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Courant R., Hilbert D. Ч Methods of Mathematical Physics, Vol. 2
Courant R., Hilbert D. Ч Methods of Mathematical Physics, Vol. 2



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Ќазвание: Methods of Mathematical Physics, Vol. 2

јвторы: Courant R., Hilbert D.

јннотаци€:

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.


язык: en

–убрика: ‘изика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

»здание: 1st edition

√од издани€: 1991

 оличество страниц: 852

ƒобавлена в каталог: 04.07.2008

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ѕредметный указатель
"Finite parts" of an integral      786
"Finite parts" of divergent integrals      740 743
"Improperly posed" problems      230 Ч 231
"Outward" derivative      134
"Tangential" differentiation;      577; see Interior differentiation TaylorТs theorem
"Weakly" space-like surface      654
$J_{0}$ (BesselТs function of order zero)      694
(F, G) pseudoanalytic function      386
A priori estimates      see Estimates
AbelТs integral equation      535
Adjoint operators      234 Ч 237 770
Alfven speed      614
Alfven waves      613
Analytic function of two complex variables      501
Analyticity of harmonic functions      269
Analyticity of solutions of elliptic equations      501 Ч 505
Approximate progressing wave      623
ArzelaТs theorem      310 381
AsgeirssonТs mean value theorem      744 Ч 752
Associated Legendre functions      242
Asymptotic solutions for a linear hyperbolic system      637 Ч 642
Backward ray cone      564 569
Balayage method of Poincare      296 (ftn)
Banach space      333 403
Barrier function(s)      31
Barrier strong      341 Ч 343
Beltrami equations      178
Beltrami equations solution of      350 Ч 357
Bernstein, theorem of      400
Bessel equation      244
Bessel function(s)      194 289
Bessel function(s) integral theorem for      534
Bessel function(s) of order zero      185
Bicharacteristic rays      551 730
Bicharacteristic rays along multiple characteristics      627
Bicharacteristic rays differentiation along      597 Ч 599
Bicharacteristic strip(s)      558 583
Bicharacteristics (or bicharacteristic curves)      558 Ч 559
Bicharacteristics in hydrodynamics      601
Bicharacteristics invariance under transformation of independent variables of      562
Bicharacteristics of a second order differential equation      559
Bilinear form      770
Boundary continuity method      336 Ч 339
Boundary counter-example of Lebesgue      303 Ч 305
Boundary for general elliptic differential equations      320 Ч 331
Boundary for Poisson equation      261 ff
Boundary for the equation of minimal surfaces      see PlateauТs problem
Boundary for the potential equation      21 Ч 22
Boundary in multiply-connected domains      292 293
Boundary nonlinear      395 Ч 399
Boundary space, maximal non-negative      658
Boundary uniqueness theorems for      320 Ч 324
Boundary value problem(s)      222 226 290 357
Boundary with nonunique solutions      324 Ч 326
Branch curves      65
Branch lines      408
Branch strip      415
Brouwer fixed point theorem      403
Calculus of variations      113 Ч 131
Canonical equations      131
Canonical form      182
Canonical perturbation theory      130 Ч 131
Canonical system of differential equations      107
Canonical transformation      127 Ч 129
Canonically conjugate variables      115
Capacity      305 Ч 306
Cauchy data      408
Cauchy double integral representation      501 (ftn)
Cauchy initial uniqueness of solutions      438 Ч 449
Cauchy initial value problem      407 Ч 766 (see also Initial value problem)
Cauchy method of the Fourier integral      210 Ч 221
Cauchy problems      see Initial value problems
Cauchy Ч Kowalewski for linear equations      54
Cauchy Ч Kowalewski theorem      39 Ч 56 363 669 670
Cauchy Ч Riemann      176 178
Cauchy Ч Riemann differential equations      58
Caustic      673 674
Caustic curves      83 89
Caustic surface      123
Characteristic      see also Bicharacteristic
Characteristic coordinates      see Characteristics as coordinate curves
Characteristic, base curve(s)      63 70 411
Characteristic, basic manifold      138
Characteristic, condition      136 Ч 139 412 419 554 555
Characteristic, cone      591
Characteristic, cone, in HuyghensТ principle      690
Characteristic, curve(s)      29 70 158 172 407 411 415
Characteristic, curve(s), for the general first order differential equation in n independent variables      103 Ч 105
Characteristic, curve(s), on an integral surface      414 Ч 416
Characteristic, curve(s),for quasi-linear differential equations in n variables      69 70 73
Characteristic, derivative      136 Ч 139
Characteristic, determinant      172
Characteristic, differential equations, canonical form of      106 Ч 107
Characteristic, differentiation      135
Characteristic, element      410
Characteristic, form(s)      156 175 412 554 579
Characteristic, initial value problem(s)      56 450
Characteristic, initial value problem(s),of the wave equation      644 Ч 646
Characteristic, line element      33
Characteristic, matrix      173 530
Characteristic, normal cone      582 663
Characteristic, parameters      427 428
Characteristic, ray cone, for elastic waves      708
Characteristic, rays, of a second order equation      559
Characteristic, sheets, multiple      595 Ч 596
Characteristic, speed      430
Characteristic, strips      77 79 97 103 411 422
Characteristic, surface(s)      174
Characteristic, surface(s), as wave fronts      632
Characteristic, surface(s), in hydrodynamics      600 Ч 601
Characteristic, surface(s), in magnetohydrodynamics      613
Characteristic, surface(s), of a differential operator      579 Ч 581
Characteristic, surface(s), of a second order linear equation      557
Characteristic,condition, interpretation of, in time and space      581 Ч 583
Characteristics      55 Ч 56 170 175 553
Characteristics as branch elements      82
Characteristics, as coordinate curves      490 Ч 507
Characteristics, as curves of discontinuity      416 Ч 418
Characteristics, for quasi-linear systems      176
Characteristics, for second or er equations in two independent variables      408
Characteristics, geometry of      552 Ч 569
Characteristics, geometry, for higher order operators      577 Ч 599
Characteristics, invariance of      423 562 589
Characteristics, with constant multiplicity      626 Ч 627
ClairautТs equation      27 Ч 28 37 94
Classification for equations of second order      181 Ч 184
Compactness theorem for harmonic functions      275
Compatibility condition      60
Complete, integral      24 Ч 25 37 84 103
Complete, progressing wave      623
Complete, space      403
Compressible fluids      429 Ч 437
Compressible fluids, linearized equations      436 Ч 437
Cone      124 Ч 125 233
Conoid of dependence      654 730
Conservation, laws      147 Ч 153 488
Conservation, of energy      665
Conservative operators of first order      653
Consistency conditions for Cauchy data      472
Continuable initial conditions      470
Continuation of a real solution into the complex plane      499 Ч 507
Continuation of solutions      505
Continuity method      336
Continuous linear functional      771
Convex set      404
Convex, hull      589
Convex, hull,of a local ray cone      589 591
Convolution of ideal functions      791 Ч 792
Crystal optics      599 602 649
Crystal optics, conical refraction in      628 629
Crystal optics, solution by method of plane mean values      718 Ч 727
Cylindrical waves      193 Ч 194
Damped waves      192
Darboux, differential equation for mean values      699 Ч 700
Darboux, equation      646 Ч 647
Darboux, equation, connection with the wave equation      700 Ч 703
Decomposition of ideal functions      781
Delta function      456 776
Delta function, as a weak limit      778
Delta function, decomposition of      730 Ч 733
Delta function, homogeneity      791
Delta function, in polar coordinates      791
Delta function, three dimensional      696
Delta function, transformations of      790 Ч 791
Derivative, directional      132
Derivative, normal      133
Descent, method of      205 Ч 206 686
Developable surfaces      10
Differentiability properties of solutions of boundary value problems      343 Ч 350
Differentiation of fractional order      see Fractional differentiation
Dipole potentials      251
Dirac equations      179
Dirichlet, integral      254 681
Dirichlet, problem      261 Ч 264 (see also Boundary value problem GreenТs Poisson
Dirichlet, problem for general homogeneous elliptic equations      342
Discontinuities      585
Discontinuities, of solutions      486 Ч 490
Discontinuities, propagation of      see also Transport equations
Discontinuities, propagation of, along rays      573 Ч 574
Discontinuities, propagation of, for first order systems      624 Ч 631
dispersion      191
Distributions      621 727 740 743 766
Domain of dependence      209 227 228 438
Domain of dependence, for crystal optics      723 Ч 727
Domain of dependence, for elastic wave equation      711
Domain of dependence, for hyperbolic equations      649 Ч 654 (see also Uniqueness theorems)
Domain of dependence, gaps in      726
Domain of dependence, in HuyghensТ principle      690
Domain of determinacy      439
Domain of influence      227 649
Double layer potential(s)      251 Ч 261 298 299
Dual cone      see Normal cone
Dual, generating pair      386
Dual, spaces      793 796
Duhamel(Сs) integral      512 Ч 515 691
Duhamel(Сs) principle      202 Ч 204 668
Duhamel(Сs) principle, for systems of first order      204
Duhamel(Сs) representation for solutions of inhomogeneous problems      552
Duhamel(Сs) theorem      512 520
Dupin cyclides      763
Edge of regression      66
Edge of regression, of an integral surface      83
Eiconal      116 Ч 118
Eiconal, equation      123 (see also Hamilton Ч Jacobi equation)
Elastic constants      707
Elastic waves      706 Ч 711
Elimination from a linear system with constant coefficients      14 Ч 15
Elliptic      156 172 174 177 240
Elliptic, equations, transition to the hyperbolic case through complex domains      499 Ч 501
Elliptic, equations, uniformly elliptic      331 Ч 332
Elliptic, theta function      200
Energy conservation of      see Conservation of energy flux
Energy, identities      662 Ч 663
Energy, inequalities      449 655 656 661 662
Energy, integral(s)      443 447 618 654 665
Energy, integral(s), for linear symmetric hyperbolic systems of first order      652 Ч 661
Energy, integral(s), for single second order equations      659
Energy, integral(s), method of      642
Equation of mixed type      161
Equicontinuous functions, lemma on the convergence of      310
Estimate(s)      see also Energy inequalities HamackТs Maximum
Estimate(s), for solutions of Neumann problem      330 331
Estimate(s), for solutions of quasi-linear elliptic equations      359
Estimate(s), of Schauder      331 Ч 350
Estimate(s), of Schauder, for solutions of linear elliptic equations      331 Ч 336
Estimate(s), of Schauder, interior      333 Ч 335
Estimate(s), of Schauder, up to the boundary      335 Ч 336
Euler equations      114 Ч 115
Euler homogeneity relation      11
Existence of solutions, (quasi-linear equations)      357 Ч 362
Existence of solutions, of boundary value problems      336 Ч 341
Existence theorem(s) for solutions of partial differential equations      see Cauchy- Kowalewsky theorem Holmgren
Existence theorem(s) of ordinary differential equations      30
Exponential operators      524
Extremal field      122
Factorial of a vector      577(ftn) 768
Families of functions, differential equations for      8 Ч 12
FermatТs principle      117
Field functions      116
First boundary value problem      see Dirichlet problem Boundary
Fixed point method of Leray and Schauder      357 Ч 362
Foca, curves      76 83 86 89
Foca, strip      77 83
Focussing      673 Ч 674 736
Folium of Descartes      588
Forward ray cone      564 569
Fourier, integral      210ff
Fourier, transformation of ideal functions      793 Ч 795
Fractional differentiation      523 702
Fredholm integral equation method      299
Free, boundary problem(s)      225 Ч 226 348
Free, curve(s)      55 172
Free, initial surface for second order quasi-linear differential equations      554
Free, strip      410
Free, surface      173
Free, surface, for a differential operator      579
Fresnel surfaces      605
Functional      see Linear functional
Fundamental solution(s)      184 Ч 187 727-
Fundamental solution(s), for elliptic equations      363 Ч 367
Fundamental solution(s), for single equations of second order      740 Ч 744
Fundamental solution(s), in crystal optics      723 Ч 726
Fundamental solution(s), of the heat equation      21
Fundamental solution(s), of the potential equation      244
Fundamental solution(s), of the reduced wave equation      314
Gaps, theorem on      736 Ч 737
GaussТ integral theorem      252 699
GaussТ integral theorem, for ideal functions      787 Ч 788
Generalized solution(s)      418 486 669
Generators      386
Geodesics on an ellipsoid      111 Ч 113
Geodetic distance      116 117 121 565
Geometrical optics, M      640 Ч 642
GreenТs formulas      252 Ч 258
GreenТs function      261 Ч 268 290 383
HaarТs uniqueness proof      145 Ч 147
HadamardТs conjecture      765
HadamardТs example of improperly posed problem      229
HadamardТs theory of hyperbolic equations      740
Hamilton Ч Jacobi equation      118
Hamilton Ч Jacobi equation, for geodetic distance      109
Hamilton Ч Jacobi equation, theorem      127 129
Hamiltonian function      106
Hamiltonian function, for the two body problem      109
Hankel function      194
Harmonic function(s)      240
Harmonic function(s), regular      241 244
HarnackТs inequality      269
HarnackТs theorem      273
Heat equation      20 Ч 22 154 188 217
Heat equation, initial value problems for      197 Ч 201 525
Heaviside function      622
Heaviside unit function      see Unit function
Heaviside, calculus (Heaviside method of operators)      507 Ч 550
Heaviside, calculus (Heaviside method of operators), justification of      528 Ч 535
Heaviside, calculus (Heaviside method of operators), method      517 Ч 535
Heaviside, method of operators      508 517
Heaviside, operator, definition      528 Ч 529
Hilbert, invariant integral      125 Ч 126
Hilbert, space      668
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