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

Foreword to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
1 Speech, Symbols and Sets
1.1 Speech Patterns and Symbols in Mathematics
1.1.1 Junctors
1.1.2 Quantifiers
1.2 Summation and Product Symbol
1.3 Powers and Roots
1.4 Symbols of Set Theory
1.5 Exercises
2 The Natural Numbers, Integers and Rational Numbers
2.1 The Natural Numbers
2.2 The Integers
2.3 The Rational Numbers
2.4 Exercises
3 The Real Numbers
3.1 Basics
3.2 Real Intervals
3.3 The Absolute Value of a Real Number
3.4 n-th Roots
3.5 Solving Equations and Inequalities
3.6 Maximum, Minimum, Supremum and Infimum
3.7 Exercises
4 Machine Numbers
4.1 b-adic Representation of Real Numbers
4.2 Floating Point Numbers
4.2.1 Machine Numbers
4.2.2 Machine Epsilon, Rounding and Floating Point Arithmetic
4.2.3 Loss of Significance
4.3 Exercises
5 Polynomials
5.1 Polynomials: Multiplication and Division
5.2 Factorization of Polynomials
5.3 Evaluating Polynomials
5.4 Partial Fraction Decomposition
5.5 Exercises
6 Trigonometric Functions
6.1 Sine and Cosine
6.2 Tangent and Cotangent
6.3 The Inverse Functions of the Trigonometric Functions
6.4 Exercises
7 Complex Numbers: Cartesian Coordinates
7.1 Construction of C
7.2 The Imaginary Unit and Other Terms
7.3 The Fundamental Theorem of Algebra
7.4 Exercises
8 Complex Numbers: Polar Coordinates
8.1 The Polar Representation
8.2 Applications of the Polar Representation
8.3 Exercises
9 Linear Systems of Equations
9.1 The Gaussian Elimination Method
9.2 The Rank of a Matrix
9.3 Homogeneous Linear Systems of Equations
9.4 Exercises
10 Calculating with Matrices
10.1 Definition of Matrices and Some Special Matrices
10.2 Arithmetic Operations
10.3 Inverting Matrices
10.4 Calculation Rules
10.5 Exercises
11 LR-Decomposition of a Matrix
11.1 Motivation
11.2 The LR-Decomposition: Simplified Variant
11.3 The LR-Decomposition: General Variant
11.4 The LR-Decomposition-with Column Pivot Search
11.5 Exercises
12 The Determinant
12.1 Definition of the Determinant
12.2 Calculation of the Determinant
12.3 Applications of the Determinant
12.4 Exercises
13 Vector Spaces
13.1 Definition and Important Examples
13.2 Subspaces
13.3 Exercises
14 Generating Systems and Linear (In)Dependence
14.1 Linear Combinations
14.2 The Span of X
14.3 Linear (In)Dependence
14.4 Exercises
15 Bases of Vector Spaces
15.1 Bases
15.2 Applications to Matrices and Systems of Linear Equations
15.3 Exercises
16 Orthogonality I
16.1 Scalar Products
16.2 Length, Distance, Angle and Orthogonality
16.3 Orthonormal Bases
16.4 Orthogonal Decomposition and Linear Combination with Respect to an ONB
16.5 Orthogonal Matrices
16.6 Exercises
17 Orthogonality II
17.1 The Orthonormalization Method of Gram and Schmidt
17.2 The Vector Product and the (Scalar) Triple Product
17.3 The Orthogonal Projection
17.4 Exercises
18 The Linear Equalization Problem
18.1 The Linear Equalization Problem and Its Solution
18.2 The Orthogonal Projection
18.3 Solution of an Over-Determined Linear System of Equations
18.4 The Method of Least Squares
18.5 Exercises
19 The QR-Decomposition of a Matrix
19.1 Full and Reduced QR-Decomposition
19.2 Construction of the QR-Decomposition
19.3 Applications of the QR-Decomposition
19.3.1 Solving a System of Linear Equations
19.3.2 Solving the Linear Equalization Problem
19.4 Exercises
20 Sequences
20.1 Terms
20.2 Convergence and Divergence of Sequences
20.3 Exercises
21 Calculation of Limits of Sequences
21.1 Determining Limits of Explicit Sequences
21.2 Determining Limits of Recursive Sequences
21.3 Exercises
22 Series
22.1 Definition and Examples
22.2 Convergence Criteria
22.3 Exercises
23 Mappings
23.1 Terms and Examples
23.2 Composition, Injective, Surjective, Bijective
23.3 The Inverse Mapping
23.4 Bounded and Monotone Functions
23.5 Exercises
24 Power Series
24.1 The Domain of Convergence of Real Power Series
24.2 The Domain of Convergence of Complex Power Series
24.3 The Exponential and the Logarithmic Function
24.4 The Hyperbolic Functions
24.5 Exercises
25 Limits and Continuity
25.1 Limits of Functions
25.2 Asymptotes of Functions
25.3 Continuity
25.4 Important Theorems about Continuous Functions
25.5 The Bisection Method
25.6 Exercises
26 Differentiation
26.1 The Derivative and the Derivative Function
26.2 Derivation Rules
26.3 Numerical Differentiation
26.4 Exercises
27 Applications of Differential Calculus I
27.1 Monotonicity
27.2 Local and Global Extrema
27.3 Determination of Extrema and Extremal Points
27.4 Convexity
27.5 The Rule of L'Hospital
27.6 Exercises
28 Applications of Differential Calculus II
28.1 The Newton Method
28.2 Taylor Expansion
28.3 Remainder Estimates
28.4 Determination of Taylor Series
28.5 Exercises
29 Polynomial and Spline Interpolation
29.1 Polynomial Interpolation
29.2 Construction of Cubic Splines
29.3 Exercises
30 Integration I
30.1 The Definite Integral
30.2 The Indefinite Integral
30.3 Exercises
31 Integration II
31.1 Integration of Rational Functions
31.2 Rational Functions in Sine and Cosine
31.3 Numerical Integration
31.4 Volumes and Surfaces of Solids of Revolution
31.5 Exercises
32 Improper Integrals
32.1 Calculation of Improper Integrals
32.2 The Comparison Test for Improper Integrals
32.3 Exercises
33 Separable and Linear Differential Equations of First Order
33.1 First Differential Equations
33.2 Separable Differential Equations
33.2.1 The Procedure for Solving a Separable Differential Equation
33.2.2 Initial Value Problems
33.3 The Linear Differential Equation of First Order
33.4 Exercises
34 Linear Differential Equations with Constant Coefficients
34.1 Homogeneous Linear Differential Equations with Constant Coefficients
34.2 Inhomogeneous Linear Differential Equations with Constant Coefficients
34.2.1 Variation of Parameters
34.2.2 Approach of the Right-Hand Side Type
34.3 Exercises
35 Some Special Types of Differential Equations
35.1 The Homogeneous Differential Equation
35.2 The Euler Differential Equation
35.3 Bernoulli's Differential Equation
35.4 The Riccati Differential Equation
35.5 The Power Series Approach
35.6 Exercises
36 Numerics of Ordinary Differential Equations I
36.1 First Procedure
36.2 Runge-Kutta Method
36.3 Multistep Methods
36.4 Exercises
37 Linear Mappings and Transformation Matrices
37.1 Definitions and Examples
37.2 Image, Kernel and the Dimensional Formula
37.3 Coordinate Vectors
37.4 Transformation Matrices
37.5 Exercises
38 Base Transformation
38.1 The Tansformation Matrix of the Composition of Linear Mappings
38.2 Base Transformation
38.3 The Two Methods for Determining Transformation Matrices
38.4 Exercises
39 Diagonalization: Eigenvalues and Eigenvectors
39.1 Eigenvalues and Eigenvectors of Matrices
39.2 Diagonalizing Matrices
39.3 The Characteristic Polynomial of a Matrix
39.4 Diagonalization of Real Symmetric Matrices
39.5 Exercises
40 Numerical Calculation of Eigenvalues and Eigenvectors
40.1 Gerschgorin Circles
40.2 Vector Iteration
40.3 The Jacobian Method
40.4 The QR-Method
40.5 Exercises
41 Quadrics
41.1 Terms and First Examples
41.2 Transformation to Normal Form
41.3 Exercises
42 Schur Decomposition and Singular Value Decomposition
42.1 The Schur Decomposition
42.2 Calculation of the Schur Decomposition
42.3 Singular Value Decomposition
42.4 Determination of the Singular Value Decomposition
42.5 Exercises
43 The Jordan Normal Form I
43.1 Existence of the Jordan Normal Form
43.2 Generalized Eigenspaces
43.3 Exercises
44 The Jordan Normal Form II
44.1 Construction of a Jordan Base
44.2 Number and Size of Jordan Boxes
44.3 Exercises
45 Definiteness and Matrix Norms
45.1 Definiteness of Matrices
45.2 Matrix Norms
45.2.1 Norms
45.2.2 Induced Matrix Norm
45.3 Exercises
46 Functions of Several Variables
46.1 The Functions and Their Representations
46.2 Some Topological Terms
46.3 Consequences, Limits, Continuity
46.4 Exercises
47 Partial Differentiation: Gradient, Hessian Matrix, Jacobian Matrix
47.1 The Gradient
47.2 The Hessian Matrix
47.3 The Jacobian Matrix
47.4 Exercises
48 Applications of Partial Derivatives
48.1 The (Multidimensional) Newton Method
48.2 Taylor Development
48.2.1 The Zeroth, First and Second Taylor Polynomial
48.2.2 The General Taylor polynomial
48.3 Exercises
49 Extreme Value Determination
49.1 Local and Global Extrema
49.2 Determination of Extrema and Extremal Points
49.3 Exercises
50 Extreme Value Determination Under Constraints
50.1 Extrema Under Constraints
50.2 The Substitution Method
50.3 The Method of Lagrange Multipliers
50.4 Extrema Under Multiple Constraints
50.5 Exercise
51 Total Differentiation, Differential Operators
51.1 Total Differentiability
51.2 The Total Differential
51.3 Differential Operators
51.4 Exercises
52 Implicit Functions
52.1 Implicit Functions: The Simple Case
52.2 Implicit Functions: The General Case
52.3 Exercises
53 Coordinate Transformations
53.1 Transformations and Transformation Matrices
53.2 Polar, Cylindrical and Spherical Coordinates
53.3 The Differential Operators in Cartesian Cylindrical and Spherical Coordinates
53.4 Conversion of Vector Fields and Scalar Fields
53.5 Exercises
54 Curves I
54.1 Terms
54.2 Length of a Curve
54.3 Exercises
55 Curves II
55.1 Reparameterization of a Curve
55.2 Frenet–Serret Frame, Curvature and Torsion
55.3 The Leibniz Sector Formula
55.4 Exercises
56 Line Integrals
56.1 Scalar and Vector Line Integrals
56.2 Applications of the Line Integrals
56.3 Exercises
57 Gradient Fields
57.1 Definitions
57.2 Existence of a Primitive Function
57.3 Determination of a Primitive Function
57.4 Exercises
58 Multiple Integrals
58.1 Integration Over Rectangles or Cuboids
58.2 Normal Domains
58.3 Integration Over Normal Domains
58.4 Exercises
59 Substitution for Multiple Variables
59.1 Integration via Polar, Cylindrical, Spherical and Other Coordinates
59.2 Application: Mass and Center of Gravity
59.3 Exercises
60 Surfaces and Surface Integrals
60.1 Regular Surfaces
60.2 Surface Integrals
60.3 Overview of the Integrals
60.4 Exercises
61 Integral Theorems I
61.1 Green's Theorem
61.2 The Plane Theorem of Gauss
61.3 Exercises
62 Integral Theorems II
62.1 The Divergence Theorem of Gauss
62.2 Stokes' Theorem
62.3 Exercises
63 General Information on Differential Equations
63.1 The Directional Field
63.2 Existence and Uniqueness of Solutions
63.3 Transformation to 1st Order Systems
63.4 Exercises
64 The Exact Differential Equation
64.1 Definition of Exact ODEs
64.2 The Solution Procedure
64.2.1 Integrating Factors: Euler's Multiplier
64.3 Exercises
65 Linear Differential Equation Systems I
65.1 The Exponential Function for Matrices
65.2 The Exponential Function as a Solution of Linear ODE Systems
65.3 The Solution for a Diagonalizable A
65.4 Exercises
66 Linear Differential Equation Systems II
66.1 The Exponential Function as a Solution of Linear ODE Systems
66.2 The Solution for a Non-Diagonalizable A
66.3 Exercises
67 Linear Differential Equation Systems III
67.1 Solving ODE Systems
67.2 Stability
67.2.1 Stability of Nonlinear Systems
67.3 Exercises
68 Boundary Value Problems
68.1 Types of Boundary Value Problems
68.2 First Solution Methods
68.3 Linear Boundary Value Problems
68.4 The Method with Green's Function
68.5 Exercises
69 Basic Concepts of Numerics
69.1 Condition
69.2 The Big O Notation
69.3 Stability
69.4 Exercises
70 Fixed Point Iteration
70.1 The Fixed Point Equation
70.2 The Convergence of Iteration Methods
70.3 Implementation
70.4 Rate of Convergence
70.5 Exercises
71 Iterative Methods for Systems of Linear Equations
71.1 Solving Systems of Equations by Fixed Point Iteration
71.2 The Jacobian Method
71.3 The Gauss-Seidel Method
71.4 Relaxation
71.5 Exercises
72 Optimization
72.1 The Optimum
72.2 The Gradient Method
72.3 Newton's Method
72.4 Exercises
73 Numerics of Ordinary Differential Equations II
73.1 Solution Methods for ODE Systems
73.2 Consistency and Convergence of One-Step Methods
73.2.1 Consistency of One-Step Methods
73.2.2 Convergence of One-Step Method
73.3 Stiff Differential Equations
73.4 Boundary Value Problems
73.4.1 Reduction of a BVP to an IVP
Remark
73.4.2 Difference Method
73.4.3 Shooting Method
73.5 Exercises
74 Fourier Series: Calculation of Fourier Coefficients
74.1 Periodic Functions
74.2 The Admissible Functions
74.3 Expanding in Fourier Series—Real Version
74.4 Application: Calculation of Series Values
74.5 Expanding in Fourier Series: Complex Version
74.6 Exercises
75 Fourier Series: Background, Theorems and Application
75.1 The Orthonormal System 1/2, cos(kx), sin(kx)
75.2 Theorems and Rules
75.3 Application to Linear Differential Equations
75.4 Exercises
76 Fourier Transform I
76.1 The Fourier Transform
76.2 The Inverse Fourier Transform
76.3 Exercise
77 Fourier Transform II
77.1 The Rules and Theorems for the Fourier Transform
77.2 Application to Linear Differential Equations
77.3 Exercises
78 Discrete Fourier Transform
78.1 Approximate Determination of the Fourier Coefficients
78.2 The Inverse Discrete Fourier Transform
78.3 Trigonometric Interpolation
78.4 Exercise
79 The Laplace Transformation
79.1 The Laplacian Transformation
79.2 The Rules and Theorems for the Laplace Transformation
79.3 Applications
79.3.1 Solving IVPs with Linear ODEs
79.3.2 Solving IVPs with Linear ODE Systems
79.3.3 Solving Integral Equations
79.4 Exercises
80 Holomorphic Functions
80.1 Complex Functions
80.1.1 Domains
80.1.2 Examples of Complex Functions
80.1.3 Visualization of Complex Functions
80.1.4 Realification of Complex Functions
80.2 Complex Differentiability and Holomorphy
80.3 Exercises
81 Complex Integration
81.1 Complex Curves
81.2 Complex Line Integrals
81.3 The Cauchy Integral Theorem and the Cauchy Integral Formula
81.4 Exercises
82 Laurent Series
82.1 Singularities
82.2 Laurent Series
82.3 Laurent Series Development
82.4 Exercises
83 The Residual Calculus
83.1 The Residue Theorem
83.2 Calculation of Real Integrals
83.3 Exercises
84 Conformal Mappings
84.1 Generalities of Conformal Mappings
84.2 Möbius Transformations
84.3 Exercises
85 Harmonic Functions and the Dirichlet Boundary Value Problem
85.1 Harmonic Functions
85.2 The Dirichlet Boundary Value Problem
85.3 Exercises
86 Partial Differential Equations of First Order
86.1 Linear PDEs of First Order with Constant Coefficients
86.2 Linear PDEs of First Order
86.3 The First Order Quasi Linear PDE
86.4 The Characteristics Method
86.5 Exercises
87 Partial Differential Equations of Second Order: General
87.1 First Terms
87.1.1 Linear-Nonlinear, Stationary-Nonstationary
87.1.2 Boundary Value and Initial Boundary Value Conditions
87.1.3 Well-Posed and Ill-Posed Problems
87.2 The Type Classification
87.3 Solution Methods
87.3.1 The Separation Method
87.3.2 Numerical Solution Methods
87.4 Exercises
88 The Laplace or Poisson Equation
88.1 Boundary Value Problems for the Poisson Equation
88.2 Solutions of the Laplace Equation
88.3 The Dirichlet Boundary Value Problem for a Circle
88.4 Numerical Solution
88.5 Exercises
89 The Heat Conduction Equation
89.1 Initial Boundary Value Problems for the Heat Conduction Equation
89.2 Solutions of the Equation
89.3 Zero Boundary Condition: Solution with Fourier Series
89.4 Numerical Solution
89.5 Exercises
90 The Wave Equation
90.1 Initial Boundary Value Problems for the Wave Equation
90.2 Solutions of the Equation
90.3 The Vibrating String: Solution with Fourier Series
90.4 Numerical Solution
90.5 Exercises
91 Solving PDEs with Fourier and Laplace Transforms
91.1 An Introductory Example
91.2 The General Procedure
91.3 Exercises
Index