Mathematical Methods in the Physical Sciences, 3rd Edition

by Boas, Mary L.

• ISBN13: 9780471198260
• ISBN10: 0471198269
• eBook ISBN(s): 9781118048887
• Edition: 3rd
• Format: Hardcover
• Copyright: 2005-07-22
• Publisher: Wiley
• More Book Details

Summary

Now in its third edition, Mathematical Concepts in the Physical Sciences provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts into one compact, clearly written reference.

Author Biography

Mary L. Boas is currently professor emeritus in the physics department at DePaul University.

Table of Contents

Infinite Series, Power Series	1	(45)
The Geometric Series	1	(3)
Definitions and Notation	4	(2)
Applications of Series	6	(1)
Convergent and Divergent Series	6	(3)
Testing Series for Convergence; the Preliminary Test	9	(1)
Convergence Tests for Series of Positive Terms: Absolute Convergence	10	(7)
The Comparison Test	10	(1)
The Integral Test	11	(2)
The Ratio Test	13	(2)
A Special Comparison Test	15	(2)
Alternating Series	17	(1)
Conditionally Convergent Series	18	(1)
Useful Facts About Series	19	(1)
Power Series; Interval of Convergence	20	(3)
Theorems About Power Series	23	(1)
Expanding Functions in Power Series	23	(2)
Techniques for Obtaining Power Series Expansions	25	(8)
Multiplying a Series by a Polynomial or by Another Series	26	(1)
Division of Two Series or of a Series by a Polynomial	27	(1)
Binomial Series	28	(1)
Substitution of a Polynomial or a Series for the Variable in Another Series	29	(1)
Combination of Methods	30	(1)
Taylor Series Using the Basic Maclaurin Series	30	(1)
Using a Computer	31	(2)
Accuracy of Series Approximations	33	(3)
Some Uses of Series	36	(8)
Miscellaneous Problems	44	(2)
Complex Numbers	46	(36)
Introduction	46	(1)
Real and Imaginary Parts of a Complex Number	47	(1)
The Complex Plane	47	(2)
Terminology and Notation	49	(2)
Complex Algebra	51	(5)
Simplifying to x+iy form	51	(1)
Complex Conjugate of a Complex Expression	52	(1)
Finding the Absolute Value of z	53	(1)
Complex Equations	54	(1)
Graphs	54	(1)
Physical Applications	55	(1)
Complex Infinite Series	56	(2)
Complex Power Series; Disk of Convergence	58	(2)
Elementary Functions of Complex Numbers	60	(1)
Euler's Formula	61	(3)
Powers and Roots of Complex Numbers	64	(3)
The Exponential and Trigonometric Functions	67	(3)
Hyperbolic Functions	70	(2)
Logarithms	72	(1)
Complex Roots and Powers	73	(1)
Inverse Trigonometric and Hyperbolic Functions	74	(2)
Some Applications	76	(4)
Miscellaneous Problems	80	(2)
Linear Algebra	82	(106)
Introduction	82	(1)
Matrices; Row Reduction	83	(6)
Determinants; Cramer's Rule	89	(7)
Vectors	96	(10)
Lines and Planes	106	(8)
Matrix Operations	114	(10)
Linear Combinations, Linear Functions, Linear Operators	124	(8)
Linear Dependence and Independence	132	(5)
Special Matrices and Formulas	137	(5)
Linear Vector Spaces	142	(6)
Eigenvalues and Eigenvectors; Diagonalizing Matrices	148	(14)
Applications of Diagonalization	162	(10)
A Brief Introduction to Groups	172	(7)
General Vector Spaces	179	(5)
Miscellaneous Problems	184	(4)
Partial Differentiation	188	(53)
Introduction and Notation	188	(3)
Power Series in Two Variables	191	(2)
Total Differentials	193	(3)
Approximations using Differentials	196	(3)
Chain Rule or Differentiating a Function of a Function	199	(3)
Implicit Differentiation	202	(1)
More Chain Rule	203	(8)
Application of Partial Differentiation to Maximum and Minimum Problems	211	(3)
Maximum and Minimum Problems with Constraints; Lagrange Multipliers	214	(9)
Endpoint or Boundary Point Problems	223	(5)
Change of Variables	228	(5)
Differentiation of Integrals; Leibniz' Rule	233	(5)
Miscellaneous problems	238	(3)
Multiple Integrals	241	(35)
Introduction	241	(1)
Double and Triple Integrals	242	(7)
Applications of Integration; Single and Multiple Integrals	249	(9)
Change of Variables in Integrals; Jacobians	258	(12)
Surface Integrals	270	(3)
Miscellaneous Problems	273	(3)
Vector Analysis	276	(64)
Introduction	276	(1)
Applications of Vector Multiplication	276	(2)
Triple Products	278	(7)
Differentiation of Vectors	285	(4)
Fields	289	(1)
Directional Derivative; Gradient	290	(6)
Some Other Expressions Involving	296	(3)
Line Integrals	299	(10)
Green's Theorem in the Plane	309	(5)
The Divergence and the Divergence Theorem	314	(10)
The Curl and Stokes' Theorem	324	(12)
Miscellaneous Problems	336	(4)
Fourier Series and Transforms	340	(50)
Introduction	340	(1)
Simple Harmonic Motion and Wave Motion; Periodic Functions	340	(5)
Applications of Fourier Series	345	(2)
Average Value of a Function	347	(3)
Fourier Coefficients	350	(5)
Dirichlet Conditions	355	(3)
Complex Form of Fourier Series	358	(2)
Other Intervals	360	(4)
Even and Odd Functions	364	(8)
An Application to Sound	372	(3)
Parseval's Theorem	375	(3)
Fourier Transforms	378	(8)
Miscellaneous Problems	386	(4)
Ordinary Differential Equations	390	(82)
Introduction	390	(5)
Separable Equations	395	(6)
Linear First-Order Equations	401	(3)
Other Methods for First-Order Equations	404	(4)
Second-Order Linear Equations with Constant Coefficients and Zero Right-Hand Side	408	(9)
Second-Order Linear Equations with Constant Coefficients and Right-Hand Side Not Zero	417	(13)
Other Second-Order Equations	430	(7)
The Laplace Transform	437	(3)
Solution of Differential Equations by Laplace Transforms	440	(4)
Convolution	444	(5)
The Dirac Delta Function	449	(12)
A Brief Introduction to Green Functions	461	(5)
Miscellaneous Problems	466	(6)
Calculus of Variations	472	(24)
Introduction	472	(2)
The Euler Equation	474	(4)
Using the Euler Equation	478	(4)
The Brachistochrone Problem; Cycloids	482	(3)
Several Dependent Variables; Lagrange's Equations	485	(6)
Isoperimetric Problems	491	(2)
Variational Notation	493	(1)
Miscellaneous Problems	494	(2)
Tensor Analysis	496	(41)
Introduction	496	(2)
Cartesian Tensors	498	(4)
Tensor Notation and Operations	502	(3)
Inertia Tensor	505	(3)
Kronecker Delta and Levi-Civita Symbol	508	(6)
Pseudovectors and Pseudotensors	514	(4)
More About Applications	518	(3)
Curvilinear Coordinates	521	(4)
Vector Operators in Orthogonal Curvilinear Coordinates	525	(4)
Non-Cartesian Tensors	529	(6)
Miscellaneous Problems	535	(2)
Special Functions	537	(25)
Introduction	537	(1)
The Factorial Function	538	(1)
Definition of the Gamma Function; Recursion Relation	538	(2)
The Gamma Function of Negative Numbers	540	(1)
Some Important Formulas Involving Gamma Functions	541	(1)
Beta Functions	542	(1)
Beta Functions in Terms of Gamma Functions	543	(2)
The Simple Pendulum	545	(2)
The Error Function	547	(2)
Asymptotic Series	549	(3)
Stirling's Formula	552	(2)
Elliptic Integrals and Functions	554	(6)
Miscellaneous Problems	560	(2)
Series Solutions of Differential Equations; Legendre, Bessel, Hermite, and Laguerre Functions	562	(57)
Introduction	562	(2)
Legendre's Equation	564	(3)
Leibniz' Rule for Differentiating Products	567	(1)
Rodrigues' Formula	568	(1)
Generating Function for Legendre Polynomials	569	(6)
Complete Sets of Orthogonal Functions	575	(2)
Orthogonality of the Legendre Polynomials	577	(1)
Normalization of the Legendre Polynomials	578	(2)
Legendre Series	580	(3)
The Associated Legendre Functions	583	(2)
Generalized Power Series or the Method of Frobenius	585	(2)
Bessel's Equation	587	(3)
The Second Solution of Bessel's Equation	590	(1)
Graphs and Zeros of Bessel Functions	591	(1)
Recursion Relations	592	(1)
Differential Equations with Bessel Function Solutions	593	(2)
Other Kinds of Bessel Functions	595	(3)
The Lengthening Pendulum	598	(3)
Orthogonality of Bessel Functions	601	(3)
Approximate Formulas for Bessel Functions	604	(1)
Series Solutions; Fuchs's Theorem	605	(2)
Hermite Functions; Laguerre Functions; Ladder Operators	607	(8)
Miscellaneous Problems	615	(4)
Partial Differential Equations	619	(47)
Introduction	619	(2)
Laplace's Equation; Steady-State Temperature in a Rectangular Plate	621	(7)
The Diffusion or Heat Flow Equation; the Schrodinger Equation	628	(5)
The Wave Equation; the Vibrating String	633	(5)
Steady-state Temperature in a Cylinder	638	(6)
Vibration of a Circular Membrane	644	(3)
Steady-state Temperature in a Sphere	647	(5)
Poisson's Equation	652	(7)
Integral Transform Solutions of Partial Differential Equations	659	(4)
Miscellaneous Problems	663	(3)
Functions of a Complex Variable	666	(56)
Introduction	666	(1)
Analytic Functions	667	(7)
Contour Integrals	674	(4)
Laurent Series	678	(4)
The Residue Theorem	682	(1)
Methods of Finding Residues	683	(4)
Evaluation of Definite Integrals by Use of the Residue Theorem	687	(15)
The Point at Infinity; Residues at Infinity	702	(3)
Mapping	705	(5)
Some Applications of Conformal Mapping	710	(8)
Miscellaneous Problems	718	(4)
Probability and Statistics	722	(57)
Introduction	722	(2)
Sample Space	724	(5)
Probability Theorems	729	(7)
Methods of Counting	736	(8)
Random Variables	744	(6)
Continuous Distributions	750	(6)
Binomial Distribution	756	(5)
The Normal or Gaussian Distribution	761	(6)
The Poisson Distribution	767	(3)
Statistics and Experimental Measurements	770	(6)
Miscellaneous Problems	776	(3)
References	779	(2)
Answers to Selected Problems	781	(30)
Index	811