Calculus and Linear Algebra in Recipes

This book provides a clear and easy-to-understand introduction to higher mathematics with numerous examples. The author shows how to solve typical problems in a recipe-like manner and divides the material into short, easily digestible learning units.

Have you ever cooked a 3-course meal based on a recipe? That generally works quite well, even if you are not a great cook. What does this have to do with mathematics? Well, you can solve a lot of math problems recipe-wise: Need to solve a Riccati's differential equation or the singular value decomposition of a matrix? Look it up in this book, you'll find a recipe for it here. Recipes are available for problems from the field of:

· Calculus in one and more variables,

· Linear algebra,

· Vector analysis,

· Theory on differential equations, ordinary and partial,

· Theory of integral transformations,

· Function theory.

Other features of this book include:

· The division of Higher Mathematics into approximately 100 chapters of roughly equal length. Each chapter covers approximately the material of a 90-minute lecture.

· Numerous exercises and solutions

· Many problems in higher mathematics can be solved with computers. We always indicate how it works with MATLAB®.

This 2nd English edition has been completely revised and numerous examples, illustrations, explanations and further exercises have been added.

Preface. - 1 Terminology, Symbols and Sets. - 2 The Natural Numbers, Integers and Rational Numbers. - 3 The Real Numbers. - 4 Machine Numbers. - 5 Polynomials. - 6 Trigonometric Functions. - 7 Complex Numbers Cartesian Coordinates. - 8 Complex Numbers Polar Coordinates. - 9 Linear Equation Systems. - 10 Calculating with Matrices. - 11
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-Decomposition of a Matrix. - 12 The Determinant. - 13 Vector Spaces. - 14 Generating Systems and Linear (In-)Dependence. - 15 Bases of Vector Spaces. - 16 Orthogonality I. - 17 Orthogonality II. - 18 The Linear Least Squares Problem. - 19 The
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-Decomposition of a Matrix. - 20 Sequences. - 21 Calculation of Limits of Sequences. - 22 Series. - 23 Mappings. - 24 Power Series. - 25 Limits and Continuity. - 26 Differentiation. - 27 Applications of Differential Calculus I. -28 Applications of Differential Calculus II. - 29 Polynomial and Spline Interpolation. - 30 Integration I. - 31 Integration II. - 32 Improper Integrals. - 33 Separable and Linear First Order Differential Equations. - 34 Linear Differential Equations with Constant Coefficients. - 35 Some Special Types of Differential Equations. - 36 Numerics of Ordinary Differential Equations I. - 37 Linear Mappings and Representation Matrices. - 38 Basic Transformation. - 39 Diagonalization Eigenvalues and Eigenvectors. - 40 Numerical Calculation of Eigenvalues and Eigenvectors. - 41 Quadrics. - 42 Schur Decomposition and Singular Value Decomposition. - 43 The Jordan Normal Form I. - 44 The Jordan Normal Form II. - 45 Definiteness and Matrix Norms. - 46 Functions of Several Variables. - 47 Partial Differentiation Gradient, Hessian Matrix, Jacobian Matrix. - 48 Applications of Partial Derivatives. - 49 Determination of Extreme Values. - 50 Determination of Extreme Values under Constraints. - 51 Total Differentiation, Differential Operators. - 52 Implicit Functions. - 53 Coordinate Transformations. - 54 Curves I. - 55 Curves II. - 56 Curve Integrals. - 57 Gradient Fields. - 58 Area Integrals. - 59 The Transformation Formula. - 60 Surfaces and Surface Integrals. - 61 Integral Theorems I. - 62 Integral Theorems II. - 63 Generalities on Differential Equations. - 64 The Exact Differential Equation. - 65 Linear Differential Equations Systems I. - 66 Linear Differential Equations Systems II. - 67 Linear Differential Equations Systems III. - 68 Boundary Value Problems. - 69 Basic Concepts of Numerics. - 70 Fixed Point Iteration. - 71 Iterative Methods for Linear Equation Systems. - 72 Optimization. - 73 Numerics of Ordinary Differential Equations II. - 74 Fourier Series - Calculation of Fourier Coefficients. - 75 Fourier Series Background, Theorems and Application. - 76 Fourier Transformation I. - 77 Fourier Transformation II. - 78 Discrete Fourier Transformation. - 79 The Laplace Transformation. - 80 Holomorphic Functions. - 81 Complex Integration. - 82 Laurent Series. - 83 The Residue Calculus. - 84 Conformal Mappings. - 85 Harmonic Functions and the Dirichlet Boundary Value Problem. - 86 First Order Partial Differential Equations. - 87 Second Order Partial Differential Equations General. - 88 The Laplace or Poisson Equation. - 89 The Heat Conduction Equation. - 90 The Wave Equation. - 91 Solving pDEs with Fourier- and Laplace Transformations. - Index.