Methods of Mathematical Physics, Volume 1

by Courant, Richard; Hilbert, David

• ISBN13: 9780471504474
• ISBN10: 0471504475
• Edition: 1st
• Format: Paperback
• Copyright: 1991-01-08
• Publisher: Wiley-VCH

Summary

Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953.

Author Biography

David Hilbert (1862 -+ 1943) received his PhD from the University of K+¦nigsberg, Prussia (now Kaliningrad, Russia) in 1884. He remained there until 1895, after which he was appointed Professor of Mathematics at the University of G+¦ttingen. He held this professorship for most of his life. Hilbert is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. His own discoveries alone would have given him that honour, yet it was his leadership in the field of mathematics throughout his later life that distinguishes him. Hilbert's name is given to Infinite-Dimensional space, called Hilbert space, used as a conception for the mathematical analysis of the kinetic gas theory and the theory of radiations.

Richard Courant (1888 -+ 1972) obtained his doctorate at the University of G+¦ttingen in 1910. Here, he became Hilbert-+s assistant. He returned to G+¦ttingen to continue his research after World War I, and founded and headed the university-+s Mathematical Institute. In 1933, Courant left Germany for England, from whence he went on to the United States after a year. In 1936, he became a professor at the New York University. Here, he headed the Department of Mathematics and was Director of the Institute of Mathematical Sciences - which was subsequently renamed the Courant Institute of Mathematical Sciences. Among other things, Courant is well remembered for his achievement regarding the finite element method, which he set on a solid mathematical basis and which is nowadays the most important way to solve partial differential equations numerically.

Table of Contents

I. The Algebra of Linear Transformations and Quadratic Forms	1	(47)
Linear equations and linear transformations	1	(16)
Vectors	1	(2)
Orthogonal systems of vectors. Completeness	3	(2)
Linear transformations. Matrices	5	(6)
Bilinear, quadratic, and Hermitian forms	11	(3)
Orthogonal and unitary transformations	14	(3)
Linear transformations with a linear parameter	17	(6)
Transformation to principal axes of quadratic and Hermitian forms	23	(8)
Transformation to principal axes on the basis of a maximum principle	23	(3)
Eigenvalues	26	(2)
Generalization to Hermitian forms	28	(1)
Intertial theorem for quadratic forms	28	(1)
Representation of the resolvent of a form	29	(1)
Solution of systems of linear equations associated with forms	30	(1)
Minimum-maximum property of eigenvalues	31	(3)
Characterization of eigenvalues by a minimum-maximum problem	31	(2)
Applications. Constraints	33	(1)
Supplement and problems	34	(14)
Linear Independence and the Gram determinant	34	(2)
Hadamard's inequality for determinants	36	(1)
Generalized treatment of canonical transformations	37	(4)
Bilinear and quadratic forms of infinitely many variables	41	(1)
Infinitesimal linear transformations	41	(1)
Perturbations	42	(2)
Constraints	44	(1)
Elementary divisors of a matrix or a bilinear form	45	(1)
Spectrum of a unitary matrix	46	(2)
References	47	(1)
II. Series Expansions of Arbitrary Functions	48	(64)
Orthogonal systems of functions	49	(8)
Definitions	49	(1)
Orthogonalization of functions	50	(1)
Bessel's inequality. Completeness relation. Approximation in the mean	51	(4)
Orthogonal and unitary transformations with infinitely many variables	55	(1)
Validity of the results for several independent variables. More general assumptions	56	(1)
Construction of complete systems of functions of several variables	56	(1)
The accumulation principle for functions	57	(4)
Convergence in function space	57	(4)
Measure of independence and dimension number	61	(4)
Measure of independence	61	(2)
Asymptotic dimension of a sequence of functions	63	(2)
Weierstrass's approximation theorem. Completeness of powers and of trigonometric functions	65	(4)
Weierstrass's approximation theorem	65	(3)
Extension to functions of several variables	68	(1)
Simultaneous approximation of derivatives	68	(1)
Completeness of the trigonometric functions	68	(1)
Fourier series	69	(8)
Proof of the fundamental theorem	69	(4)
Multiple Fourier series	73	(1)
Order of magnitude of Fourier coefficients	74	(1)
Change in length of basic interval	74	(1)
Examples	74	(3)
The Fourier integral	77	(4)
The fundamental theorem	77	(2)
Extension of the result to several variables	79	(1)
Reciprocity formulas	80	(1)
Examples of Fourier integrals	81	(1)
Legendre Polynomials	82	(5)
Construction of the Legendre polynomials by orthogonaliza-of the powers 1, x, x2	82	(3)
The generating function	85	(1)
Other properties of the Legendre polynomials	86	(1)
Examples of other orthogonal systems	87	(10)
Generalization of the problem leading to Legendre polynomials	87	(1)
Tehebycheff polynomials	88	(2)
Jacobi polynomials	90	(1)
Hermite polynomials	91	(2)
Laguerre polynomials	93	(2)
Completeness of the Laguerre and Hermite functions	95	(2)
Supplement and problems	97	(15)
Hurwitz's solution of the isoperimetric problem	97	(1)
Reciprocity formulas	98	(1)
The Fourier integral and convergence in the mean	98	(1)
Spectral decomposition by Fourier series and integrals	99	(1)
Dense systems of functions	100	(2)
A Theorem of H. Muntz on the completeness of powers	102	(1)
Fejer's summation theorem	102	(1)
The Mellin inversion formulas	103	(2)
The Gibbs Phenomenon	105	(2)
A Theorem on Gram's determinant	107	(1)
Application of the Lebesgue integral	108	(4)
References	111	(1)
III. Linear Integral Equations	112	(52)
Introduction	112	(3)
Notation and basic concepts	112	(1)
Functions in integral representation	113	(1)
Degenerate kernels	114	(1)
Fredholm's theorems for degenerate kernels	115	(3)
Fredholm's theorems for arbitrary kernels	118	(4)
Symmetric kernels and their eigenvalues	122	(12)
Existence of an eigenvalue of a symmetric kernel	122	(4)
The totality of eigenfunctions and eigenvalues	126	(6)
Maximum-minimum property of eigenvalues	132	(2)
The expansion theorem and its applications	134	(6)
Expansion theorem	134	(2)
Solution of the inhomogeneous linear integral equation	136	(1)
Bilinear formula for iterated kernels	137	(1)
Mercer's theorem	138	(2)
Neumann series and the reciprocal kernel	140	(2)
The Fredholm formulas	142	(5)
Another derivation of the theory	147	(5)
A Lemma	147	(1)
Eigenfunctions of a symmetric kernel	148	(2)
Unsymmetric Kernels	150	(1)
Continuous dependence of eigenvalues and eigenfunctions on the kernel	151	(1)
Extensions of the theory	152	(1)
Supplement and problems for Chapter III	153	(11)
Problems	153	(1)
Singular integral equations	154	(1)
E. Schmidt's derivation of the Fredholm theorems	155	(1)
Enskog's method for solving symmetric integral equations	156	(1)
Kellogg's method for the determination of eigenfunctions	156	(1)
Symbolic functions of a kernel and their eigenvalues	157	(1)
Examples of an unsymmetric kernel without null solutions	157	(1)
Volterra integral equation	158	(1)
Abel's integral equation	158	(1)
Adjoint orthogonal systems belonging to an unsymmetric kernel	159	(1)
Integral equations of the first kind	159	(1)
Method of infinitely many variables	160	(1)
Minimum properties of eigenfunctions	161	(1)
Polar integral equations	161	(1)
Symmetrizable kernels	161	(1)
Determination of the resolvent kernel by functional equations	162	(1)
Continuity of definite kernels	162	(1)
Hammerstein's theorem	162	(2)
References	162	(2)
IV. The Calculus of Variations	164	(111)
Problems of the calculus of variations	164	(10)
Maxima and minima of functions	164	(3)
Functionals	167	(2)
Typical problems of the calculus of variations	169	(4)
Characteristic difficulties of the calculus of variations	173	(1)
Direct solutions	174	(9)
The Isoperimetric problem	174	(1)
The Rayleigh-Ritz method. Minimizing Sequences	175	(1)
Other Direct methods. Method of finite differences. Infinitely many variables	176	(6)
General remarks on direct methods of the calculus of variations	182	(1)
The Euler equations	183	(23)
``Simplest problem'' of the variational calculus	184	(3)
Several unknown functions	187	(3)
Higher derivatives	190	(1)
Several independent variables	191	(2)
Identical vanishing of the Euler differential expression	193	(3)
Euler equations in homogeneous form	196	(3)
Relaxing of conditions. Theorems of du Bois-Reymond and Haar	199	(6)
Variational problems and functional equations	205	(1)
Integration of the Euler differential equation	206	(2)
Boundary conditions	208	(6)
Natural boundary conditions for free boundaries	208	(3)
Geometrical problems. Transversality	211	(3)
The second variation and the Legendre condition	214	(2)
Variational problems with subsidiary conditions	216	(6)
Isoperimetric problems	216	(3)
Finite subsidiary conditions	219	(2)
Differential equations as subsidiary conditions	221	(1)
Invariant character of the Euler equations	222	(9)
The Euler expression as a gradient in function space. Invariance of the Euler expression	222	(2)
Transformation of Δu. Spherical coordinates	224	(2)
Ellipsoidal coordinates	226	(5)
Transformation of variational problems to canonical and involutory form	231	(11)
Transformation of an ordinary minimum problem with subsidiary conditions	231	(2)
Involutory transformation of the simplest variational problems	233	(5)
Transformation of variational problems to canonical form	238	(2)
Generalizations	240	(2)
Variational calculus and the differential equations of mathematical physics	242	(10)
General remarks	242	(2)
The vibrating string and the vibrating rod	244	(2)
Membrane and plate	246	(6)
Reciprocal quadratic variational problems	252	(5)
Supplementary remarks and exercises	257	(18)
Variational problem for a given differential equation	257	(1)
Reciprocity for isoperimetric problems	258	(1)
Circular light rays	258	(1)
The problem of Dido	258	(1)
Examples of problems in space	258	(1)
The indicatrix and applications	258	(2)
Variable domains	260	(2)
E. Noether's theorem on invariant variational problems. Integrals in particle mechanics	262	(4)
Transversality for multiple integrals	266	(1)
Euler's differential expressions on surfaces	267	(1)
Thomson's principle in electrostatics	267	(1)
Equilibrium problems for elastic bodies. Castigliano's principle	268	(4)
The Variational problem of buckling	272	(3)
References	274	(1)
V. Vibration and Eigenvalue Problems	275	(122)
Preliminary remarks about linear differential equations	275	(6)
Principle of superposition	275	(2)
Homogeneous and nonhomogeneous problems. Boundary conditions	277	(1)
Formal relations. Adjoint differential expressions. Green's formulas	277	(3)
Linear functional equations as limiting cases and analogues of systems of linear equations	280	(1)
Systems of a finite number of degrees of freedom	281	(5)
Normal modes of vibration. Normal coordinates. General theory of motion	282	(3)
General properties of vibrating systems	285	(1)
The vibrating string	286	(9)
Free motion of the homogeneous string	287	(2)
Forced motion	289	(2)
The general nonhomogeneous string and the Sturm-Liouville eigenvalue problem	291	(4)
The vibrating rod	295	(2)
The vibrating membrane	297	(10)
General eigenvalue problem for the homogeneous membrane	297	(3)
Forced motion	300	(1)
Nodal lines	300	(1)
Rectangular membrane	300	(2)
Circular membrane. Bessel functions	302	(4)
Nonhomogeneous membrane	306	(1)
The Vibrating plate	307	(1)
General remarks	307	(1)
Circular boundary	307	(1)
General remarks on the eigenfunction method	308	(5)
Vibration and equilibrium problems	308	(3)
Heat conduction and eigenvalue problems	311	(2)
Vibration of three-dimensional continua. Separation of variables	313	(2)
Eigenfunctions and the boundary value problem of potential theory	315	(9)
Circle, sphere, and spherical shell	315	(4)
Cylindrical domain	319	(1)
The Lame problem	319	(5)
Problems of the Sturm-Liouville type. Singular boundary points	324	(7)
Bessel functions	324	(1)
Legendre functions of arbitrary order	325	(2)
Jacobi and tehebycheff polynomials	327	(1)
Hermite and Laguerre polynomials	328	(3)
The asymptotic behavior of the solutions of Strum-Liouville equations	331	(8)
Boundedness of the solution as the independent variables tends to infinity	331	(1)
A sharper result. (Bessel functions)	332	(2)
Boundedness as the parameter increases	334	(1)
Asymptotic representation of the solutions	335	(1)
Asymptotic representation of Sturm-Liouville eigenfunctions	336	(3)
Eigenvalue problems with a continuous spectrum	339	(4)
Trigonometric functions	340	(1)
Bessel functions	340	(1)
Eigenvalue problem of the membrane equation for the infinite plane	341	(1)
The Schrodinger eigenvalue problem	341	(2)
Perturbation theory	343	(8)
Simple eigenvalues	344	(2)
Multiple eigenvalues	346	(2)
An example	348	(3)
Green's function (influence function) and reduction of differential equations to integral equations	351	(20)
Green's function and boundary value problem for ordinary differential equations	351	(3)
Construction of Green's function; Green's function in the generalized sense	354	(4)
Equivalence of integral and differential equations	358	(4)
Ordinary differential equations of higher order	362	(1)
Partial differential equations	363	(8)
Examples of Green's function	371	(17)
Ordinary differential equations	371	(6)
Green's function for Δu: circle and sphere	377	(1)
Green's function and conformal mapping	377	(1)
Green's function for the potential equation on the surface of a sphere	378	(1)
Green's function for Δ u = 0 in a rectangular parallelepiped	378	(6)
Green's function for Δ u in the interior of a rectangle	384	(2)
Green's function for a circular ring	386	(2)
Supplement to Chapter V	388	(9)
Examples for the vibrating string	388	(2)
Vibrations of a freely suspended rope; Bessel functions	390	(1)
Examples for the explicit solution of the vibration equation. Mathieu functions	391	(1)
Boundary conditions with parameters	392	(1)
Green's tensors for systems of differential equations	393	(2)
Analytic continuation of the solutions of the equation Δ u + λu = 0	395	(1)
A theorem on the nodal curves of the solutions of Δu + λu = 0	395	(1)
An example of eigenvalues of infinite multiplicity	395	(1)
Limits for the validity of the expansion theorems	395	(2)
References	396	(1)
VI. Application of the Calculus of Variations to Eigenvalue Problems	397	(69)
Extremum properties of eigenvalues	398	(9)
Classical extremum properties	398	(4)
Generalizations	402	(3)
Eigenvalue problems for regions with separate components	405	(1)
The maximum-minimum property of eigenvalues	405	(2)
General consequences of the extremum properties of the eigenvalues	407	(17)
General Theorems	407	(5)
Infinite growth of the eigenvalues	412	(2)
Asymptotic behavior of the eigenvalues in the Sturm-Liouville problem	414	(1)
Singular differential equations	415	(1)
Further remarks concerning the growth of eigenvalues. Occurrence of negative eigenvalues	416	(2)
Continuity of eigenvalues	418	(6)
Completeness and expansion theorems	424	(5)
Completeness of the eigenfunctions	424	(2)
The expansion theorem	426	(1)
Generalization of the expansion theorem	427	(2)
Asymptotic distribution of eigenvalues	429	(16)
The equation Δu + λu = 0 for a rectangle	429	(2)
The equation Δu + λu = 0 for domains consisting of a finite number of squares or cubes	431	(3)
Extension to the general differential equation L[u] + λpu = 0	434	(2)
Asymptotic distribution of eigenvalues for an arbitrary domain	436	(7)
Sharper form of the laws of asymptotic distribution of eigenvalues for the differential equation Δu + λu = 0	443	(2)
Eigenvalue problems of the Schrodinger type	445	(6)
Nodes of eigenfunctions	451	(4)
Supplementary remarks and problems	455	(11)
Minimizing properties of eigenvalues. Derivation from completeness	455	(3)
Characterization of the first eigenfunction by absence of nodes	458	(1)
Further minimizing properties of eigenvalues	459	(1)
Asymptotic distribution of eigenvalues	460	(1)
Parameter eigenvalue problems	460	(1)
Boundary conditions containing parameters	461	(1)
Eigenvalue problems for closed surfaces	461	(1)
Estimates of eigenvalues when singular points occur	461	(2)
Minimum theorems for the membrane and plate	463	(1)
Minimum problems for variable mass distribution	463	(1)
Nodal points for the Sturm-Liouville problem. Maximum-minimum principle	463	(3)
References	464	(2)
VII. Special Functions Defined by Eigenvalue Problems	466	(80)
Preliminary discussion of linear second order differential equations	466	(1)
Bessel functions	467	(34)
Application of the integral transformation	468	(1)
Hankel functions	469	(2)
Bessel and Neumann functions	471	(3)
Integral representations of Bessel functions	474	(2)
Another integral representation of the Hankel and Bessel functions	476	(6)
Power series expansion of Bessel functions	482	(3)
Relations between Bessel functions	485	(7)
Zeros of Bessel functions	492	(4)
Neumann functions	496	(5)
Legendre functions	501	(5)
Schlafi's integral	501	(2)
Integral representations of Laplace	503	(1)
Legendre functions of the second kind	504	(1)
Associated Legendre functions. (Legendre functions of higher order.)	505	(1)
Application of the method of integral transformation to Legendre, Tchebycheff, Hermite, and Laguerre equations	506	(4)
Legendre functions	506	(1)
Tehebycheff functions	507	(1)
Hermite functions	508	(1)
Laguerre functions	509	(1)
Laplace spherical harmonics	510	(12)
Determination of 2n + 1 spherical harmonics of n-th order	511	(1)
Completeness of the system of functions	512	(1)
Expansion theorem	513	(1)
The Poisson integral	513	(1)
The Maxwell-Sylvester representation of spherical harmonics	514	(8)
Asymptotic expansions	522	(13)
Stirling's formula	522	(2)
Asymptotic calculation of Hankel and Bessel functions for large values of the arguments	524	(2)
The Saddle point method	526	(1)
Application of the saddle point method to the calculation of Hankel and Bessel functions for large parameter and large argument	527	(5)
General remarks on the saddle point method	532	(1)
The Darboux method	532	(1)
Application of the Darboux method to the asymptotic expansion of Legendre polynomials	533	(2)
Appendix to Chapter VII. Transformation of Spherical Harmonics	535	(11)
Introduction and notation	535	(1)
Orthogonal transformations	536	(3)
A generating function for spherical harmonics	539	(3)
Transformation formula	542	(1)
Expressions in terms of angular coordinates	543	(3)
Additional bibliography	546	(5)
Index	551