Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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Index
J\bel,63,
177,
183,233,416,462,653
biography, 662
abelian group, 654
abelian Lie algebra, 834
absolute
convergence,
252
additive identity, 20
adjoint
formal, 560
adjoint action, 827
adjoint algebra, 840
adjoint
Be,563
adjoint DO,
364-367
adjoint Green's function, 566
adjoint map, 825
adjoint of a matrix, 87
adjoint
of an
operator,
61
adjoint
representation,
679
character, 684
affine group, 876
algebra,
41-45
definition, 41
derivation
of an,42
dimensionof an,41
homomorphism,
43
ideal,44
isomorphism,
43
of
operators, 49--76
quaternions,
44
structure
constants,
43
synunettic, 737
tensor,
731
algebraic equation, 937
synunetry gronp of, 937
analytic cootinuation,
302-309
analytic function,
228-235
definition, 233
derivative, 228
derivatives
as
integrals.
249
entire, 233
poles of, 273
roots (zeros) of, 262
angular momentum, 831, 869
additiontheorem, 866
eigenvalues, 337
operator, 331
construction
of eigenvalues, 334
annihi1ator, 39
anti-hermitian
operator,
63
anticommutator,
59
antisymmetric
representation,
679
antisymmetrizer,
739
arc length, 890
associated
Legendre
functions,
341
associative
algebra,
41
atlas, 765
automorphism, 36
averaging
operator,
71
azimuthal symmetry, 343
1008
INDEX
backward
difference
operator.
70,
384-387
Baker-Campbell-Hausdorff, 59
Banach space, 148
basis, 22
dual
of
a, 38, 729
oriented,745
orthouormal, 26
transformation
matrix,
91
Becquerel, 799
Bernoulli, 423, 981
Bessel, 177, 620, 982
biography, 422
Bessel
equation
Liouville
substitution,
514
Bessel functiun,
423-426
asymptotic
behavior,
443
large argument, 444
large order, 443
confluent hypergeomettic, 424
generating
function,
440
integral
representation
of,438
modified, 425
second kind, 425
oscillation
of, 363
second kind, 424
third kind, 425
Bessel
functions
spherical, 542
Bessel
imaginary (bei), 539
Bessel inequality, 149
Bessel real (ber), 539
beta function,
309-312
definition, 310
Bianchi identity, 889
bijective map, 6
binary
operation,
7
binomial theorem, 13
Birkhoff's theorem, 925
Bolzano,l1
Balzano-Weierstrass
theorem,
461
Boole,704
boundary
conditions
Dirichlet, 591
homogeneous, 559
mixed, 564
Neumann, 591
periodic, SIS
separated,
510
unrnixed,564
boundery functionals, 559
boundery point, 459
boundery value problem, 560, 583
DiricWet,591
Neumann, 591
numerical
solution,
383
hounded operator,
453-457
continnity,454
branch cut, 296
Brouwer, 850
bundle
of
tensors,
786
BWHB theorem, 853
canonical basis, 747, 802
second order tinear DE, 404
canonical
coordinates,
802
canonical
transformation,
802,806
Cantor, 463, 739, 799
biography, 10
Cantorse!, 12
cardina1ity, 12
Cartan,798,841,942
biugraphy, 743
Cartan mettic tensor, 843, 862
Cartan subalgebra, 845
Cartan's lemma, 743
Cartesian
product,
2
Casimir operator, 861-863, 868-870, 872,
873
Cauchy, 96, 271, 296, 416, 478, 528, 588,
653,891
biography, 233
Cauchy data, 584
Cauchyintegral formula, 245
for operators, 482
Cauchy problem, 584
ill-posed, 590
Cauchy sequence, 10, 146
Cauchy-Goursat theorem, 242
Cauchy-Riemann conditions, 230
differentiability, 232
Cayley, 704, 744
center of a group, 658
central
difference
operator,
70
centralizer, 658
Champollion, 198
character
adjoint
representation,
684
compound, 683, 685
conjugacy class, 684
group and its subgroup, 692
of a
representation,
683
simple, 683, 685
character table, 691
character
table
for S2,694
character
table
for S3,694
characteristic hypersorface,
585-589
definition, 586
characteristic polynomial, 115
linear DEs, 376
chart, 764
Chebyshev, 588
Chebyshev polynomials, 185
Chevalley's theorem, 862
Chevalley, biography, 864
Christoffel symbol, 890
Christoffel, biography, 890
circnit matrix of a DE, 403
circularheat-conducting plate, 537
Clebsch, 178, 704
biography, 702
Clebsch-Gordan
coefficients, 705, 706
decompo~tion,701-705,721,866
series, 703, 706
Clifford, 744
closed form, 797
closed snbset, 459
closure, 459
codomain,S
cofactor, 95
commntative algebra, 41
commutative
group,
654
commutative
Lie
algebra,
834
commutator
diagona1izability, 124
commutator subgroup, 658
commutators, 55
compactLie algebra, 843
INDEX
1009
compact
Lie
group
representation,
845-855
compact operator,
464-466
definition, 464
spectturn,467-473
compact
resolvent,
508
compact set, 458-463
compact subset, 461
compact support, 165n,801
comparison
theorem,
361-363
complement of aset,2
complete
0.0.
sequence,
149
cornpletenessrelatipn,68,150
complex
coordinate
space,
21
complex exponential function, 234
complex
FOillEs,
401-402
complex function, 227
integration,
241
complex GL(V), 822
complexplane
contour
in the,242
curvein
the,
242
multiply connected region, 245
path in the, 242
simply connectedregion, 245
complex potential, 236
complex
series,
252
complexSOLDE, 404-410
complex
vector
space,
20
composition of
maps,
6
compound character, 685
conducting cylindrical can, 535
confluenthypergeometric
function
definition, 420
integral representation of, 437
conformal gronp, 912
in 2
dimensions,
912
conformal Killing vector, 911
conformal map, 236-241
definition, 237
translation, 238
conformal
transformation,
911
special, 912
conicsections, 132
conjngacy class, 661, 684
conjngate,661
conjugate subgronp, 657
1010
INDEX
conjugation of operators, 61-63
conjunct, 560,
563,576
connection
coefficients, 886,900
conservation
law,855,992
characteristic, 994
trivial, 993
trivial of the first kind, 992
trivial of the second kind, 993
conserved
current
density,
992
constant
of the
motion,
992
constrained
systems,8040
continuous
index,
159
contraction,
735
contravariant degree, 731
contravariant
tensor,
731
convex
subset,
474
convolution
theorem,
223
coordinate curve, 774
coordinate frame, 774
coordinate
functions,
764
coordinate
representation
of L
8
*, 820
coordinate
transformation
orientation
preserving,
800
orientation
reversing,
800
cosel,658
cosmological
constant,
921
Coulomb, 985
countabiy infinite set, 12
covariant degree, 731
covariant
derivative,
897-908
and curvature, 903
Lie derivative, 902
covariant
tensor,
731
criterion for irredncibility, 685
crystallography, 675
current density, 992
energy
momentum,
998
curvature
and gravity, 914
as
relative
acceleration,
913
matrix, 888
two-form, 885
curve
coordinate, 774
differentiabie, 770
curvilinear
coordinates,
892
cycle, 666
cyclic permutation, 666
cyclic subgronp, 657
d' Alembert, 984
biography, 330
Darboux, 744, 942
Darbonx inequality, 245
Darbonx theorem, 802
de Broglie, 807
Dedekind, 11,
711,739,891
degeneracy, 697
energy, 605
degenerate kernels,
501-504
delta function, 160, 592
and step function, 163
expansion
Fourier,
204
general, 189
limit of seqnence, 161, 163
potential, 601
dense snhset, 460
densityfunction, 833
density of states, 532
derivation,
840
derivation algebra, 840
derivation of an algebra, 42
derivation property
tangent
vector,
772
derivative
covarianl, 897
derivative
operator
nnboundedness of, 455
Descartes, 738
detenrdnant,
93-101, 744
analytic definition of, 137
connection
with
trace,
102
definition, 93
derivative of, 102
expansion of, 95
exponential of trace, 103
of an operator, 101
products
of
matrices,
97
relation
to
trace,
101
diffeomorphism, 768
dUferentiablecurve, 770
tangent
vector,
772
differentiable manifold, 763-769
dimension
of
a, 764
differentiable map, 768
coordinate
expression
of,768
differential
of
a constant map, 777
ofamap,
776
one-form, 786
real-valued maps, 778
differential
equation
analytic, 401
analytic properties, 401
associated Legendre, 343
Bessel,407
confiuent
hypergeometric,
420
constant
coefficients, 376-379
Euler, 411
first-order
existence, 350
linear, 351
uormal form, 351
Peanoexistence
theorem,
351
uniqueness,
350
uniqueness
theorem,
352
first-order
linear
irregular singularpoint, 402
regular singularpoint, 402
Fuchsian, 411
higher-order
numerical
solution,
393
homogeneons, 349
hypergeometric,
407
definition, 413
Jacobi
fUnctions,
418
Kummer's
solutions, 418
inhomogeneous,
349
linear, 349
superposition principle,
354
numerical
solution,
384
Adam's
method, 386
Euler's method, 386
Riemann, 412
second
order
linear
hehavior at infinity, 410
canonical basis,404
characteristic
exponents,
406
second-order
linear
adjoint, 365
INDEX
1011
Frobenius method, 371
generalproperties,
352
integrating factor, 364
regular, 353
uniqueness
theorem,
354
Wronskian, 356
symmetry group of,
941-950
differential form, 791
Maxwell's
equations,
793
differential
forms
Lorentz force law, 794
differential operator, 349, 863
differentiation
operator,
73
diffusion equation, 528, 591, 621
time-dependent,
530
diffusion
operator
Green's
function for, 632
dilation, 238
dilitation, 912
dimension
theorem,
35
Dirac, 850
biography, 167
Diracdelta
function,
160
direct products, 661-{j63
direct sum, 109-111
definition, 110
directional derivative, 976
DUichlet,
177,738,888,891
biography, 614
Dirichlet boundary condition, 591
DUichlet BVP, 591,
613-{j19
in twodimensions, 638
discrete Fouriertransform, 217-219
dispersion
relation
withone
subtraction,
308
dispersion relations, 306-309
distribution, 166
density, 166
derivative of, 169
Fouriertransform
of
a, 219
limit
of
functions, 168
divergence, 987
oftensors,
903
domain,5
dual
basis, 38, 729
of an
operator,
39
1012
INDEX
space, 34
eigenfunction
expansion
technique,
6364/1
2D Laplacian, 638
eigenspace,115
eigenvalue,
114-116
characteristic polynomial, I IS
definition, 114
of angular momentum, 337
simple, 115
eigenvector,
114-116
angular momentum,
338-346
definition, 114
generalized, 468
Einstein, 799, 849, 889, 891, 899, 920,
921,996
Einstein
tensor,
919
Einstein's equation, 918-921
Schwarzschild
solution,
924
Einstein's
summation
convention. 728
electromagnetic field tensor, 796, 797
elementary
column
operation,
99
elementary
row
operation,
99
elliptic PDE, 589, 613-fj21
elsewhere, 838
empty set, 2
endomorphism, 33
energy
function,
805
entirefunction,
233
equivalence
class,3
equivalence
relation,
3
equivalent
representations,
674
essential singularity, 273
essentially idempotent, 689
~
-orthogonal matrices, 837
Euclid,
151,807
Euclidean metric, 892
Euclidean space, 754
Euler, 233, 390, 415, 423,
514,888,984
biography, 981
Euler angles, 89, 865
Eulerkernel, 434
Euler operator, 980
Eulertheorem, 865
Eulertransform, 433
Euler-Lagrange equations, 981
Euler-Mascheroni constant, 310
evaluation function, 977
event, 837
evolution
operator,
58, 626
exactform,797
expectation
value,64
exponential
function
complex, 234
exponential map, 824
exterior a1gehra, 739--749
exteriorcalculus,
791--801
exterior
derivative,
791
exterior
product,
740
F-relatedvector fields, 780
faithful representation, 674
Fermi energy, 532
Feynman diagram, 602
Feymnan propagator, 636
field, 20
finite-difference operators,
70-73
finite-rank operator, 464
first integral, 992
first variation, 983
fiat manifold, 895, 896
fiat map, 747, 802
fiow,784
form factor, 216
formal adjoint, 560
formally self-adjoint, 560
forward
difference
operator,
70
Fourier, 614, 653, 982
biography, 198
Fourier
series,196-208
angnlarvariable, 197
fundamental cell, 198
general variable, 200
group theory, 853
higher
dimensions,
207
main theorem, 202
sawtooth, 201
square
wave,
200
to Fourier transform, 208-210
Fourier transform, 208-220, 433
and derivatives,
216-217
and GF, 628-fj36
and quark model, 216