The Mathematics Companion: Mathematical Methods for Physicists and Engineers

✍ Scribed by Anthony Craig Fischer-Cripps

• Publisher: CRC Press
• Year: 2005
• Tongue: English
• Leaves: 202
• Edition: 1
• Category: Library

✦ Synopsis

Following the style of The Physics Companion and The Electronics Companion, this book is a revision aid and study guide for undergraduate students in physics and engineering. It consists of a series of one-page-per-topic descriptions of the key concepts covered in a typical first-year "mathematics for physics" course. The emphasis is placed on relating the mathematical principles being introduced to real-life physical problems. In common with the other companions, there is strong use of figures throughout to help in understanding of the concepts under consideration. The book will be an essential reference and revision guide, particularly for those students who do not have a strong background in mathematics when beginning their degree.

✦ Table of Contents

The Mathematics Companion
Contents
Preface
Part 1: Mathematics Essentials
1.1 Numbers, Trigonometry and Analytical Geometry
1.1.1 Real Numbers
1.1.2 Complex numbers
1.1.3 Coordinate systems
1.1.4 Vectors
1.1.5 The unit vectors
1.1.6 Trigonometry
1.1.7 Straight line
1.1.8 Circle and ellipse
1.1.9 Parabola
1.1.10 Hyperbola
1.2 Limits and Functions
1.2.1 Functions
1.2.2 Quadratic function
1.2.3 Limits
1.2.4 Theorems on limits
1.3 Differentiation
1.3.1 Derivative
1.3.2 Rules for calculating the derivative
1.3.3 Higher order derivatives
1.3.4 Maxima and minima
1.3.5 The 2nd derivative
1.3.6 Curve sketching
1.3.7 Time rate of change
1.3.8 Anti-derivatives
1.4 Integration
1.4.1 Definite integral
1.4.2 Fundamental theorem of calculus
1.4.3 Properties of the definite integral
1.4.4 Indefinite integral
1.4.5 Numerical integration
1.5 Exponential and Logarithmic Functions
1.5.1 Logarithms
1.5.2 The natural logarithm
1.5.3 The natural exponential
1.5.4 Differentiation and integration of ex
1.5.5 Exponential law of growth and decay
1.6 Trigonometric and Hyperbolic Functions
1.6.1 Circular measure
1.6.2 Derivatives and integrals of trigonometric functions
1.6.3 Inverse trigonometric functions
1.6.4 Derivatives of trigonometric functions
1.6.5 Hyperbolic functions
1.6.6 Properties of hyperbolic functions
1.6.7 Derivative of hyperbolic functions
1.6.8 Inverse hyperbolic functions
1.7 Methods of Integration
1.7.1 Integration by substitution
1.7.2 Integration by parts
1.7.3 Trigonometric substitutions
1.7.4 Integration by partial fractions
1.7.5 Quadratic expressions
1.7.6 Indeterminate forms
1.7.7 Improper integrals
1.8 Infinite Series Summary
1.8.1 Sequences
1.8.2 Series
1.8.3 d’Alembert’s ratio test
1.8.4 Power series
1.8.5 Binomial series
1.9 Probability
1.9.1 Mean, median, mode
1.9.2 Permutations and combinations
1.9.3 Probabilities, odds and expectation
1.9.4 Probability distribution
1.9.5 Expected value
1.9.6 Binomial distribution
1.9.7 Normal distribution
1.9.8 Sampling
1.9.9 t distribution
1.9.10 Chi-squared distribution
1.10 Matrices
1.10.1 Matrices
1.10.2 Determinants
1.10.3 Systems of equations
1.10.4 Eigenvalues and eigenvectors
1.10.5 Cayley-Hamilton theorem
1.10.6 Tensors
Part 2: Advanced Mathematics
2.1 Ordinary Differential Equations
2.1.1 Ordinary differential equations
2.1.2 Separation of variables
2.1.3 Homogenous equations
2.1.4 Exact equations
2.1.5 Linear equations
2.1.6 Linear equations with constant coefficients
2.1.7 Method of undetermined coefficients
2.1.8 Systems of equations
2.1.9 Complex eigenvalues
2.1.10 Power series
2.2 Laplace Transforms
2.2.1 Laplace transform
2.2.2 Laplace transform of derivatives
2.2.3 Step functions
2.2.4 Laplace transforms to solve differential equations
2.2.5 Laplace transforms and partial fractions
2.3 Vector Analysis
2.3.1 Vectors
2.3.2 Direction cosines
2.3.4 Vector dot product
2.3.5 Equation of a line in space
2.3.6 Equation of a plane
2.3.7 Distance from a point to a plane
2.3.8 Vector cross product
2.3.9 Distance from a point to a line
2.3.10 Distance between two skew lines
2.3.11 Vector differentiation
2.3.12 Motion of a body
2.4 Partial Derivatives
2.4.1 Partial differentiation
2.4.2 Chain rule for partial derivatives
2.4.3 Increments and differentials
2.4.4 Directional derivatives
2.4.5 Tangent planes and normal vector
2.4.6 Gradient, divergence and curl
2.4.7 Maxima and minima
2.4.8 Lagrange multipliers
2.4.9 Multiple least squares analysis
2.4.10 Constraints
2.5 Multiple Integrals
2.5.1 Line integrals
2.5.2 Electrical potential
2.5.3 Work done by a force
2.5.4 Double integral
2.5.5 Triple integral
2.5.6 Surface integrals
2.5.7 Gauss’ law
2.5.8 Divergence theorem
2.5.9 Stokes’ theorem
2.5.10 Green’s theorem
2.5.11 Vector representations of Green’s theorem
2.5.12 Application of Green’s theorem
2.5.13 Maxwell’s equations (integral form)
2.5.14 Maxwell’s equations (differential form)
2.6 Fourier Series
2.6.1 Fourier series
2.6.2 Fourier transform
2.6.3 Sampling
2.6.4 Discrete Fourier transform
2.6.5 Odd and even functions
2.6.6 Convolution
2.7 Partial Differential Equations
2.7.1 Partial differential equations
2.7.2 General wave equation
2.7.3 Solution to the general wave equation
2.7.4 d’Alembert’s solution to the wave equation
2.7.5 Heat conduction equation
2.7.6 Solution to the heat conduction equation
2.7.7 Heat equation for a thin rod of infinite length
2.8 Complex Functions
2.8.1 Complex functions
2.8.2 Quantum mechanics
2.8.3 Solutions to the wave equation
2.8.4 Zero potential
2.8.5 Infinite square well
2.8.6 Harmonic oscillator
2.9 Numerical methods
2.9.1 Newton’s method
2.9.2 Interpolating polynomial
2.9.3 Linear least squares
2.9.4 Non-linear least squares
2.9.5 Error propagation through equations
2.9.6 Cubic spline
2.9.7 Differentiation
2.9.8 Integration
2.9.9 1st Order ordinarydifferential equations
2.9.10 Runge-Kutta method
2.9.11 Finite element method
Appendix
A.1 Useful information
A.2 Some standard integrals
A.3 Special functions