Calculus and Linear Algebra in Recipes: Terms, phrases and numerous examples in short learning units

Length: 1064 pages
Edition: 1
Language: English
Publisher: Springer
Publication Date: 2022-12-10
ISBN-10: 3662654571
ISBN-13: 9783662654576
Sales Rank: #4218264 (See Top 100 Books)

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

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.
Many tasks, the solutions to which can be found in the accompanying workbook.
Many problems in higher mathematics can be solved with computers. We always indicate how it works with MATLAB®.

For the present 3rd edition, the book has been completely revised and supplemented by a section on the solution of boundary value problems for ordinary differential equations, by the topic of residue estimates for Taylor expansions and by the characteristic method for partial differential equations of the 1st order, as well as by several additional problems.

Cover
Front Matter
1. Speech, 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 Systems of Equations
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 Equalization 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 Differential Equations of First Order
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 Transformation Matrices
38. Base 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. Extreme Value Determination
50. Extreme Value Determination Under Constraints
51. Total Differentiation, Differential Operators
52. Implicit Functions
53. Coordinate Transformations
54. Curves I
55. Curves II
56. Line Integrals
57. Gradient Fields
58. Multiple Integrals
59. Substitution for Multiple Variables
60. Surfaces and Surface Integrals
61. Integral Theorems I
62. Integral Theorems II
63. General Information on Differential Equations
64. The Exact Differential Equation
65. Linear Differential Equation Systems I
66. Linear Differential Equation Systems II
67. Linear Differential Equation Systems III
68. Boundary Value Problems
69. Basic Concepts of Numerics
70. Fixed Point Iteration
71. Iterative Methods for Systems of Linear Equations
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 Transform I
77. Fourier Transform II
78. Discrete Fourier Transform
79. The Laplace Transformation
80. Holomorphic Functions
81. Complex Integration
82. Laurent Series
83. The Residual Calculus
84. Conformal Mappings
85. Harmonic Functions and the Dirichlet Boundary Value Problem
86. Partial Differential Equations of First Order
87. Partial Differential Equations of Second Order: General
88. The Laplace or Poisson Equation
89. The Heat Conduction Equation
90. The Wave Equation
91. Solving PDEs with Fourier and Laplace Transforms
Back Matter