Mathematical Methods in the Physical Sciences

by Mary L. Boas (DePaul Univ.)

• ISBN13: 9780471044093
• ISBN10: 0471044091
• Edition: 2nd
• Format: Hardcover
• Copyright: 1983-01-01
• Publisher: John Wiley & Sons Inc

Summary

Updates the original, comprehensive introduction to the areas of mathematical physics encountered in advanced courses in the physical sciences. Intuition and computational abilities are stressed. Original material on DE and multiple integrals has been expanded.

Table of Contents

Infinite Series, Power Series	1	(42)
The geometric series	1	(2)
Definitions and notation	3	(2)
Applications of series	5	(1)
Convergent and divergent series	5	(2)
Testing series for convergence; the preliminary test	7	(1)
Tests for convergence of series of positive terms; absolute convergence	8	(7)
Alternating series	15	(1)
Conditionally convergent series	16	(1)
Useful facts about series	17	(1)
Power series; interval of convergence	18	(3)
Theorems about power series	21	(1)
Expanding functions in power series	22	(2)
Techniques for obtaining power series expansions	24	(5)
Questions of convergence and accuracy in computation	29	(4)
Some uses of series	33	(8)
Miscellaneous problems	41	(2)
Complex Numbers	43	(38)
Introduction	43	(1)
Real and imaginary parts of a complex number	44	(1)
The complex plane	45	(1)
Terminology and notation	46	(2)
Complex algebra	48	(6)
Complex infinite series	54	(2)
Complex power series; circle of convergence	56	(2)
Elementary functions of complex numbers	58	(2)
Euler's formula	60	(3)
Powers and roots of complex numbers	63	(3)
The exponential and trigonometric functions	66	(3)
Hyperbolic functions	69	(2)
Logarithms	71	(1)
Complex roots and powers	72	(2)
Inverse trigonometric and hyperbolic functions	74	(2)
Some applications	76	(3)
Miscellaneous problems	79	(2)
Linear Equations; Vectors, Matrices, And Determinants	81	(64)
Introduction	81	(1)
Sets of linear equations, row reducation	82	(5)
Determinants; Cramer's rule	87	(8)
Vectors	95	(10)
Lines and planes	105	(8)
Matrix operations	113	(14)
Linear combinations, linear functions, linear operators	127	(3)
General theory of sets of linear equations	130	(9)
Special matrices	139	(3)
Miscellaneous problems	142	(3)
Partial Differentiation	145	(56)
Introduction and notation	145	(3)
Power series in two variables	148	(2)
Total differentials	150	(4)
Approximate calculations using differentials	154	(2)
Chain rule or differentiating a function of a function	156	(3)
Implicit differentiation	159	(2)
More chain rule	161	(8)
Application of partial differentiation to maximum and minimum problems	169	(3)
Maximum and minimum problems with constraints; Lagrange multipliers	172	(9)
Endpoint or boundary point problems	181	(5)
Change of variables	186	(6)
Differentiation of integrals; Leibniz' rule	192	(5)
Miscellaneous problems	197	(4)
Multiple Integrals; Applications Of Integration	201	(34)
Introduction	201	(1)
Double and triple integrals	201	(7)
Applications of integration; single and multiple integrals	208	(9)
Change of variables in integrals; Jacobians	217	(11)
Surface integrals	228	(3)
Miscellaneous problems	231	(4)
Vector Analysis	235	(62)
Introduction	235	(1)
Applications of vector multiplication	235	(2)
Triple products	237	(7)
Differentiation of vectors	244	(4)
Fields	248	(1)
Directional derivative; gradient	249	(5)
Some other expressions involving	254	(3)
Line integrals	257	(9)
Green's theorem in the plane	266	(5)
The divergence and the divergence theorem	271	(10)
The curl and Stokes' theorem	281	(12)
Miscellaneous problems	293	(4)
Fourier Series	297	(40)
Introduction	297	(1)
Simple harmonic motion and wave motion; periodic functions	297	(5)
Applications of Fourier series	302	(2)
Average value of a function	304	(3)
Fourier coefficients	307	(6)
Dirichlet conditions	313	(2)
Complex form of Fourier series	315	(2)
Other intervals	317	(4)
Even and odd functions	321	(7)
An application to sound	328	(3)
Parseval's theorem	331	(3)
Miscellaneous problems	334	(3)
Ordinary Differential Equations	337	(46)
Introduction	337	(4)
Separable equations	341	(5)
Linear first-order equations	346	(4)
Other methods for first order equations	350	(2)
Second-order linear equations with constant coefficients and zero right-hand side	352	(9)
Second-order linear equations with constant coefficients and right-hand side not zero	361	(13)
Other second-order equations	374	(5)
Miscellaneous problems	379	(4)
Calculus Of Variations	383	(24)
Introduction	383	(3)
The Euler equation	386	(3)
Using the Euler equation	389	(4)
The brachistochrone problem; cycloids	393	(3)
Several dependent variables; Lagrange's equations	396	(5)
Isoperimetric problems	401	(2)
Variational notation	403	(1)
Miscellaneous problems	404	(3)
Coordinate Transformations; Tensor Analysis	407	(50)
Introduction	407	(2)
Linear transformations	409	(1)
Orthogonal transformations	410	(3)
Eigenvalues and eigenvectors; diagonalizing matrices	413	(7)
Applications of diagonalization	420	(6)
Curvilinear coordinates	426	(2)
Scale factors and basis vectors for orthogonal systems	428	(1)
General curvilinear coordinates	429	(2)
Vector operators in orthogonal curvilinear coordinates	431	(4)
Tensor analysis---introduction	435	(2)
Cartesian tensors	437	(4)
Uses of tensors; dyadics	441	(6)
General coordinate systems	447	(5)
Vector operations in tensor notation	452	(1)
Miscellaneous problems	453	(4)
Gamma, Beta, And Error Functions; Asymptotic Series; Striling's Formula; Elliptic Integrals And Functions	457	(26)
Introduction	457	(1)
The factorial function	457	(1)
Definition of the gamma function; recursion relation	458	(2)
The gamma function of negative numbers	460	(1)
Some important formulas involving gamma functions	461	(1)
Beta functions	462	(1)
The relation between the beta and gamma functions	463	(2)
The simple pendulum	465	(2)
The error function	467	(2)
Asymptotic series	469	(3)
Stirling's formula	472	(2)
Elliptic integrals and functions	474	(7)
Miscellaneous problems	481	(2)
Series Solutions Of Differential Equations; Legendre Polynomials; bessel Functions; Sets Of Orthogonal Functions	483	(58)
Introduction	483	(2)
Legendre's equation	485	(3)
Leibniz' rule for differentiating products	488	(1)
Rodrigues' formula	489	(1)
Generating function for Legendre polynomials	490	(6)
Complete sets of orthogonal functions	496	(3)
Orthogonality of the Legendre polynomials	499	(1)
Normalization of the Legendre polynomials	500	(2)
Legendre series	502	(2)
The associated Legendre functions	504	(2)
Generalized power series or the method of Frobenius	506	(3)
Bessel's equation	509	(3)
The second solution of Bessel's equation	512	(2)
Tables, graphs, and zeros of bessel functions	514	(1)
Recursion relations	514	(2)
A general differential equation having Bessel functions as solutions	516	(1)
Other kinds of Bessel functions	517	(2)
The lengthening pendulum	519	(3)
Orthogonality of Bessel functions	522	(3)
Approximate formulas for Bessel functions	525	(1)
Some general comments about series solutions	526	(4)
Hermite functions; Laguerre functions; ladder operators	530	(7)
Miscellaneous problems	537	(4)
Partial Differential Equations	541	(38)
Introduction	541	(2)
Laplace's equation; steady-state temperature in a rectangular plate	543	(7)
The diffusion or heat flow equation; heat flow in a bar or slab	550	(4)
The wave equation; the vibrating string	554	(4)
Steady-state temperature in a cylinder	558	(6)
Vibration of a circular membrane	564	(3)
Steady-state temperature in a sphere	567	(3)
Poisson's equation	570	(6)
Miscellaneous problems	576	(3)
Functions Of A Complex Variable	579	(56)
Introduction	579	(1)
Analytic functions	580	(8)
Contour integrals	588	(4)
Laurent series	592	(4)
The residue theorem	596	(2)
Methods of finding residues	598	(4)
Evaluation of definite integrals by use of the residue theorem	602	(12)
The point at infinity; residues at infinity	614	(3)
Mapping	617	(5)
Some applications of conformal mapping	622	(8)
Miscellaneous problems	630	(5)
Integral Transforms	635	(50)
Introduction	635	(4)
The Laplace transform	639	(3)
Solution of differential equations by Laplace transforms	642	(5)
Fourier transforms	647	(8)
Convolution; Parseval's theorem	655	(7)
Inverse Laplace transform (Bromwich integral)	662	(3)
The Dirac delta function	665	(5)
Green functions	670	(6)
Integral transform solutions of partial differential equations	676	(5)
Miscellaneous problems	681	(4)
Probability	685	(56)
Introduction; definition of probability	685	(2)
Sample space	687	(5)
Probability theorems	692	(7)
Methods of counting	699	(8)
Random variables	707	(5)
Continuous distributions	712	(6)
Binomial distribution	718	(5)
The normal or Gaussian distribution	723	(5)
The Poisson distribution	728	(3)
Applications to experimental measurements	731	(6)
Miscellaneous problems	737	(4)
References	741	(2)
Bibliography	743	(4)
Answers To Selected Problems	747	(28)
Index	775