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