Mathematical Methods
 for  Scientists and Engineers

Donald A. McQuarrie
University of California, Davis

Detailed Table of Contents

Chapter 1:  Functions of a Single Variable

1-1.           Functions

1-2.           Limits

1-3.           Continuity

1-4.           Differentiation

1-5.           Differentials

1-6.           Mean Value Theorems

1-7.           Integration

1-8.           Improper Integrals

1-9.           Uniform Convergence of Integrals


Chapter 2:  Infinite Series

2-1.           Infinite Sequences

2-2.           Convergence and Divergence of Infinite Series

2-3.           Tests for Convergence

2-4.           Alternating Series

2-5.           Uniform Convergence

2-6.           Power Series

2-7.           Taylor Series

2-8.           Applications of Taylor Series

2-9.           Asymptotic Expansions


Chapter 3:  Functions Defined As Integrals

3-1.           The Gamma Function

3-2.           The Beta Function

3-3.           The Error Function

3-4.           The Exponential Integral

3-5.           Elliptic Integrals

3-6.           The Dirac Delta Function

3-7.           Bernoulli Numbers and Bernoulli Polynomials


Chapter 4:  Complex Numbers and Complex Functions

4-1.           Complex Numbers and the Complex Plane

4-2.           Functions of a Complex Variable

4-3.           Euler’s Formula and the Polar Form of Complex Numbers

4-4.           Trigonometric and Hyperbolic Functions

4-5.            The Logarithms of Complex Numbers

4-6.           Powers of Complex Numbers


Chapter 5:  Vectors

5-1.           Vectors in Two Dimensions

5-2.           Vector Functions in Two Dimensions

5-3.           Vectors in Three Dimensions

5-4.           Vector Functions in Three Dimensions

5-5.           Lines and Planes in Space


Chapter 6:  Functions of Several Variables

6-1.           Functions

6-2.           Limits and Continuity

6-3.           Partial Derivatives

6-4.           Chain Rules for Partial Differentiation

6-5.           Differentials and the Total Differential

6-6.           The Directional Derivative and the Gradient

6-7.           Taylor’s Formula in Several Variables

6-8.           Maxima and Minima

6-9.           The Method of Lagrange Multipliers

6-10.          Multiple Integrals


Chapter 7:  Vector Calculus

7-1.           Vector Fields

7-2.           Line Integrals

7-3.           Surface Integrals

7-4.           The Divergence Theorem

7-5.           Stokes’s Theorem


Chapter 8:  Curvilinear Coordinates

8-1.           Plane Polar Coordinates

8-2.           Vectors in Plane Polar Coordinates

8-3.           Cylindrical Coordinates

8-4.           Spherical Coordinates

8-5.           Curvilinear Coordinates

8-6.           Some Other Coordinate Systems


Chapter 9:  Linear Algebra and Vector Spaces

9-1.           Determinants

9-2.           Gaussian Elimination

9-3.           Matrices

9-4.           Rank of a Matrix

9-5.           Vector Spaces

9-6.           Inner Product Spaces

9-7.           Complex Inner Product Spaces


Chapter 10:  Matrices and Eigenvalue Problems

10-1.           Orthogonal and Unitary Transformations

10-2.           Eigenvalues and Eigenvectors

10-3.           Some Applied Eigenvalue Problems

10-4.           Change of Basis

10-5.           Diagonalization of Matrices

10-6.           Quadratic Forms


Chapter 11:  Ordinary Differential Equations

11-1.           Differential Equations of First Order and First Degree

11-2.           Linear First-Order Differential Equations

11-3.           Homogeneous Linear Differential Equations with Constant Coefficients

11-4.           Nonhomogeneous Linear Differential Equations with Constant Coefficients

11-5.           Some Other Types of Higher-Order Differential Equations

11-6.           Systems of First-Order Differential Equations

11-7.           Two Invaluable Resources for Solutions to Differential Equations


Chapter 12:  Series Solutions of Differential Equations

12-1.          The Power Series Method

12-2.           Ordinary Points and Singular Points of Differential Equations

12-3.           Series Solutions Near an Ordinary Point: Legendre’s Equation

12-4.           Solutions Near Regular Singular Points

12-5.           Bessel’s Equation

12-6.           Bessel Functions


Chapter 13:  Qualitative Methods for Nonlinear Differential Equations

13-1.          The Phase Plane

13-2.           Critical Points in the Phase Plane

13-3.           Stability of Critical Points

13-4.           Nonlinear Oscillators

13-5.           Population Dynamics


Chapter 14:  Orthogonal Polynomials and Sturm–Liouville Problems

14-1.           Legendre Polynomials

14-2.           Orthogonal Polynomials

14-3.           Sturm–Liouville Theory

14-4.           Eigenfunction Expansions

14-5.           Green’s Functions


Chapter 15:  Fourier Series

15-1.           Fourier Series as Eigenfunction Expansions

15-2.           Sine and Cosine Series

15-3.           Convergence of Fourier Series

15-4.           Fourier Series and Ordinary Differential Equations


Chapter 16:  Partial Differential Equations

16-1.           Some Examples of Partial Differential Equations

16-2.           Laplace’s Equation

16-3.          The One-Dimensional Wave Equation

16-4.          The Two-Dimensional Wave Equation

16-5.          The Heat Equation

16-6.          The Schrödinger Equation

a.  Particle in a Box

b.  A Rigid Rotor

c.  The Electron in a Hydrogen Atom

16-7.         The Classification of Partial Differential Equations


Chapter 17:  Integral Transforms

17-1.          The Laplace Transform

17-2.          The Inversion of Laplace Transforms

17-3.           Laplace Transforms and Ordinary Differential Equations

17-4.           Laplace Transforms and Partial Differential Equations

17-5.           Fourier Transforms

17-6.           Fourier Transforms and Partial Differential Equations

17-7.          The Inversion Formula for Laplace Transforms


Chapter 18:  Functions of a Complex Variable: Theory

18-1.           Functions, Limits, and Continuity

18-2.           Differentiation. The Cauchy–Riemann Equations

18-3.           Complex Integration. Cauchy’s Theorem

18-4.           Cauchy’s Integral Formula

18-5.           Taylor Series and Laurent Series

18-6.           Residues and the Residue Theorem


Chapter 19:  Functions of a Complex Variable: Applications

19-1.          The Inversion Formula for Laplace Transforms

19-2.           Evaluation of Real, Definite Integrals

19-3.           Summation of Series

19-4.           Location of Zeros

19-5.           Conformal Mapping

19-6.           Conformal Mapping and Boundary Value Problems

19-7.           Conformal Mapping and Fluid Flow


Chapter 20:  Calculus of Variations

20-1.          The Euler’s Equation

20-2.           Two Laws of Physics in Variational Form

20-3.           Variational Problems with Constraints

20-4.           Variational Formulation of Eigenvalue Problems

20-5.           Multidimensional Variational Problems


Chapter 21:  Probability Theory and Stochastic Processes

21-1.           Discrete Random Variables

21-2.           Continuous Random Variables

21-3.           Characteristic Functions

21-4.           Stochastic Processes—General

21-5.           Stochastic Processes—Examples

a.  Poisson Process

b.  The Shot Effect


Chapter 22:  Mathematical Statistics

22-1.           Estimation of Parameters

22-2.           Three Key Distributions Used in Statistical Tests

a.     The Normal Distribution

b.     The Chi–Square Distribution

c.     Student t-Distribution

22-3.           Confidence Intervals

a.  Confidence Intervals for the Mean of a Normal Distribution Whose Variance is Known

b.  Confidence Intervals for the Mean of a Normal Distribution with Unknown Variance

c.  Confidence Intervals for the Variance of a Normal Distribution

22-4.           Goodness of Fit

22-5.           Regression and Correlation