Description
From best-selling author Donald McQuarrie comes his newest text, Mathematical Methods for Scientists and Engineers. Intended for upper-level undergraduate and graduate courses in chemistry, physics, math and engineering, this book will also become a must-have for the personal library of all advanced students in the physical sciences. Comprising more than 2000 problems and 700 worked examples that detail every single step, this text is exceptionally well adapted for self study as well as for course use. Famous for his clear writing, careful pedagogy, and wonderful problems and examples, McQuarrie has crafted yet another tour de force.
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
Index
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