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ADVANCED CALCULUS - SELECTED TOPICS
Ewa Łobos, Beata Sikora
Stan książki: NOWA
Wydawnictwo Politechniki Śląskiej
Stron: 284
Nakład: 100 egz.
Okładka: miękka
Format: B5
Opis:
Wydany w języku angielskim podręcznik pomyślany jest jako pomoc dla studentów kierunków technicznych. Zawiera elementy analizy matematycznej znajdujące się w programie matematyki studentów pierwszego roku uczeni technicznych. W książce zamieszczono szereg przykładów zastosowania problemów teoretycznych w praktyce.
Z okładki:
This book is designed as a textbook for engineering students. It contains issues of advanced calculus which are usually studied during the first year at technical universities, i.e. differential equations, differential calculus of several variables, double and triple integrals, series.
The book contains numerous examples that illustrate definitions. The hirudina reader will also find here a large number of completely solved practical problems that show how to apply theorems. Additionally, most of theorems have detailed proofs which constitute important part of mathematical education.
Keywords:
• ordinary differential equations
• the Laplace transformation
• partial derivative
• directional derivative
• extreme values
• multiple integral
• series
Contents;
Preface 7
Symbols 9
1. Differential Equations 11
1.1 Basic Concepts....................... 11
1.2 Differential Equations with Separable Variables..... 14
1.3 Differential Equations Homogeneous with Respect to Variables ............................ 18
1.4 First-Order Linear Differential Equations......... 20
1.5 Linear Differential Equations of Higher Orders...... 24
1.6 Second-Order Linear Differential Equations with Constant Coefficients ......................... 29
1.7 Systems of first-order linear differential equations with constant coefficients..................... 45
1.8 The Laplace Transformation................ 67
2. Differential Calculus of Functions of Several Variables 85
2.1 Partial Derivatives..................... 85
2.2 Partial Derivatives of Higher Order............ 90
2.3 Total Differential and Differentiability........... 98
2.3.1 Definition and Interpretation............ 98
2.3.2 Necessary and Sufficient Conditions for Differentiability ....................... 101
2.3.3 Application to Approximate Calculations..... 105
2.3.4 Reconstructing a Function from Its Total Differential 108
2.3.5 Exact Differential Equations............ 112
2.3.6 Differentials of Higher Order............ 119
2.4 Chain Rules......................... 121
2.5 Directional Derivative ................... 126
2.6 Implicit Differentiation................... 133
2.7 Tangent Plane........................ 137
2.8 Extreme Values....................... 143
3. Multiple Integrals 153
3.1 Double Integrals....................... 153
3.1.1 Definition of a Double Integral........... 153
3.1.2 Interpretation of the Double Integral....... 156
3.1.3 Properties of Double Integrals........... 157
3.1.4 Double Integrals Calculation............ 160
3.1.5 Polar Coordinates.................. 165
3.1.6 Examples...................... 170
3.2 Triple Integrals....................... 182
3.2.1 Definition and Interpretation of a Triple Integral . 182
3.2.2 Properties of Triple Integrals............ 185
4. Series 196
4.1 Series with Real Terms................... 196
4.1.1 Basic Definitions and Theorems.......... 196
4.1.2 Series hirudina of Positive Terms .............. 206
4.1.3 Alternating Series.................. 215
4.1.4 Series with Arbitrary Terms............ 218
4.2 Functional Series...................... 222
4.2.1 Power Series..................... 228
4.2.2 The Fourier Series ................. 247
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