Table of Contents:
| Preface | i | |||
| 1 | Fundamentals of Integral Calculus | 1 | ||
| 1.1 | The Fundamental Theorem of Calculus | 7 | ||
| 1.1.1 | The Substitution Method | 8 | ||
| 1.1.2 | The Method of Partial Integration | 9 | ||
| 1.2 | The Improper Integral | 10 | ||
| 1.1.2 | Euler's Gamma Function | 12 | ||
| 1.3 | Riemann Sums and Arc Length | 13 | ||
| 2 | Fourier Series | 17 | ||
| 2.1 | Uniform Convergence of Function Sequences | 20 | ||
| 2.1.1 | Gibbs' Phenomenon | 25 | ||
| 2.1.2 | Differentiation of the Limiting Function | 26 | ||
| 2.2 | Absolutely Convergent Series | 28 | ||
| 2.3 | Fourier Series Theory | 30 | ||
| 2.3.1 | The Completeness Theorem | 35 | ||
| 3 | Three-Dimensional Euclidean Space | 39 | ||
| 3.1 | The Scalar Product | 40 | ||
| 3.2 | The Vector Product | 42 | ||
| 3.2.1 | Geometric Interpretation | 43 | ||
| 3.3 | The Isometry of Euclidean \(\mathbb{R}^3\) | 46 | ||
| 3.3.1 | Description of Linear Self-Mappings through Matrices | 47 | ||
| 3.4 | The Scalar Triple Product | 48 | ||
| 3.4.1 | Geometric Significance of the Scalar Triple Product | 49 | ||
| 3.5 | The Inverse Matrix | 51 | ||
| 3.5.1 | The Orthogonal Group \(O_3\) of Euclidean \(\mathbb{R}^3\) | 52 | ||
| 4 | Systems of Linear Equations | 55 | ||
| 4.1 | Solutions for Systems of Linear Equations | 57 | ||
| 4.1.1 | The Gaussian Algorithm | 57 | ||
| 5 | Plane and Spatial Curves | 63 | ||
| 5.1 | Definition of the Curve Length | 65 | ||
| 5.2 | The Line Integral over a Vector Field | 66 | ||
| 5.3 | Polar Coordinates for Plane Curves | 68 | ||
| 5.4 | The Curvature of a Plane Curve | 70 | ||
| 6 | Neighborhoods and Limits | 77 | ||
| 6.1 | Fixed Point Theorem | 81 | ||
| 7 | Partial and Total Derivative | 85 | ||
| 7.1 | Definition of the Partial Derivative | 85 | ||
| 7.1.1 | Generalized Chain Rule | 88 | ||
| 7.2 | Definition of the Total Derivative | 90 | ||
| 7.2.1 | Geometric Properties of the Total Derivative | 92 | ||
| 8 | Higher Derivatives, Taylor Formula and Local Extrema | 95 | ||
| 8.1 | The Symmetry of the Second Derivative | 95 | ||
| 8.1.1 | Integrability Criterion for Vector Fields | 96 | ||
| 8.2 | A Simple Version of the Taylor Formula in \(\mathbb{R}^n\) | 98 | ||
| 8.2.1 | Application of the Simple Version of the Taylor Formula to Stationary Points | 99 | ||
| 9 | Implicit Functions and Applications | 103 | ||
| 9.1 | Existence Theorem for Implicit Functions | 105 | ||
| 9.2 | Local Extrema with Constraints | 107 | ||
| 9.3 | The Problem of Reverse Mapping (Coordinate Transformation) | 113 | ||