Mathematics for Physicists and Engineers: Fundamentals and Interactive Study Guide

Main Authors of the International Version
Preface and Introduction
The Combination of Textbook and Study Guid
Preface to the Third Edition
How to Access the Study Guide
Contents
1 Vector Algebra I: Scalars and Vectors
1.1 Scalars and Vectors
1.2 Addition of Vectors
1.2.1 Sum of Two Vectors: Geometrical Addition
1.3 Subtraction of Vectors
1.4 Components and Projection of a Vector
1.5 Component Representation in Coordinate Systems
1.5.1 Position Vector
1.5.2 Unit Vectors
1.5.3 Component Representation of a Vector
1.5.4 Representation of the Sum of Two Vectors in Terms of Their Components
1.5.5 Subtraction of Vectors in Terms of Their Components
1.6 Multiplication of a Vector by a Scalar
1.7 Magnitude of a Vector
2 Vector Algebra II: Scalar and Vector Products
2.1 Scalar Product
2.1.1 Application: Equation of a Line and a Plane
2.1.2 Special Cases
2.1.3 Commutative and Distributive Laws
2.1.4 Scalar Product in Terms of the Components of the Vectors
2.2 Vector Product
2.2.1 Torque
2.2.2 Torque as a Vector
2.2.3 Definition of the Vector Product
2.2.4 Special Cases
2.2.5 Anti-commutative Law for Vector Products
2.2.6 Components of the Vector Product
3 Functions
3.1 The Mathematical Concept of Functions and Its Meaning in Physics and Engineering
3.1.1 Introduction
3.1.2 The Concept of a Function
3.2 Graphical Representation of Functions
3.2.1 Coordinate System, Position Vector
3.2.2 The Linear Function: The Straight Line
3.2.3 Graph Plotting
3.3 Quadratic Equations
3.4 Parametric Changes of Functions and Their Graphs
3.5 Inverse Functions
3.6 Trigonometric or Circular Functions
3.6.1 Unit Circle
3.6.2 Sine Function
3.6.3 Cosine Function
3.6.4 Relationships Between the Sine and Cosine Functions
3.6.5 Tangent and Cotangent
3.6.6 Addition Formulae
3.7 Inverse Trigonometric Functions
3.8 Function of a Function (Composition)
4 Exponential, Logarithmic and Hyperbolic Functions
4.1 Powers, Exponential Function
4.1.1 Powers
4.1.2 Laws of Indices or Exponents
4.1.3 Binomial Theorem
4.1.4 Exponential Function
4.2 Logarithm, Logarithmic Function
4.2.1 Logarithm
4.2.2 Operations with Logarithms
4.2.3 Logarithmic Functions
4.3 Hyperbolic Functions and Inverse Hyperbolic Functions
4.3.1 Hyperbolic Functions
4.3.2 Inverse Hyperbolic Functions
5 Differential Calculus
5.1 Sequences and Limits
5.1.1 The Concept of Sequence
5.1.2 Limit of a Sequence
5.1.3 Limit of a Function
5.1.4 Examples for the Practical Determination of Limits
5.2 Continuity
5.3 Series
5.3.1 Geometric Series
5.4 Differentiation of a Function
5.4.1 Gradient or Slope of a Line
5.4.2 Gradient of an Arbitrary Curve
5.4.3 Derivative of a Function
5.4.4 Physical Application: Velocity
5.4.5 The Differential
5.5 Calculating Differential Coefficients
5.5.1 Derivatives of Power Functions; Constant Factors
5.5.2 Rules for Differentiation
5.5.3 Differentiation of Fundamental Functions
5.6 Higher Derivatives
5.7 Extreme Values and Points of Inflexion; Curve Sketching
5.7.1 Maximum and Minimum Values of a Function
5.7.2 Further Remarks on Points of Inflexion (Contraflexure)
5.7.3 Curve Sketching
5.8 Applications of Differential Calculus
5.8.1 Extreme Values
5.8.2 Increments
5.8.3 Curvature
5.8.4 Determination of Limits by Differentiation: L'Hôpital's Rule
5.9 Further Methods for Calculating Differential Coefficients
5.9.1 Implicit Functions and their Derivatives
5.9.2 Logarithmic Differentiation
5.10 Parametric Functions and their Derivatives
5.10.1 Parametric Form of an Equation
5.10.2 Derivatives of Parametric Functions
6 Integral Calculus
6.1 The Primitive Function
6.1.1 Fundamental Problem of Integral Calculus
6.2 The Area Problem: The Definite Integral
6.3 Fundamental Theorem of the Differential and Integral Calculus
6.4 The Definite Integral
6.4.1 Calculation of Definite Integrals from Indefinite Integrals
6.4.2 Examples of Definite Integrals
6.5 Methods of Integration
6.5.1 Principle of Verification
6.5.2 Standard Integrals
6.5.3 Constant Factor and the Sum of Functions
6.5.4 Integration by Parts: Product of Two Functions
6.5.5 Integration by Substitution
6.5.6 Substitution in Particular Cases
6.5.7 Integration by Partial Fractions
6.6 Rules for Solving Definite Integrals
6.7 Mean Value Theorem
6.8 Improper Integrals
6.9 Line Integrals
7 Applications of Integration
7.1 Areas
7.1.1 Areas for Parametric Functions
7.1.2 Areas in Polar Coordinates
7.1.3 Areas of Closed Curves
7.2 Lengths of Curves
7.2.1 Lengths of Curves in Polar Coordinates
7.3 Surface Area and Volume of a Solid of Revolution
7.4 Applications to Mechanics
7.4.1 Basic Concepts of Mechanics
7.4.2 Center of Mass and Centroid
7.4.3 The Theorems of Pappus
7.4.4 Moments of Inertia; Second Moment of Area
8 Taylor Series and Power Series
8.1 Introduction
8.2 Expansion of a Function in a Power Series
8.3 Interval of Convergence of Power Series
8.4 Approximate Values of Functions
8.5 Expansion of a Function f(x) at an Arbitrary Position
8.6 Applications of Series
8.6.1 Polynomials as Approximations
8.6.2 Integration of Functions When Expressed as Power Series
8.6.3 Expansion in a Series by Integrating
9 Complex Numbers
9.1 Definition and Properties of Complex Numbers
9.1.1 Imaginary Numbers
9.1.2 Complex Numbers
9.1.3 Fields of Application
9.1.4 Operations with Complex Numbers
9.2 Graphical Representation of Complex Numbers
9.2.1 Gauss Complex Number Plane: Argand Diagram
9.2.2 Polar Form of a Complex Number
9.3 Exponential Form of Complex Numbers
9.3.1 Euler's Formula
9.3.2 Exponential Form of the Sine and Cosine Functions
9.3.3 Complex Numbers as Powers
9.3.4 Multiplication and Division in Exponential Form
9.3.5 Raising to a Power, Exponential Form
9.3.6 Periodicity of r ejα
9.3.7 Transformation of a Complex Number From One Form Into Another
9.4 Operations with Complex Numbers Expressed in Polar Form
9.4.1 Multiplication and Division
9.4.2 Raising to a Power
9.4.3 Roots of a Complex Number
10 Differential Equations
10.1 Concept and Classification of Differential Equations
10.2 Preliminary Remarks
10.3 General Solution of First- and Second-Order DEs with Constant Coefficients
10.3.1 Homogeneous Linear DE
10.3.2 Non-homogeneous Linear DE
10.4 Boundary Value Problems
10.4.1 First-Order DEs
10.4.2 Second-Order DEs
10.5 Some Applications of DEs
10.5.1 Radioactive Decay
10.5.2 The Harmonic Oscillator
10.6 General Linear First-Order DEs
10.6.1 Solution by Variation of the Constant
10.6.2 A Straightforward Method Involving the Integrating Factor
10.7 Some Remarks on General First-Order DEs
10.7.1 Bernoulli's Equations
10.7.2 Separation of Variables
10.7.3 Exact Equations
10.7.4 The Integrating Factor—General Case
10.8 Simultaneous DEs
10.9 Higher-Order DEs Interpreted as Systems of First-Order Simultaneous DEs
10.10 Some Advice on Intractable DEs
11 Laplace Transforms
11.1 Introduction
11.2 The Laplace Transform Definition
11.3 Laplace Transform of Standard Functions
11.4 Solution of Linear DEs with Constant Coefficients
11.5 Solution of Simultaneous DEs with Constant Coefficients
12 Functions of Several Variables; Partial Differentiation; and Total Differentiation
12.1 Introduction
12.2 Functions of Several Variables
12.2.1 Representing the Surface by Establishing a Table of z-Values
12.2.2 Representing the Surface by Establishing Intersecting Curves
12.2.3 Obtaining a Functional Expression for a Given Surface
12.3 Partial Differentiation
12.3.1 Higher Partial Derivatives
12.4 Total Differential
12.4.1 Total Differential of Functions
12.4.2 Application: Small Tolerances
12.4.3 Gradient
12.5 Total Derivative
12.5.1 Explicit Functions
12.5.2 Implicit Functions
12.6 Maxima and Minima of Functions of Two or More Variables
12.7 Applications: Wave Function and Wave Equation
12.7.1 Wave Function
12.7.2 Wave Equation
13 Multiple Integrals; Coordinate Systems
13.1 Multiple Integrals
13.2 Multiple Integrals with Constant Limits
13.2.1 Decomposition of a Multiple Integral into a Product of Integrals
13.3 Multiple Integrals with Variable Limits
13.4 Coordinate Systems
13.4.1 Polar Coordinates
13.4.2 Cylindrical Coordinates
13.4.3 Spherical Coordinates
13.5 Application: Moments of Inertia of a Solid
14 Transformation of Coordinates; Matrices
14.1 Introduction
14.2 Parallel Shift of Coordinates: Translation
14.3 Rotation
14.3.1 Rotation in a Plane
14.3.2 Successive Rotations
14.3.3 Rotations in Three-Dimensional Space
14.4 Matrix Algebra
14.4.1 Addition and Subtraction of Matrices
14.4.2 Multiplication of a Matrix by a Scalar
14.4.3 Product of a Matrix and a Vector
14.4.4 Multiplication of Two Matrices
14.5 Rotations Expressed in Matrix Form
14.5.1 Rotation in Two-Dimensional Space
14.5.2 Special Rotation in Three-Dimensional Space
14.6 Special Matrices
14.7 Inverse Matrix
15 Sets of Linear Equations; Determinants
15.1 Introduction
15.2 Sets of Linear Equations
15.2.1 Gaussian Elimination: Successive Elimination of Variables
15.2.2 Gauss–Jordan Elimination
15.2.3 Matrix Notation of Sets of Equations and Determination of the Inverse Matrix
15.2.4 Existence of Solutions
15.3 Determinants
15.3.1 Preliminary Remarks on Determinants
15.3.2 Definition and Properties of An n-Row Determinant
15.3.3 Rank of a Determinant and Rank of a Matrix
15.3.4 Applications of Determinants
16 Eigenvalues and Eigenvectors of Real Matrices
16.1 Two Case Studies: Eigenvalues of 2times2 Matrices
16.2 General Method for Finding Eigenvalues
16.3 Worked Example: Eigenvalues of a 3times3 Matrix
16.4 Important Facts on Eigenvalues and Eigenvectors
17 Vector Analysis: Surface Integrals, Divergence, Curl and Potential
17.1 Flow of a Vector Field Through a Surface Element
17.2 Surface Integral
17.3 Special Cases of Surface Integrals
17.3.1 Flow of a Homogeneous Vector Field Through a Cuboid
17.3.2 Flow of a Spherically Symmetrical Field Through a Sphere
17.3.3 Application: The Electrical Field of a Point Charge
17.4 General Case of Computing Surface Integrals
17.5 Divergence of a Vector Field
17.6 Gauss's Theorem
17.7 Curl of a Vector Field
17.8 Stokes' Theorem
17.9 Potential of a Vector Field
17.10 Short Reference on Vector Derivatives
18 Fourier Series; Harmonic Analysis
18.1 Expansion of a Periodic Function into a Fourier Series
18.1.1 Evaluation of the Coefficients
18.1.2 Odd and Even Functions
18.2 Examples of Fourier Series
18.3 Expansion of Functions of Period 2L
18.4 Fourier Spectrum
19 Fourier Integrals and Fourier Transforms
19.1 Transition from Fourier Series to Fourier Integral
19.2 Fourier Transforms
19.2.1 Fourier Cosine Transform
19.2.2 Fourier Sine Transform, General Fourier Transform
19.2.3 Complex Representation of the Fourier Transform
19.3 Shift Theorem
19.4 Discrete Fourier Transform, Sampling Theorems
19.5 Fourier Transform of the Gaussian Function
20 Probability Calculus
20.1 Introduction
20.2 Concept of Probability
20.2.1 Random Experiment, Outcome Space and Events
20.2.2 The Classical Definition of Probability
20.2.3 The Statistical Definition of Probability
20.2.4 General Properties of Probabilities
20.2.5 Probability of Statistically Independent Events. Compound Probability
20.3 Permutations and Combinations
20.3.1 Permutations
20.3.2 Combinations
21 Probability Distributions
21.1 Discrete and Continuous Probability Distributions
21.1.1 Discrete Probability Distributions
21.1.2 Continuous Probability Distributions
21.2 Mean Values of Discrete and Continuous Variables
21.3 The Normal Distribution as the Limiting Value of the Binomial Distribution
21.3.1 Properties of the Normal Distribution
21.3.2 Derivation of the Binomial Distribution
22 Theory of Errors
22.1 Purpose of the Theory of Errors
22.2 Mean Value and Variance
22.2.1 Mean Value
22.2.2 Variance and Standard Deviation
22.2.3 Mean Value and Variance in a Random Sample and Parent Population
22.3 Mean Value and Variance of Continuous Distributions
22.4 Error in Mean Value
22.5 Normal Distribution: Distribution of Random Errors
22.6 Law of Error Propagation
22.7 Weighted Average
22.8 Curve Fitting: Method of Least Squares, Regression Line
22.9 Correlation and Correlation Coefficient
Appendix Answers
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Index