Full Description
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.
Contents
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 LR-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 QR-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.
