z),theta=0..2*Pi,z=-2..2, axes=box);

> V:=Pi*int(f(z)^2,z=-2..2); V :=

> S:=2*Pi*int(f(z)*sqrt(1+diff(f(z),z)^2), z=-2..2);

10 π 3

S := 8 π

8 Ɋȿɒȿɇɂȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ ȼ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɧɚɭɱɧɭɸ ɢɥɢ ɬɟɯɧɢɱɟɫɤɭɸ ɩɪɨɛɥɟɦɭ ɦɨɠɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɢ, ɧɚɢɛɨɥɟɟ ɜɟɪɨɹɬɧɨ, ɱɬɨ ɡɚɞɚɱɚ ɫɜɟɞɟɬɫɹ ɤ ɨɞɧɨɦɭ ɢɥɢ ɧɟɫɤɨɥɶɤɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɹɦ. ɗɬɨ ɜɫɟɝɞɚ ɢɦɟɟɬ ɦɟɫɬɨ ɞɥɹ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ ɩɪɨɛɥɟɦ, ɫɜɹɡɚɧɧɵɯ ɫ ɫɢɥɚɦɢ ɢ ɞɜɢɠɟɧɢɟɦ. ȼ ɝɢɞɪɨ- ɢ ɚɷɪɨɞɢɧɚɦɢɤɟ, ɬɟɩɥɨɬɟɯɧɢɤɟ, ɪɚɞɢɨɬɟɯɧɢɤɟ ɢ ɦɧɨɝɢɯ ɞɪɭɝɢɯ ɨɛɥɚɫɬɹɯ ɧɚɭɤɢ ɢ ɬɟɯɧɢɤɢ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɡɚɞɚɱ ɫɜɨɞɢɬɫɹ ɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɹɦ. Ɉɞɧɚɤɨ, ɧɟɫɦɨɬɪɹ ɧɚ ɛɨɥɶɲɢɟ

ɭɫɢɥɢɹ, ɤɨɬɨɪɵɟ ɛɨɥɟɟ ɞɜɭɯ ɫɬɨɥɟɬɢɣ ɩɪɢɥɚɝɚɸɬ ɦɧɨɝɢɟ ɦɚɬɟɦɚɬɢɤɢ ɦɢɪɚ, ɱɢɫɥɨ ɬɢɩɨɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɪɚɡɪɟɲɟɧɧɵɯ ɜ ɡɚɦɤɧɭɬɨɦ ɜɢɞɟ ɢɥɢ ɜ ɤɜɚɞɪɚɬɭɪɚɯ, ɨɫɬɚɟɬɫɹ ɨɱɟɧɶ ɨɝɪɚɧɢɱɟɧɧɵɦ. ɉɨɷɬɨɦɭ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɫɭɳɟɫɬɜɭɟɬ ɦɧɨɠɟɫɬɜɨ ɩɪɨɛɥɟɦ, ɬɨɱɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɯ ɜ ɜɢɞɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɪɟɲɟɧɢɟ ɤɨɬɨɪɵɯ ɟɳɟ ɧɟ ɧɚɣɞɟɧɵ. ȼɫɟ ɷɬɨ ɩɪɢɜɟɥɨ ɤ ɬɨɦɭ, ɱɬɨ ɧɚɪɹɞɭ ɫ ɚɧɚɥɢɬɢɱɟɫɤɢɦɢ ɢ ɩɪɢɛɥɢɠɟɧɧɵɦɢ ɦɟɬɨɞɚɦɢ ɧɚɱɚɥɢ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɬɶɫɹ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɪɨɥɶ ɤɨɬɨɪɵɯ ɨɫɨɛɟɧɧɨ ɜɨɡɪɨɫɥɚ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɗȼɆ. Maple ɢɦɟɟɬ ɫɪɟɞɫɬɜɚ ɞɥɹ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ, ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɢ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɤɚɤ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ (ordinary differential equations – ODEs), ɬɚɤ ɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ (partial differential equations – PDEs), ɚ ɬɚɤɠɟ ɢɯ ɫɢɫɬɟɦ. 8.1 Ɋȿɒȿɇɂȿ ɈȻɕɄɇɈȼȿɇɇɕɏ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ Ⱦɥɹ ɪɟɲɟɧɢɹ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢɫɩɨɥɶɡɭɟɬɫɹ ɮɭɧɤɰɢɹ: dsolve(ODE, y(x), [ɩɚɪɚɦɟɬɪɵ]), ɝɞɟ ODE – ɨɛɵɤɧɨɜɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ; y (x) – ɢɫɤɨɦɚɹ ɮɭɧɤɰɢɹ. ɇɟɨɛɹɡɚɬɟɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ ɡɚɞɚɸɬɫɹ ɜ ɮɨɪɦɟ = . ɦɨɠɟɬ ɛɵɬɶ ɨɩɭɳɟɧɨ, ɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɩɚɪɚɦɟɬɪɨɦ ɛɭɞɟɬ ɹɜɥɹɬɶɫɹ ɬɨɥɶɤɨ . Ɏɭɧɤɰɢɹ dsolve ɩɨɡɜɨɥɹɟɬ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɦɧɨɝɢɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ. ɉɨ ɭɦɨɥɱɚɧɢɸ dsolve ɩɵɬɚɟɬɫɹ ɧɚɣɬɢ ɬɨɱɧɨɟ (ɚɧɚɥɢɬɢɱɟɫɤɨɟ) ɪɟɲɟɧɢɟ. Ɉɞɧɚɤɨ, ɟɫɥɢ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ, ɬɨ ɦɨɠɧɨ ɩɨɩɵɬɚɬɶɫɹ ɧɚɣɬɢ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɫ ɩɨɦɨɳɶɸ ɪɚɡɥɨɠɟɧɢɹ ɜ ɪɹɞ (ɩɚɪɚɦɟɬɪ type = series) ɢɥɢ ɱɢɫɥɟɧɧɵɦ ɦɟɬɨɞɨɦ (ɩɚɪɚɦɟɬɪ type = numeric). ɇɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ d y ( x) 3 dx + 3xy ( x) = e − 2 x 2x

.

> ODE:=diff(y(x),x)/(2*x)+3*x*y(x)=exp(-2*x^3); d y( x ) 3 1 dx ( −2 x ) ODE := + 3 x y( x ) = e x 2

> dsolve(ODE);

y( x ) = ( x 2 + _C1 ) e

3 ( −2 x )

ɇɚɣɞɟɧɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɢɫɯɨɞɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫɨɞɟɪɠɚɥɨ ɬɨɥɶɤɨ ɨɞɧɭ ɮɭɧɤɰɢɸ – y (x), ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɜɨɡɦɨɠɧɨ ɪɟɲɟɧɢɟ, ɩɨɷɬɨɦɭ ɜ ɮɭɧɤɰɢɢ dsolve ɜɬɨɪɨɣ ɩɚɪɚɦɟɬɪ ɦɨɠɟɬ ɛɵɬɶ ɨɩɭɳɟɧ. ɇɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ y (0) = 5. Ɇɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨɫɬɨɹɧɧɭɸ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ (Maple ɨɛɨɡɧɚɱɢɥ ɟɟ _C1), ɢɫɩɨɥɶɡɭɹ ɭɠɟ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɭɫɥɨɜɢɟ. ɇɨ ɟɫɥɢ ɜ ɤɚɱɟɫɬɜɟ ɩɟɪɜɨɝɨ ɩɚɪɚɦɟɬɪɚ ɜ ɮɭɧɤɰɢɢ dsolve ɩɨɞɫɬɚɜɢɬɶ ɦɧɨɠɟɫɬɜɨ ɢɥɢ ɫɩɢɫɨɤ ɢɡ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ ɭɫɥɨɜɢɹ, ɬɨ Maple ɫɪɚɡɭ ɧɚɣɞɟɬ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ.

> R1:=dsolve({ODE,y(0)=5});

R1 := y( x ) = ( x 2 + 5 ) e

3 ( −2 x )

ɇɚɣɞɟɦ ɪɟɲɟɧɢɟ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɜɵɲɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɩɨɦɨɳɶɸ ɪɹɞɨɜ. ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ ɭɫɬɚɧɨɜɢɦ ɦɚɤɫɢɦɚɥɶɧɭɸ ɫɬɟɩɟɧɶ ɪɹɞɚ – 16, ɧɚɩɨɦɧɢɦ, ɱɬɨ ɩɨ ɭɦɨɥɱɚɧɢɸ ɷɬɨ ɡɧɚɱɟɧɢɟ ɪɚɜɧɨ 6. > Order:=16: R2:=dsolve({ODE,y(0)=5},y(x),series); R2 := y( x ) = 5 + x − 10 x − 2 x + 10 x + 2 x − 2

+

3

5

6

8

10 12 2 14 4 15 16 x + x − x + O( x ) 3 3 3

20 9 4 11 x − x + 3 3

ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɨɟ ɜ ɜɢɞɟ ɪɹɞɚ ɪɟɲɟɧɢɟ, ɨɪɝɚɧɢɡɭɟɦ ɫɩɢɫɨɤ ɢɡ ɤɨɨɪɞɢɧɚɬ ɬɨɱɟɤ ɜ ɞɢɚɩɚɡɨɧɟ x ɨɬ 0 ɞɨ 1, ɞɥɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɜɵɜɨɞɚ ɷɬɢɯ ɬɨɱɟɤ ɮɭɧɤɰɢɟɣ plot. > R2p:=[seq([i/25,subs(x=i/25, op(2,convert(R2,polynom)))], i=0..25)]: ɇɚɣɞɟɦ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɱɢɫɥɟɧɧɵɦ ɦɟɬɨɞɨɦ. Maple ɭɦɟɟɬ ɪɟɲɚɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɪɚɡɥɢɱɧɵɦɢ ɱɢɫɥɟɧɧɵɦɢ ɦɟɬɨɞɚɦɢ. ɉɨ ɭɦɨɥɱɚɧɢɸ ɢɫɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɊɭɧɝɟɄɭɬɬɚ ɱɟɬɜɟɪɬɨɝɨ-ɩɹɬɨɝɨ ɩɨɪɹɞɤɚ. > R3:=dsolve({ODE,y(0)=5},numeric); R3 := proc (x_rkf45 ) ... end proc

ɉɪɢ ɱɢɫɥɟɧɧɨɦ ɪɟɲɟɧɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɮɭɧɤɰɢɢ dsolve ɫɨɡɞɚɟɬ ɩɪɨɰɟɞɭɪɭ. ɉɪɢ ɜɵɡɨɜɟ ɩɪɨɰɟɞɭɪɵ, ɩɨɞɫɬɚɜɥɹɹ ɜ ɤɚɱɟɫɬɜɟ ɩɚɪɚɦɟɬɪɚ ɡɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɚ, ɜɵɜɨɞɢɬɫɹ ɫɩɢɫɨɤ, ɫɨɫɬɨɹɳɢɣ ɢɡ ɚɪɝɭɦɟɧɬɚ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɪɢ ɱɢɫɥɟɧɧɨɦ ɪɟɲɟɧɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ n-ɝɨ ɩɨɪɹɞɤɚ ɬɚɤɠɟ ɜɵɜɨɞɹɬɫɹ ɡɧɚɱɟɧɢɹ ɜɫɟɯ ɩɪɨɢɡɜɨɞɧɵɯ ɞɨ n-1 ɩɨɪɹɞɤɚ. > R3(0.12);

[ x = 0.12 , y ( x ]) = 4.99709897276009496 ]

ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɨɟ ɜ ɜɢɞɟ ɩɪɨɰɟɞɭɪɵ ɪɟɲɟɧɢɟ, ɨɪɝɚɧɢɡɭɟɦ ɫɩɢɫɨɤ ɢɡ ɤɨɨɪɞɢɧɚɬ ɬɨɱɟɤ ɜ ɞɢɚɩɚɡɨɧɟ x ɨɬ 0 ɞɨ 1, ɞɥɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɜɵɜɨɞɚ ɷɬɢɯ ɬɨɱɟɤ ɮɭɧɤɰɢɟɣ plot. > R3p:=[seq([i/25+0.02,op(2,op(2,R3(i/25+0.02)))], i=0..25)]: ɋɨɜɦɟɫɬɢɦ ɧɚ ɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ, ɪɟɲɟɧɢɟ ɩɪɢ ɩɨɦɨɳɢ ɪɹɞɚ ɢ ɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ. > plot([rhs(R1), R2p, R3p],x=0..1, style=[line,point,point], color=[red,blue,black],symbol=[box, circle], symbolsize=[17,17],legend=["ɚɧɚɥɢɬɢɱɟɫɨɟ ɪɟɲɟɧɢɟ", "ɪɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ","ɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ"]);

Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɫɥɟɞɭɸɳɢɟ ɩɚɪɚɦɟɬɪɵ ɮɭɧɤɰɢɢ plot: • symbol – ɫɩɨɫɨɛ ɨɬɨɛɪɚɠɟɧɢɹ ɬɨɱɟɤ; • symbolsize – ɪɚɡɦɟɪ ɬɨɱɟɤ (ɩɨ ɭɦɨɥɱɚɧɢɸ – 10); • legend – ɩɨɞɪɢɫɭɧɨɱɧɚɹ ɧɚɞɩɢɫɶ (ɥɟɝɟɧɞɚ). Ⱦɥɹ ɜɢɡɭɚɥɢɡɚɰɢɢ ɱɢɫɥɟɧɧɵɯ ɪɟɲɟɧɢɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɩɚɤɟɬɟ plots ɢɦɟɟɬɫɹ ɮɭɧɤɰɢɹ odeplot. ȼ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ, ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɪɟɲɟɧɢɹ ɧɚ ɢɧɬɟɪɜɚɥɟ ɨɬ 0 ɞɨ 1, ɮɭɧɤɰɢɸ odeplot ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ > plots[odeplot](R3,0..1); Ⱦɥɹ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɟɪɜɵɦ ɩɚɪɚɦɟɬɪɨɦ ɮɭɧɤɰɢɢ dsolve ɞɨɥɠɧɨ ɛɵɬɶ ɦɧɨɠɟɫɬɜɨ ɢɥɢ ɫɩɢɫɨɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ, ɚ ɜ ɫɥɭɱɚɟ ɧɚɯɨɠɞɟɧɢɹ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɬɭɞɚ ɠɟ ɞɨɥɠɧɵ ɜɯɨɞɢɬɶ ɢ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɫɥɨɜɢɹ. ɇɚɣɞɟɦ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ  dx(t )  dt = y (t ) − x(t )   dy (t ) = − x(t ) − 3 y (t )  dt

ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɭɫɥɨɜɢɸ: x(1) = 0, y(1) = 1. > SYS:={diff(x(t),t)=y(t)-x(t), diff(y(t),t)=-x(t)-3*y(t), x(1)=0,y(1)=1}; SYS := {

d d x( t ) = y( t ) − x( t ), y( t ) = −x( t ) − 3 y( t ), x( 1 ) = 0, y( 1 ) = 1 } dt dt

Ⱥɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ > R1:=dsolve(SYS,{x(t),y(t)});

R1 := { y( t ) = −e

( −2 t )

 − 2 + t , x( t ) = e ( −2 t )  − 1 + t  }      e ( -2 ) e ( -2 )   e ( -2 ) e ( -2 )     

> A:=plot([rhs(R1[1]),rhs(R1[2])],t=0..2, color=[blue,red]): ɑɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ > R2:=dsolve(SYS,{x(t),y(t)},numeric); R2 := proc (x_rkf45 ) ... end proc

> B:=plots[odeplot](R2,[[t,x(t),color=blue, style=point],[t,y(t),color=red,style=point]],0..2): > plots[display](A,B);

ɇɚ ɝɪɚɮɢɤɟ ɫɩɥɨɲɧɵɦɢ ɥɢɧɢɹɦɢ ɩɨɤɚɡɚɧɨ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɚ ɬɨɱɤɚɦɢ – ɱɢɫɥɟɧɧɨɟ. ɉɪɢ ɪɟɲɟɧɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɵɲɟ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɛɵɜɚɟɬ ɧɟɨɛɯɨɞɢɦɨ, ɜ ɤɚɱɟɫɬɜɟ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɥɨɜɢɣ, ɡɚɞɚɬɶ ɧɟ ɬɨɥɶɤɨ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɜ ɤɚɤɨɣ-ɥɢɛɨ ɬɨɱɤɟ, ɧɨ ɢ ɡɧɚɱɟɧɢɹ ɩɪɨɢɡɜɨɞɧɵɯ. Ⱦɥɹ ɷɬɨɝɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ – D. ȿɫɥɢ, ɧɚɩɪɢɦɟɪ, ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ y (t), ɜ ɬɨɱɤɟ t = 0 ɪɚɜɧɨ 5, ɬɨ ɧɟɨɛɯɨɞɢɦɨ ɡɚɩɢɫɚɬɶ – D(y)(0) = 5. Ɍɨɠɟ ɞɥɹ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ – D (D(y))(0) = 5 ɢɥɢ (D@@2)(y)(0) = 5, ɞɥɹ ɬɪɟɬɶɟɣ – D (D (D(y)))(0) = 5 ɢɥɢ (D@@3)(y)(0) = 5 ɢ ɬ.ɞ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɨɢɡɜɨɞɧɭɸ n-ɝɨ ɩɨɪɹɞɤɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɤɚɤ (D@@n)(y)(0). Ɋɟɲɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ y′′′ + 2 y′ + 12 y = 0 ɩɪɢ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɥɨɜɢɹɯ: y(0) = 4, y′(0) = 0, y′′(0) = 0. > de:=diff(y(t),t$3)+2*diff(y(t),t)+12*y(t)=0; 3 d d  de :=  3 y( t )  + 2  y( t )  + 12 y( t ) = 0  dt  d   t  

> init:=y(0)=4,D(y)(0)=0,(D@@2)(y)(0)=0; init := y( 0 ) = 4, D( y )( 0 ) = 0, ( D

(2)

)( y ) ( 0 ) = 0

> dsolve({de,init}); y( t ) =

12 ( −2 t ) 8 16 t e e cos ( 5 t ) 5 e t sin( 5 t ) + + 7 35 7

8.2 Ɋȿɒȿɇɂȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ ȼ ɑȺɋɌɇɕɏ ɉɊɈɂɁȼɈȾɇɕɏ Ⱦɥɹ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɢɫɩɨɥɶɡɭɟɬɫɹ ɮɭɧɤɰɢɹ: pdsolve(PDE, u(x, y), [ɩɚɪɚɦɟɬɪɵ]), ɝɞɟ PDE – ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ; u(x, y) – ɢɫɤɨɦɚɹ ɮɭɧɤɰɢɹ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ. ɇɟɨɛɹɡɚɬɟɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ ɜɵɩɨɥɧɹɸɬ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɬɭɠɟ ɪɨɥɶ, ɱɬɨ ɢ ɜ ɮɭɧɤɰɢɢ dsolve ɢ ɛɨɥɟɟ ɩɨɞɪɨɛɧɨ ɨ ɧɢɯ ɛɭɞɟɬ ɪɚɫɫɤɚɡɚɧɨ ɧɢɠɟ.

Ⱦɥɹ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɜ ɤɚɱɟɫɬɜɟ ɩɟɪɜɵɯ ɞɜɭɯ ɩɚɪɚɦɟɬɪɨɜ ɫɥɟɞɭɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɧɨɠɟɫɬɜɚ ɢɥɢ ɫɩɢɫɤɢ ɫ ɭɪɚɜɧɟɧɢɹɦɢ ɢ ɢɫɤɨɦɵɦɢ ɮɭɧɤɰɢɹɦɢ. Ɍɢɩɢɱɧɨɣ ɨɫɨɛɟɧɧɨɫɬɶɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɢ ɢɯ ɫɢɫɬɟɦ ɹɜɥɹɟɬɫɹ ɬɨ, ɱɬɨ ɞɥɹ ɨɞɧɨɡɧɚɱɧɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɡɞɟɫɶ ɬɪɟɛɭɟɬɫɹ ɡɚɞɚɧɢɟ ɧɟ ɡɧɚɱɟɧɢɣ ɬɨɝɨ ɢɥɢ ɢɧɨɝɨ ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɩɚɪɚɦɟɬɪɨɜ, ɚ ɧɟɤɨɬɨɪɵɯ ɮɭɧɤɰɢɣ. ɇɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ∂ 2 u ( x, t ) ∂ 2 u ( x, t ) = ∂x 2 ∂t 2

> PDE:=diff(u(x,t),t,t)= diff(u(x,t),x,x); PDE :=

> pdsolve(PDE,u(x,t));

∂2 ∂2 u ( x , t ) u( x, t ) = ∂t 2 ∂x 2

u( x, t ) = _F1( t + x ) + _F2( t − x )

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɚɫɫɦɨɬɪɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɥɢɲɶ ɜ ɬɨɣ ɦɟɪɟ ɨɝɪɚɧɢɱɢɜɚɟɬ ɩɪɨɢɡɜɨɥ ɜ ɜɵɛɨɪɟ ɮɭɧɤɰɢɣ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ u, ɱɬɨ ɟɟ ɭɞɚɟɬɫɹ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɫɭɦɦɭ ɞɜɭɯ ɮɭɧɤɰɢɣ _F1 ɢ _F2 ɨɬ ɨɞɧɨɝɨ ɩɟɪɟɦɟɧɧɨɝɨ, ɤɨɬɨɪɵɟ ɨɫɬɚɸɬɫɹ ɩɪɨɢɡɜɨɥɶɧɵɦɢ, ɟɫɥɢ ɧɟ ɞɚɧɨ ɤɚɤɢɯ-ɥɢɛɨ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɥɨɜɢɣ. ȿɫɥɢ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɭɧɤɰɢɣ, ɬɨ ɫɥɟɞɭɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɚɪɚɦɟɬɪ HINT = `*`. > pdsolve(PDE,u(x,t),HINT=`*`);

( u( x, t ) = _F1( _ξ1 ) _F2( _ξ2 ) ) &where  { d _F2( _ξ2 ) = _c , 2  d _F1( _ξ1 )  _c = 0 },   2  2  d_ξ2   d_ξ1 x t &and { _ξ1 = t − x, _ξ2 = + }   2 2  

ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɩɚɪɚɦɟɬɪɚ build – Maple ɩɨɩɵɬɚɟɬɫɹ ɡɚɩɢɫɚɬɶ ɪɟɲɟɧɢɟ ɜ ɹɜɧɨɦ ɜɢɞɟ. > pdsolve(PDE,u(x,t),HINT=`*`, build);

u ( x, t ) =

1 1 _C1 _c 2 x + _C1 _c2 t + _C1 _C2 2 2

ɝɞɟ _C1, _C2 ɢ _c2 – ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ ɤɪɚɟɜɵɯ ɢɥɢ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ, ɩɨɡɜɨɥɹɸɳɢɯ ɨɞɧɨɡɧɚɱɧɨ ɜɵɞɟɥɢɬɶ ɢɧɬɟɪɟɫɭɸɳɟɟ ɪɟɲɟɧɢɟ. ȼ ɬɨ ɜɪɟɦɹ ɤɚɤ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ ɡɚɞɚɸɬɫɹ ɢɫɤɥɸɱɢɬɟɥɶɧɨ ɧɚ ɝɪɚɧɢɱɧɵɯ ɬɨɱɤɚɯ ɨɛɥɚɫɬɢ, ɝɞɟ ɢɳɟɬɫɹ ɪɟɲɟɧɢɟ, ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɦɨɝɭɬ ɨɤɚɡɚɬɶɫɹ ɡɚɞɚɧɧɵɦɢ ɧɚ ɨɩɪɟɞɟɥɟɧɧɨɦ ɦɧɨɠɟɫɬɜɟ ɬɨɱɟɤ ɜɧɭɬɪɢ ɨɛɥɚɫɬɢ. ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ F(x, y), ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɭɪɚɜɧɟɧɢɸ ∂ 2 F ( x, y ) ∂ 2 F ( x, y ) = 0 , ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɧɚ ɨɤɪɭɠɧɨɫɬɢ x 2 + y 2 =16 ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɦɨɝɭɬ ɛɵɬɶ ɪɚɫ+ 2 2 ∂y ∂x

ɫɱɢɬɚɧɵ ɩɨ ɜɵɪɚɠɟɧɢɸ x 2 y 2 . ɍɱɢɬɵɜɚɹ, ɱɬɨ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɡɚɞɚɧɵ ɧɚ ɨɤɪɭɠɧɨɫɬɢ, ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɪɟɲɚɬɶ ɡɚɞɚɱɭ ɜ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ. Ⱦɥɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɩɨɥɹɪɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɢɫɩɨɥɶɡɭɟɦ ɮɭɧɤɰɢɸ PDEchangecoords ɢɡ ɩɚɤɟɬɚ Detools, ɚ ɞɥɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ – ɮɭɧɤɰɢɹ dchange ɢɡ ɩɚɤɟɬɚ PDEtools. 2  ∂ PDE :=  2 F( x, y )  +  ∂x   

> PDE:=diff(F(x,y),x,x)+diff(F(x,y),y,y);

2  ∂  2 F( x, y )   ∂y   

> PDE:=DEtools[PDEchangecoords](PDE,[x,y],polar, [r,phi]); 2 2  ∂ F( r, φ )  r +  ∂ F( r, φ )  r 2 +  ∂ F( r, φ )    2    ∂φ2  ∂r   ∂r     PDE := 2 r

> dp:={x = r*cos(phi), y = r*sin(phi)};

dp := { x = r cos( φ ), y = r sin( φ ) }

> f:=x^2*y^2; f := x 2 y 2

> f:=PDEtools[dchange](dp,f); f := r 4 cos ( φ ) 2 sin ( φ ) 2

ɉɨɥɭɱɚɟɦ ɪɟɲɟɧɢɟ ɜ ɹɜɧɨɦ ɜɢɞɟ. > R:=rhs(pdsolve(PDE,HINT=`*`,build));

R := _C3 sin( _c1 φ ) _C1 r

(

_c ) 1

+

_C3 sin( _c1 φ ) _C2 (

r

+ _C4 cos( _c1 φ ) _C1 r

(

_c ) 1

+

_c ) 1

+

_C4 cos( _c1 φ ) _C2 (

r

_c ) 1

Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɢɫɤɨɦɚɹ ɮɭɧɤɰɢɹ ɨɩɪɟɞɟɥɟɧɚ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɜɤɥɸɱɚɹ ɰɟɧɬɪ ɤɨɨɪɞɢɧɚɬ, ɬ.ɟ. ɩɪɢ r = 0. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, _C2 = 0, ɬɚɤ ɤɚɤ ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɡɧɚɦɟɧɚɬɟɥɶ ɜ ɞɜɭɯ ɩɨɫɥɟɞɧɢɯ ɫɥɚɝɚɟɦɵɯ ɪɟɲɟɧɢɹ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ. > _C2:=0; _ 2:=0 > R;

_C3 sin( _c1 φ ) _C1 r

(

_c ) 1

+ _C4 cos( _c1 φ ) _C1 r

(

_c ) 1

ɉɟɪɜɨɟ ɢ ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɫɨɞɟɪɠɚɬ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɤɨɧɫɬɚɧɬ, ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ _C1=1 ɢ ɢɫɤɚɬɶ ɬɨɥɶɤɨ _C3 ɢ _C4. > _C1:=1; _C1 := 1

> R;

_C3 sin( _c1 φ ) r

(

_c ) 1

+ _C4 cos( _c1 φ ) r

(

_c ) 1

Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɧɫɬɚɧɬ _C3 ɢ _C4 ɢɫɩɨɥɶɡɭɟɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ. > simplify(subs({phi=0,r=4},R)) = eval(subs({phi=0,r=4},f)): > simplify(subs({phi=Pi,r=4},R)) = eval(subs({phi=Pi,r=4},f)): > solve({%,%%},{_C3,_C4}); { _C3 = 0, _C4 = 0 }

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɧɚɣɞɟɧɧɨɟ ɪɟɲɟɧɢɟ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ. Ɍɚɤɠɟ ɨɬɦɟɬɢɦ, ɮɭɧɤɰɢɹ pdsolve ɧɚɯɨɞɢɬ ɪɟɲɟɧɢɟ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɧɟɤɨɬɨɪɨɣ ɩɨɫɬɨɹɧɧɨɣ. ɉɨɷɬɨɦɭ ɞɨɛɚɜɢɦ ɤ ɩɨɥɭɱɟɧɧɨɦɭ ɪɚɧɟɟ ɪɟɲɟɧɢɸ ɩɨɫɬɨɹɧɧɭɸ Z, ɩɨɞɥɟɠɚɳɭɸ ɨɩɪɟɞɟɥɟɧɢɸ. > R:=R+Z;

R := _C3 sin( _c1 φ ) r

(

_c ) 1

+ _C4 cos( _c1 φ ) r

(

_c ) 1

+Z

ɂɫɩɨɥɶɡɭɟɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɧɫɬɚɧɬ _C3, _C4 ɢ Z, ɚ ɬɚɤɠɟ ɤɨɧɫɬɚɧɬɵ _c2. > R4:=subs(r=4,R=f); R4 := _C3 sin( _c1 φ ) 4

( _c ) 1

+ _C4 cos( _c1 φ ) 4

> simplify(subs(phi=0,R4)): U1:=simplify(%): > simplify(subs(phi=Pi/2/_c[1]^(1/2),R4)): U2:=simplify(%):

( _c ) 1

+ Z = 256 cos( φ )2 sin( φ )2

> simplify(subs(phi=Pi/4,R4)): U3:=simplify(%): > S:=solve({U1,U2,U3},{_C3,_C4,Z}): > R4:=simplify(subs(S,R4)): > _c[1]=solve(subs(phi=Pi,R4),_c[1]); _c 1 = ( 4, 16 )

Ʉɨɧɫɬɚɧɬɚ _c1 ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɞɜɚ ɡɧɚɱɟɧɢɹ 4 ɢ 16. ɉɨɞɫɬɚɜɥɹɟɦ ɜ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɧɚɣɞɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɧɫɬɚɧɬ _C3, _C4 ɢ Z. > R:=subs(S,R): ɉɨɞɫɬɚɧɨɜɤɚ _c1 = 4 ɩɪɢɜɨɞɢɬ ɤ ɨɲɢɛɤɟ – ɞɟɥɟɧɢɟ ɧɚ ɧɨɥɶ. ɂɫɩɨɥɶɡɭɟɦ ɩɨɞɫɬɚɧɨɜɤɭ _c1 = 16. > SOL:=combine(subs(_c[1]=16,R)); SOL := 32 −

1 cos ( 4 φ ) r 4 8

ɉɨɥɭɱɟɧɨ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ. ɇɟɬɪɭɞɧɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɧɚɣɞɟɧɧɨɟ ɪɟɲɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɤɚɤ ɭɪɚɜɧɟɧɢɸ , ɬɚɤ ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ. ɉɪɟɨɛɪɚɡɭɟɦ ɪɟɲɟɧɢɟ ɜ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɢɫɩɨɥɶɡɭɸ ɢɡɜɟɫɬɧɭɸ ɫɜɹɡɶ ɦɟɠɞɭ ɩɨɥɹɪɧɨɣ ɢ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɨɣ, ɚ ɬɚɤɠɟ ɩɪɢɜɟɞɟɧɧɵɟ ɧɢɠɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɬɨɠɞɟɫɬɜɚ. > pd:={r=sqrt(x^2+y^2),phi=arctan(y/x)}; y pd := { φ = arctan  , r = x

x2 + y2 }

> SOL:=PDEtools[dchange](pd,SOL); SOL := 32 −

2 1 y cos  4 arctan    ( x 2 + y 2 ) x 8   

> cos(4*arctan(y/x)) = op(3,trigsubs(cos(4*arctan(y/x))));

y y cos  4 arctan   = 2 cos  2 arctan   − 1 x     x   2

> SOL:=subs(%,SOL);

SOL := 32 −

 2 1  y 2 cos  2 arctan   − 1  ( x 2 + y 2 )  8   x   2

> cos(2*arctan(y/x))^2 = op(11,trigsubs(cos(2*arctan(y/x))^2));

   1 − tan arctan  y    2    x   y     cos  2 arctan    =  2 2   x     1 + tan arctan  y       x        2

> SOL:=simplify(subs(%,SOL)); > F:=unapply(SOL,x,y);

SOL := 32 −

2

1 4 3 2 2 1 4 x + x y − y 8 4 8

F := ( x, y ) → 32 −

1 4 3 2 2 1 4 x + x y − y 8 4 8

ɉɨɫɬɪɨɢɦ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ F(x,y) ɜ ɤɪɭɝɟ ɪɚɞɢɭɫɨɦ 4. > plot3d(F(x,y), x=-4..4, y=-sqrt(16-x^2)..sqrt(16-x^2), axes=framed, title="F(x,y)");

8.3 ɑɂɋɅȿɇɇɈȿ Ɋȿɒȿɇɂȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ ȼ ɑȺɋɌɇɕɏ ɉɊɈɂɁȼɈȾɇɕɏ Ɋɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɭɸ ɡɚɞɚɱɭ. ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɰɟɫɫ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɤɪɭɝɥɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɪɚɞɢɭɫɨɦ r = 0,002 ɦ ɢ ɞɥɢɧɨɣ L = 1 ɦ ɫ ɠɟɫɬɤɨ ɡɚɤɪɟɩɥɟɧɧɵɦɢ ɤɨɧɰɚɦɢ. ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɬɪɭɧɚ ɢɦɟɥɚ ɮɨɪɦɭ ɤɜɚɞɪɚɬɢɱɧɨɣ ɩɚɪɚɛɨɥɵ, ɫɢɦɦɟɬɪɢɱɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ ɤ ɫɟɪɟɞɢɧɟ ɫɬɪɭɧɵ ɢ ɦɚɤɫɢɦɚɥɶɧɵɦ ɨɬɤɥɨɧɟɧɢɟɦ h = 0,01 ɦ, ɚ ɡɚɬɟɦ ɨɬɩɭɳɟɧɚ ɛɟɡ ɬɨɥɱɤɚ. ɇɚɬɹɠɟɧɢɟ ɫɬɪɭɧɵ T = 100 H, ɚ ɩɥɨɬɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɫɬɪɭɧɵ ρc = 7800 ɤɝ/ɦ3. ɉɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ ɨɩɢɫɵɜɚɸɬɫɹ

ɝɞɟ a =

∂ 2 u ( x, t ) ∂ 2 u ( x, t ) = a2 , 2 ∂x 2 ∂t

T , ρl = ρS – ɥɢɧɟɣɧɚɹ ɩɥɨɬɧɨɫɬɶ ɫɬɪɭɧɵ, ɤɝ/ɦ; S = πr2 – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɪɭɧɵ. ρl

Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ u (0, t) = 0

u (l, t) = 0 Ɂɚɩɢɲɟɦ

:

ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ u (x, 0) = f (x) f (x) – ɮɨɪɦɚ ɫɬɪɭɧɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 0. ∂u ∂t

t =0 =

0

:

> PDE:=diff(u(x,t),t,t)=a^2*diff(u(x,t),x,x); PDE :=

∂2  ∂2  u( x, t ) = a 2  2 u( x, t )  2 x ∂t ∂  

> a:=sqrt(T/rho[l]);rho[l]:=rho[c]*S;S:=Pi*r^2;

T ρl ρ l := ρ c S

a :=

S := π r 2

> PDE;

ȼɜɟɞɟɦ ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ:

 ∂ 2  u( x, t )  T 2 2  ∂x ∂  u( x, t ) =  ∂ t2 ρc π r2

> T:=100;L:=1;r:=0.002;rho[c]:=7800;h:=0.01; T := 100 L := 1 r := 0.002 ρ c := 7800 h := 0.01

Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɤɜɚɞɪɚɬɢɱɧɨɣ ɩɚɪɚɛɨɥɵ: > f:=A*x^2+B*x+C; ɇɚɣɞɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ A, B ɢ C:

f := A x 2 + B x + C

> solve({subs(x=0,A*x^2+B*x+C=0), subs(x=L/2,A*x^2+B*x+C=h), subs(x=L,A*x^2+B*x+C=0)}, {A,B,C});

{ C = 0., A = -0.40000000 , B = 0.40000000 }

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ ɮɨɪɦɭ ɫɬɪɭɧɵ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ: > f:=subs(%,f);

f := −0.04000000000x 2 + 0.04000000000x

> plot(f,x=0..L);

Ɂɚɩɢɲɟɦ

:

> BC:={u(0,t)=0,u(L,t)=0,u(x,0)=f,D[2](u)(x,0)=0}; BC := { u( 0, t ) = 0, D 2( u )( x , 0 ) = 0, u( 1, t ) = 0,

ɉɨɥɭɱɢɦ ɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ ɩɨ ɜɪɟɦɟɧɢ ɢ ɤɨɨɪɞɢɧɚɬɟ:

u( x , 0 ) = −0.04000000000 x 2 + 0.04000000000 x }

ɫ ɭɱɟɬɨɦ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ, ɡɚɞɚɜ ɜɟɥɢɱɢɧɭ ɲɚɝɚ

> SOL:=pdsolve(PDE,BC,numeric,timestep=1/200, spacestep=1/200); SOL := module () export plot , plot3d , animate , value , settings ; ...end module

ɉɨɥɭɱɟɧɧɵɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɦɨɞɭɥɶ SOL ɩɨɡɜɨɥɹɟɬ ɜɢɡɭɚɥɢɡɢɪɨɜɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɚ ɬɚɤɠɟ ɩɨɥɭɱɢɬɶ ɱɢɫɥɨɜɵɟ ɡɧɚɱɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɣ ɬɨɱɟɤ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ. ɉɨɫɬɪɨɢɦ ɩɨɥɨɠɟɧɢɟ ɫɬɪɭɧɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 0,27 ɫ. > SOL:-plot(t=0.27,title="t=0.27");

ɋɨɡɞɚɞɢɦ ɩɪɨɰɟɞɭɪɭ ɞɥɹ ɪɚɫɱɟɬɚ ɩɟɪɟɦɟɳɟɧɢɣ ɬɨɱɟɤ ɫɬɪɭɧɵ: > XT:=SOL:-value(); XT := proc () ... end proc

ɉɨɥɭɱɢɦ ɩɟɪɟɦɟɳɟɧɢɟ ɫɟɪɟɞɢɧɵ ɫɬɪɭɧɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 0,27 ɫ. > XT(L/2,0.27);

[ x = 0.50000000000000 , t = 0.27 , u ( x , t ) = 0.00121214618006418979 ]

9 ɉɊɂɆȿɊɕ Ɋȿɒȿɇɂə ɂɇɀȿɇȿɊɇɕɏ ɁȺȾȺɑ 9.1 ɊȺɋɑȿɌ ȻɕɋɌɊɈ ȼɊȺɓȺɘɓɂɏɋə ȾɂɋɄɈȼ Ɂɚɞɚɱɚ ɨ ɪɚɫɱɟɬɟ ɛɵɫɬɪɨ ɜɪɚɳɚɸɳɢɯɫɹ ɞɢɫɤɨɜ ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɫɜɨɣɫɬɜɚɯ ɦɚɬɟɪɢɚɥɚ E (ɦɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ), µ (ɤɨɷɮɮɢɰɢɟɧɬ ɉɭɚɫɫɨɧɚ) ɢ ɩɟɪɟɦɟɧɧɨɣ ɬɨɥɳɢɧɟ h(r) ɨɩɢɫɵɜɚɟɬɫɹ ɫɢɫɬɟɦɨɣ ɭɪɚɜɧɟɧɢɣ:  d d  σr (r )h(r ) + r σr (r ) h(r ) + rσr (r ) h(r )  − σt (r )h(r ) + ρω2 r 2 h(r ) = 0 , (1)   dr  dr  d ε r (r ) = (2) u (r ) , dr u (r ) ε t (r ) = , (3) r σ (r ) − µσ t (r ) ε r (r ) = r + θ(r ) , (4) E σ (r ) − µσ r (r ) ε t (r ) = t (5) + θ(r ) , E ɧɚɩɪɹɠɟɧɢɹ; ε r (r ) , εt (r ) – ɪɚɞɢɚɥɶɧɵɟ ɢ ɨɤɪɭɠɧɵɟ ɞɟɮɨɪɦɚ-

ɝɞɟ σ r (r ) , σt (r ) – ɪɚɞɢɚɥɶɧɵɟ ɢ ɨɤɪɭɠɧɵɟ ɰɢɢ; u (r) – ɪɚɞɢɚɥɶɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ, θ(r) – ɬɟɦɩɟɪɚɬɭɪɧɵɟ ɞɟɮɨɪɦɚɰɢɢ; ρ – ɩɥɨɬɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɞɢɫɤɚ; ω – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ. > restart; Ɂɚɩɢɲɟɦ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ: > u1:=diff(R*sigma[r](R)*h(R),R)-sigma[t](R)*h(R)+rho*omega^2*R^2*h(R)=0; d d σr( R )  h( R ) + R σr( R )  h( R )  − σt( R ) h( R ) u1 := σr( R ) h( R ) + R  d R d R     + ρ ω2 R2 h( R ) = 0

ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɜ ɬɪɟɬɶɟ ɭɪɚɜɧɟɧɢɟ ɜ ɜɢɞɟ u ( R) = Rε t ( R) ɩɨ R, ɢ ɢɫɩɨɥɶɡɭɹ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: > u2:=epsilon[r](R)=epsilon[t](R)+R* diff(epsilon[t](R),R);

d ε ( R )  u2 := ε r( R ) = ε t( R ) + R  d  R t .

ɉɨɞɫɬɚɜɢɜ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ε ɢɡ ɱɟɬɜɟɪɬɨɝɨ ɢ ɩɹɬɨɝɨ ɭɪɚɜɧɟɧɢɣ ɜ ɩɨɥɭɱɟɧɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɩɨɥɭɱɢɦ:

> u2:=simplify(subs({epsilon[r](R)=(1/E)* (sigma[r](R)-mu*sigma[t](R))+theta(R), epsilon[t](R)=(1/E)*(sigma[t](R)-mu*sigma[r](R))+theta(R)},u2)); u2 :=

σ r( R ) − µ σ t ( R ) + θ ( R ) E E

=

d d d σ ( R )  − R µ  σ ( R )  + R  θ( R )  E σt( R ) − µ σr( R ) + θ( R ) E + R   dR t   dR r   dR  E

ɍɩɪɨɫɬɢɦ ɜɵɪɚɠɟɧɢɟ, ɪɚɡɞɟɥɢɜ ɱɥɟɧɵ ɫ σt (R) ɢ σ r (R) : > u2:=isolate(u2,sigma[t](R));

d u2 := µ σt( R ) + σt( R ) + R  σ ( R )  = d   R t d d  σr( R ) + µ σr( R ) + R µ  σr( R )  − R  θ( R )  E d R d R    

ȼɵɪɚɡɢɦ σt (R) ɢɡ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ: σt( R ) :=

> sigma[t](R):=solve(u1,sigma[t](R));

ɉɨɞɫɬɚɜɢɦ ɩɨɥɭɱɟɧɧɨɟ > u2:=simplify(u2);

d d σr( R )  h( R ) + R σr( R )  h( R )  + ρ ω2 R2 h( R ) σr( R ) h( R ) + R  d R d    R  h( R ) ɜɵɪɚɠɟɧɢɟ ɞɥɹ σt (R) ɜ ɭɪɚɜɧɟɧɢɟ u2:

 d d σ ( R )  + µ h( R ) R σr( R )  h( R )  u2 :=  µ h( R ) 2 σr( R ) + µ h( R ) 2 R    dR   dR r  d σr( R )  h( R ) 2 + µ h( R ) 2 ρ ω2 R2 + σr( R ) h( R ) 2 + 3 R  d R   d  d2    2 2 2 + 2 h( R ) R σr( R )  h( R )  + 3 ρ ω R h( R ) + R2  2 σr( R )  h( R ) 2 d R    dR  2 d d  d  + h( R ) R2  σ ( R )   h( R )  + h( R ) R2 σr( R )  2 h( R )  R d    dR  dR r  

d − R2 σr( R )  h( R )    dR

2

  

h( R ) 2 =

d d σr( R ) + µ σr( R ) + R µ  σr( R )  − R  θ( R )  E d R d R    

ɍɩɪɨɫɬɢɦ ɩɨɥɭɱɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɫɨɛɪɚɜ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɜɫɟ ɱɥɟɧɵ ɫ σ r (R) : > u2:=isolate(u2,sigma[r](R)); d d u2 := µ h( R ) σr( R )  h( R )  + 3  σ ( R )  h( R ) 2 d R d    R r  2 d d   h( R )  + R  2 σr( R )  h( R ) 2 + 2 h( R ) σr( R )  d R d R    

d − θ( R )  E h( R ) 2 − µ h( R ) 2 ρ ω2 R − 3 ρ ω2 R h( R ) 2  dR 

d d  d2  + h( R ) R  σr( R )   h( R )  + h( R ) R σr( R )  2 h( R )  d R d R     dR  d − R σr( R )  h( R )  = d  R  2

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɫɜɟɥɢ ɡɚɞɚɱɭ, ɨɩɢɫɵɜɚɟɦɭɸ ɫɢɫɬɟɦɨɣ ɢɡ ɩɹɬɢ ɭɪɚɜɧɟɧɢɣ ɤ ɪɟɲɟɧɢɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ σ r (R) . Ɉɤɪɭɠɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢɡ ɜɵɪɚɠɟɧɢɹ sigma[t](R), ɩɨɥɭɱɟɧɧɨɝɨ ɪɚɧɟɟ, ɚ ɞɟɮɨɪɦɚɰɢɢ ɢ ɩɟɪɟɦɟɳɟɧɢɟ ɢɡ ɢɫɯɨɞɧɵɯ ɫɜɹɡɟɣ. Ɋɚɫɫɦɨɬɪɢɦ, ɧɚɩɪɢɦɟɪ, ɫɥɭɱɚɣ, ɤɨɝɞɚ ɧɟɪɚɜɧɨɦɟɪɧɨɫɬɶ ɧɚɝɪɟɜɚ ɨɬɫɭɬɫɬɜɭɟɬ, ɬ.ɟ. θ(R) = const: > theta(R):=theta; > u2:=simplify(u2);

θ( R ) : =θ

d d u2 := µ h( R ) σr( R )  h( R )  + 3  σ ( R )  h( R ) 2 d R d    R r  2 d d   h( R )  + R  2 σr( R )  h( R ) 2 + 2 h( R ) σr( R )  d R R d    

d d  d2  σr( R )   h( R )  + h( R ) R σr( R )  2 h( R )  + h( R ) R  d R d R     dR  d − R σr( R )  h( R )  = −µ h( R ) 2 ρ ω2 R − 3 ρ ω2 R h( R ) 2 d  R  2

Ɂɚɞɚɞɢɦ ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɬɨɥɳɢɧɵ ɞɢɫɤɚ ɨɬ ɪɚɞɢɭɫɚ, ɩɪɢ ɷɬɨɦ ɪɚɞɢɭɫ ɜɧɭɬɪɟɧɧɟɝɨ ɨɬɜɟɪɫɬɢɹ ɩɪɢɦɟɦ ɪɚɜɧɵɦ 0,2 ɦ, ɚ ɧɚɪɭɠɧɵɣ ɪɚɞɢɭɫ ɞɢɫɤɚ 1 ɦ: > mu:=1/3; rho:=7800; omega:=100; h(R):=1/20/R; µ :=

1 3 ρ := 7800 ω := 100 1 h( R ) := 20 R

> plot({h(R)/2,-h(R)/2},R=1/5..1,color=black, titlefont=[COURIER,12],title="Ɏɨɪɦɚ ɞɢɫɤɚ");

ɍɫɥɨɜɢɹ ɧɚ ɝɪɚɧɢɰɚɯ ɞɢɫɤɚ: > ini:=sigma[r](1/5)=0,sigma[r](1)=0;

1 ini := σr  = 0, σr( 1 ) = 0  5

Ɋɟɲɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ u2 ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ini: > u3:=evalf(simplify(dsolve({u2,ini},sigma[r](R)))); u3 :=

σr( R ) = 0.5656797953 108 R0.7583057390 −

853693.8027 − 0.5571428571 108 R2 R1.758305739

ɇɚɣɞɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ R ɨɬ ɩɨɥɭɱɟɧɧɨɝɨ ɪɟɲɟɧɢɹ: > u4:=diff(rhs(u3),R); u4 :=

0.4289582352108 0.1501054713107 + − 0.1114285714109 R R0.2416942610 R2.758305739

ɉɨɞɫɬɚɜɢɦ ɧɚɣɞɟɧɧɨɟ ɪɟɲɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ σ r (R) (ɩɟɪɟɦɟɧɧɚɹ u3) ɢ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɧɟɝɨ (ɩɟɪɟɦɟɧɧɚɹ u4) ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ σt( R ) : > u5:=eval(subs({diff(sigma[r](R),R)=u4, sigma[r](R)=rhs(u3)},sigma[t](R)));

  0.2144791176107 75052.73565 + 2.758305739 − 0.1671428570107 R  R u5 := 20   R0.2416942610 R 

ɉɨɫɬɪɨɢɦ ɷɩɸɪɵ ɪɚɞɢɚɥɶɧɵɯ ɢ ɨɤɪɭɠɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɜ ɞɢɫɤɟ:

> plot([rhs(u3)/1E6,u5/1E6],R=1/5..1, color=black,labelfont=[COURIER,12],labels=["ɦ","Mɉɚ"],titlefont=[COURIER,12],title="ɗɩɸɪɵ ɪɚɞɢɚɥɶɧɵɯ\n ɢ ɨɤɪɭɠɧɵɯ ɧɚɩɪɹɠɟɧɢɣ", legend=["sigma[r]","sigma[t]"], thickness=[1,3]);

9.2 ɊȺɋɑȿɌ ȺɉɉȺɊȺɌȺ ȼɕɋɈɄɈȽɈ ȾȺȼɅȿɇɂə ɇɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɧɚɪɭɠɧɵɣ ɪɚɞɢɭɫ ɚɩɩɚɪɚɬɚ, ɬɨɥɳɢɧɭ ɫɬɟɧɤɢ, ɡɧɚɱɟɧɢɟ ɩɪɟɞɟɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ, ɞɚɜɥɟɧɢɟ ɨɩɪɟɫɫɨɜɤɢ, ɚ ɬɚɤɠɟ ɩɨɫɬɪɨɢɬɶ ɷɩɸɪɵ ɧɚɩɪɹɠɟɧɢɣ ɩɪɢ ɪɚɛɨɱɟɦ ɞɚɜɥɟɧɢɢ ɞɨ ɢ ɩɨɫɥɟ ɚɜɬɨɮɪɟɬɢɪɨɜɚɧɢɹ.

ɂɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɞɥɹ ɪɚɫɱɟɬɚ ɜɧɭɬɪɟɧɧɢɣ ɪɚɞɢɭɫ ɚɩɩɚɪɚɬɚ ɪɚɛɨɱɟɟ ɞɚɜɥɟɧɢɟ ɩɪɟɞɟɥ ɬɟɤɭɱɟɫɬɢ ɦɚɬɟɪɢɚɥɚ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɩɚɫɚ ɩɪɨɱɧɨɫɬɢ ɩɨ ɩɪɟɞɟɥɶɧɨɦɭ ɫɨɫɬɨɹɧɢɸ ɝɥɭɛɢɧɚ ɩɥɚɫɬɢɱɟɫɤɢɯ ɞɟɮɨɪɦɚɰɢɣ ɩɪɢ ɚɜɬɨɮɪɟɬɢɪɨɜɚɧɢɢ ɭɫɥɨɜɢɟ ɩɥɚɫɬɢɱɧɨɫɬɢ ɦɚɬɟɪɢɚɥ ɩɪɢ ɞɟɮɨɪɦɚɰɢɢ ɭɩɪɨɱɧɹɟɬɫɹ ɦɨɞɭɥɶ ɩɪɨɞɨɥɶɧɨɣ ɭɩɪɭɝɨɫɬɢ ɦɨɞɭɥɶ ɭɩɪɨɱɧɟɧɢɹ

0,6 ɦ 70 Ɇɉɚ 780 Ɇɉɚ 1,4 40 % ȽɭɛɟɪɚɆɢɡɟɫɚ 2,1 ⋅ 105 Ɇɉɚ 5880 Ɇɉɚ

> restart:Digits:=8; Digits := 8

> Sigma:=proc(x::string,P) local L,Zn: option operator,arrow: description " ɉɨɰɟɞɭɪɚ ɪɚɫɱɟɬɚ ɧɚɩɪɹɠɟɧɢɣ ": if (x="z" and nargs=1) then return ((Sigma("r")+Sigma("k"))/2): end if: if (x="z" and nargs=2) then return (P*R1^2/(R2^2-R1^2)) :end if: if x="r" then Zn:=-1: end if: if x="k" then Zn:=+1: end if: if nargs=1 then L:=`if`(r>R[t],0,lambda): A*sigma[tau]/2 * (2*L*ln(r/R[t])+R[t]^2/R2^2+Zn*(1-L) * R[t]^2/r^2+Zn*L): else P*R1^2/(R2^2-R1^2)*(1+Zn*R2^2/r^2):

end if: end proc: ȼɜɨɞ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ > R1:=0.6;P[rab]:=70; sigma[tau]:=780; n[pred]:=1.4; T:=0.4;E[pu]:=210000; E[u]:=5880; A:=evalf(2/sqrt(3)); R1 := 0.6 Prab := 70 στ := 780

n pred := 1.4

T := 0.4 Epu := 210000 Eu := 5880

A := 1.1547005

Ɋɚɫɱɟɬ ɩɚɪɚɦɟɬɪɚ ɭɩɪɨɱɧɟɧɢɹ > lambda:=1-E[u]/E[pu];

λ :=

243 250

Ɋɚɫɱɟɬ ɩɪɟɞɟɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ > P[pred]:=A*n[pred]*P[rab]; Ppred := 113.16065

Ɋɚɫɱɟɬ ɧɚɪɭɠɧɨɝɨ ɪɚɞɢɭɫɚ > R2:=solve(P[pred]=A*sigma[tau]/2 * (2*lambda*ln(R2/R1)+(1-lambda)*(R2^2/R1^2-1)),R2); R2 := 0.68000014

Ɋɚɫɱɟɬ ɞɚɜɥɟɧɢɹ ɨɩɪɟɫɫɨɜɤɢ > P[opres]:=A*sigma[tau]/2*(2*lambda*ln(R[t]/R1)-R[t]^2/R2^2+(1-lambda)*R[t]^2/R1^2+lambda); Popres := 108.20162

Ɍɨɥɳɢɧɚ ɫɬɟɧɤɢ ɚɩɩɚɪɚɬɚ > Delta:=R2-R1;

∆ := 0.08000014

Ɋɚɫɱɟɬ ɪɚɞɢɭɫɚ ɩɥɚɫɬɢɱɟɫɤɢɯ ɞɟɮɨɪɦɚɰɢɣ > R[t]:=R1+Delta*T; Rt := 0.63200006

Ɋɚɫɱɟɬ ɧɚɩɪɹɠɟɧɢɣ ɩɪɢ ɪɚɛɨɱɟɦ ɢ ɩɪɨɛɧɨɦ ɞɚɜɥɟɧɢɢ > sigma[r_rab]:=Sigma("r",P[rab]): sigma[k_rab]:=Sigma("k",P[rab]): sigma[z_rab]:=Sigma("z",P[rab]): > sigma[r_prob]:= Sigma("r"): sigma[k_prob]:= Sigma("k"): sigma[z_prob]:= Sigma("z"): Ɋɚɫɱɟɬ ɧɚɩɪɹɠɟɧɢɣ ɪɚɡɝɪɭɡɤɢ ɢ ɨɫɬɚɬɨɱɧɵɯ ɧɚɩɪɹɠɟɧɢɣ > sigma[r_raz]:=Sigma("r",P[opres]): sigma[k_raz]:=Sigma("k",P[opres]): sigma[z_raz]:=Sigma("z",P[opres]):

> sigma[r_ost]:=sigma[r_prob]-sigma[r_raz]: sigma[k_ost]:=sigma[k_prob]-sigma[k_raz]: sigma[z_ost]:=sigma[z_prob]-sigma[z_raz]: Ɋɚɫɱɟɬ ɧɚɩɪɹɠɟɧɢɣ ɩɪɢ ɪɚɛɨɱɟɦ ɞɚɜɥɟɧɢɢ ɩɨɫɥɟ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɚɜɬɨɮɪɟɬɢɪɨɜɚɧɢɹ > sigma[r_auto]:=sigma[r_rab]+sigma[r_ost]: sigma[k_auto]:=sigma[k_rab]+sigma[k_ost]: sigma[z_auto]:=sigma[z_rab]+sigma[z_ost]: ɋɬɪɨɢɦ ɷɩɸɪɵ ɧɚɩɪɹɠɟɧɢɣ > plot([sigma[k_rab],sigma[k_auto]],r=R1..R2, color=[black,black,black,blue,blue,blue], labelfont=[COURIER,12],labels=["ɦ","Ɇɉɚ"], titlefont=[COURIER,12],title="ɗɩɸɪɚ ɤɨɥɶɰɟɜɵɯ ɧɚɩɪɹɠɟɧɢɣ\n ɩɪɢ ɪɚɛɨɱɟɦ ɞɚɜɥɟɧɢɢ\n ɞɨ ɢ ɩɨɫɥɟ ɚɜɬɨɮɪɟɬɢɪɨɜɚɧɢɹ", legend=["sigma_ɤ","sigma_ɤ (auto)"], thickness=[1,3]);

> plot([sigma[z_rab],sigma[z_auto]],r=R1..R2, color=[black,black,black,blue,blue,blue], labelfont=[COURIER,12],labels=["ɦ","Ɇɉɚ"],titlefont=[COURIER,12],title="ɗɩɸɪɚ ɨɫɟɜɵɯ ɧɚɩɪɹɠɟɧɢɣ\n ɩɪɢ ɪɚɛɨɱɟɦ ɞɚɜɥɟɧɢɢ\n ɞɨ ɢ ɩɨɫɥɟ ɚɜɬɨɮɪɟɬɢɪɨɜɚɧɢɹ", legend=["sigma_z","sigma_z (auto)"], thickness=[1,3]);

ɗɩɸɪɚ ɪɚɞɢɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɧɟ ɩɪɢɜɟɞɟɧɚ, ɜɫɥɟɞɫɬɜɢɟ ɢɯ ɧɟɫɭɳɟɫɬɜɟɧɧɵɯ ɢɡɦɟɧɟɧɢɣ. Ɉɛɪɚɬɢɬɟ ɜɧɢɦɚɧɢɟ ɤɚɤ ɩɪɨɰɟɫɫ ɚɜɬɨɮɪɟɬɢɪɨɜɚɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɩɟɪɟɪɚɫɩɪɟɞɟɥɟɧɢɸ ɤɨɥɶɰɟɜɵɯ ɢ ɨɫɟɜɵɯ ɧɚɩɪɹɠɟɧɢɣ.

ɋɉɂɋɈɄ ɅɂɌȿɊȺɌɍɊɕ 1. Ⱦɶɹɤɨɧɨɜ ȼ. ɉ. Maple 7: ɍɱɟɛɧɵɣ ɤɭɪɫ. ɋɉɛ.: ɉɢɬɟɪ, 2002. 672 ɫ. 2. Ɇɚɬɪɨɫɨɜ Ⱥ. ȼ. Maple 6. Ɋɟɲɟɧɢɟ ɡɚɞɚɱ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ ɢ ɦɟɯɚɧɢɤɢ. ɋɉɛ.: BHV, 2001. 528 ɫ. 3. Ⱥɥɚɞɶɟɜ ȼ. Ɂ., Ȼɨɝɞɹɜɢɱɸɫ Ɇ. Ⱥ. Maple 6: Ɋɟɲɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ, ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɢ ɢɧɠɟɧɟɪɧɨ-ɮɢɡɢɱɟɫɤɢɯ ɡɚɞɚɱ. Ɇ.: Ʌɚɛɨɪɚɬɨɪɢɹ ɛɚɡɨɜɵɯ ɡɧɚɧɢɢ, 2001. 824 ɫ. 4. ɉɪɨɱɧɨɫɬɶ. ɍɫɬɨɣɱɢɜɨɫɬɶ. Ʉɨɥɟɛɚɧɢɹ. ɋɩɪɚɜɨɱɧɢɤ / ɉɨɞ ɪɟɞ. ɂ. Ⱥ. Ȼɢɪɝɟɪɚ. Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1968. Ɍ. 2. 464 ɫ. 5. Ɍɢɦɨɲɟɧɤɨ ɋ. ɉ. ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɨɜ. Ɇ.: ɇɚɭɤɚ, 1965. Ɍ. 2. 480 ɫ. 6. Ƚɠɢɪɨɜ Ɋ. ɂ. Ʉɪɚɬɤɢɣ ɫɩɪɚɜɨɱɧɢɤ ɤɨɧɫɬɪɭɤɬɨɪɚ. Ʌ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1983. 465 ɫ. 7. Ɇɚɥɢɧɢɧ ɇ. ɇ. ɉɪɢɤɥɚɞɧɚɹ ɬɟɨɪɢɹ ɩɥɚɫɬɢɱɧɨɫɬɢ ɢ ɩɨɥɡɭɱɟɫɬɢ. Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1975. 400 ɫ.

ɋɈȾȿɊɀȺɇɂȿ ȼȼȿȾȿɇɂȿ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... 1 ɂɇɌȿɊɎȿɃɋ MAPLE . . . . . . . . . . . . . . . . . . . . . . . . .......... 1.1 ɂɇɌȿɊɎȿɃɋ ɊȺȻɈɑȿȽɈ ȾɈɄɍɆȿɇɌȺ . . . . . ......... 1.2 ɂɇɌȿɊɎȿɃɋ ɋɉɊȺȼɈɑɇɈɃ ɋɂɋɌȿɆɕ . . . . . . ...........

3 4 4 6

1.3 ɂɇɌȿɊɎȿɃɋ ȾȼɍɏɆȿɊɇɈɃ ȽɊȺɎɂɑȿɋɄɈɃ ɋɂɋɌȿɆɕ 1.4 ɂɇɌȿɊɎȿɃɋ ɌɊȿɏɆȿɊɇɈɃ ȽɊȺɎɂɑȿɋɄɈɃ ɋɂɋɌȿɆɕ 2 ɋɂɇɌȺɄɋɂɋ əɁɕɄȺ MAPLE . . . . . . . . . . . . . . . . .......... 2.1 ɉɊɈɋɌɕȿ ȼɕɑɂɋɅȿɇɂə . . . . . . . . . . . . . . . . . ......... 2.2 ȼɕɑɂɋɅȿɇɂȿ ɋɍɆɆɕ ɊəȾȺ, ɉɊɈɂɁȼȿȾȿɇɂə ɂ ɉɊȿȾȿɅȺ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 2.3 ɈɋɇɈȼɇɕȿ Ɍɂɉɕ ȾȺɇɇɕɏ . . . . . . . . . . . . . .......... 2.4 ɈɉȿɊȺɐɂɂ ɋ ɎɈɊɆɍɅȺɆɂ . . . . . . . . . . . . . . ......... 2.5 ɉɊɈɂɁȼɈȾɇɕȿ ɂ ɂɇɌȿȽɊȺɅɕ . . . . . . . . . . . ......... 2.6 ɉȺɄȿɌɕ ɊȺɋɒɂɊȿɇɂɃ ɂ ɊȺȻɈɌȺ ɋ ɇɂɆɂ . . . . . . . . 3 ȾȼɍɏɆȿɊɇȺə ȽɊȺɎɂɄȺ . . . . . . . . . . . . . . . . . . . . ......... 3.1 ɋɈȼɆȿɓȿɇɂȿ ȽɊȺɎɂɄɈȼ . . . . . . . . . . . . . . . ........ 3.2 ȺɇɂɆȺɐɂə ȽɊȺɎɂɄɈȼ . . . . . . . . . . . . . . . . . ......... 3.3 ɉɈɋɌɊɈȿɇɂȿ ȽɊȺɎɂɄȺ ɇȿəȼɇɈɃ ɎɍɇɄɐɂɂ . . . . . 3.4 ɉɈɋɌɊɈȿɇɂȿ ȽɊȺɎɂɄɈȼ ɅɂɇɂəɆɂ ɊȺȼɇɈȽɈ ɍɊɈȼɇə . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 3.5 ȽɊȺɎɂɄ ɉɅɈɌɇɈɋɌɂ . . . . . . . . . . . . . . . . . . . . ......... 3.6 ȽɊȺɎɂɄ ȼȿɄɌɈɊɇɈȽɈ ɉɈɅə ȽɊȺȾɂȿɇɌȺ ......... 3.7 ȽɊȺɎɂɄ ȼȿɄɌɈɊɇɈȽɈ ɉɈɅə . . . . . . . . . . . . . ........ 3.8 ɋɈȼɆȿɓȿɇɂȿ ȽɊȺɎɂɄɈȼ ɉɈɋɌɊɈȿɇɇɕɏ ɊȺɁɅɂɑ-ɇɕɆɂ ɎɍɇɄɐɂəɆɂ . . . . . . . . . . . . . ................ 4 ɌɊȿɏɆȿɊɇȺə ȽɊȺɎɂɄȺ . . . . . . . . . . . . . . . . . . . . ......... 5 Ɋȿɒȿɇɂȿ ɍɊȺȼɇȿɇɂɃ, ɋɂɋɌȿɆ ɍɊȺȼɇȿɇɂɃ ɂ ɇȿɊȺȼȿɇɋɌȼ . . . . . . . . . . . . . . . . . . . . . . . . .................. 6 ɍɉɊȺȼɅəɘɓɂȿ ɄɈɇɋɌɊɍɄɐɂɂ . . . . . . . . . . . ......... 7 ɉɊɈɐȿȾɍɊɕ ɂ ɎɍɇɄɐɂɂ . . . . . . . . . . . . . . . . . . ......... 8 Ɋȿɒȿɇɂȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ . . . . . 8.1 Ɋȿɒȿɇɂȿ ɈȻɕɄɇɈȼȿɇɇɕɏ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ . . . . . . . . . . . . . . . . . .................... 8.2 Ɋȿɒȿɇɂȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼ-

7 9 11 11

13 15 21 25 27 31 34 35 35

37 39 39 40 41 43 47 50 53 58 58 63

ɇȿɇɂɃ ȼ ɑȺɋɌɇɕɏ ɉɊɈɂɁȼɈȾɇɕɏ . . . . . . . . . . . . . . . ......... 8.3 ɑɂɋɅȿɇɇɈȿ Ɋȿɒȿɇɂȿ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ ȼ ɑȺɋɌɇɕɏ ɉɊɈɂɁȼɈȾɇɕɏ ......... 9 ɉɊɂɆȿɊɕ Ɋȿɒȿɇɂə ɂɇɀȿɇȿɊɇɕɏ ɁȺȾȺɑ ......... 9.1 ɊȺɋɑȿɌ ȻɕɋɌɊɈ ȼɊȺɓȺɘɓɂɏɋə ȾɂɋɄɈȼ . . . . . . . 9.2 ɊȺɋɑȿɌ ȺɉɉȺɊȺɌȺ ȼɕɋɈɄɈȽɈ ȾȺȼɅȿɇɂə . . . . . . . ɋɉɂɋɈɄ ɅɂɌȿɊȺɌɍɊɕ . . . . . . . . . . . . . . . . . . . . . . . ..........

67 70 70 75 79