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Table of Contents:
Preface i  
1  Integrals with Parameters 1  
1.1  Differentiation under the Integra 2  
1.2  Swapping the Order of Integration in Double Integrals 3  
1.3  Integrals with variable Limits 4  
1.4  Improper Integrals with Parameters 5  
1.5  The Laplace Transform 7  
1.5.1  The Laplace Transform of the Derivative 9  
2  Multiple Integrals 11  
2.1  Integral for a restricted Function Class 11  
2.1.1  A Characterization of the Integral 14  
2.1.2  Transformation Formula of the Multiple Integral under Linear Mappings 15  
2.2  Extension of the Class of Integrable Functions 17  
2.2.1  Reversal of the Order of Integration 19  
2.2.2  Application: Cavalieri's Principle 20  
2.3  Formulation of the General Transformation Formula for Area Integrals 21  
3  Integral Theorems in the Plane 25  
3.1  Integrability Criterion 26  
3.2  The Cauchy-Riemann Differential Equations and the Cauchy Integral Theorem 29  
3.2.1  Cauchy's Integral Theorem 31  
4  Surfaces in Space and Surface Integrals 35  
4.1  Surface Representations 35  
4.1.1  Regularity Condition 37  
4.1.2  Geometric Significance 37  
4.2  Area Measurement of Curved Surfaces 39  
4.2.1  Invariance of the Area Content with respect to Orthogonal Transformations \(A\) of \(\mathbb{R}^3\) 41  
4.2.2  Invariance of the Area Measure under Parameter Transformations 41  
4.3  Gauß' Theorem 42  
4.4  Differential Geometric Interpretation of Gauß' Theorem 45  
4.5  Stokes' Integral Theorem 47  
5  Quadratic Matrices and Determinants 49  
5.1  Characteristic Properties of the Determinant 51  
5.1.1  Calculation Rules 53  
5.2  Product Theorem for Determinants 54  
5.2.1  Behavior of the Determinant under Elementary Row Transformations 55  
5.2.2  Rules for Row Transformations 56  
5.2.3  The Adjugate of a Matrix \(A \in M_{n}(K)\) 56  
5.2.4  The Inverse Matrix 57  
6  Vector Spaces, Linear Self-mappings, Eigenvalues 61  
6.1  Vector Space Axioms 61  
6.1.1  Linear Combinations 62  
6.1.2  Steinitz Exchange Lemma 63  
6.2  Subspaces and Dimension Formula 64  
6.3  Eigenvalues and Eigenvectors 66  
6.4  Euclidean and Unitary Scalar Products 70  
6.4.1  Orthonormalization Theorem 71  
6.4.2  The Principal Axis Transformation 73  
7  Linear Differential Equation with Constant Coefficients 79  
7.1  Differential Equation of Growth and Decay 79  
7.1.1  Matching Inhomogeneous Differential Equation 80  
7.2  Differential Equation of Damped Oscillation 80  
7.2.1  Matching Inhomogeneous Linear Differential Equation 84  
7.2.2  Rewriting the 2nd Order Differential Equation into a System of 1st Order Differential Equations 85  
7.3  Systems of 1st Order Linear Differential Equations with Constant Coefficients 87  
7.3.1  Picard-Iteration Ansatz 87  
7.4  Matrix Norms 88  
7.4.1  Exponential Function on \(M_{n}(\mathbb{K})\) 90  
7.5  General 1st Order Linear Differential Equation in \(\mathbb{K}^n\) 92  
7.6  The Jordan Normal Form for Matrices in \(M_{n}(\mathbb{C})\) 94  
7.7  Higher-order Scalar-valued Linear Differential Equations 96  
7.8  Stability Questions 98  
7.8.1  Stability Criterion for Real Polynomials 100  
7.9  Applications of the Laplace Transform 101  
8  Existence and Uniqueness Theorem for Explicit Ordinary Differential Equations 107  
8.1  The Lipschitz Condition 108  
8.1.1  Definition of the Lipschitz Condition 109  
8.1.2  Dependence of Solutions on Initial Values 110  
8.2  Picard-Lindelöf Existence Theorem 111  
8.2.1  The Runge-Kutta Method 114  
8.2.2  Simpson's Rule 114  
8.2.3  The Power Series Ansatz 115  
8.3  Ordinary Differential Equations of \(n\)-th Order 116  
8.3.1  Systems of 1-st Order Linear Differential Equations with Non-constant Coefficients 117  
8.3.2  The Case of Minimum Dimension \(n = 1\) 119  

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