Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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intersection,
2
partition
of a,4
uncountable,
12
union,
2
universal,
2
sharpmap, 747, 802
shifting
operator,
71
signatureof
9,752
similaritytransformation,91-92
orthonormal
basis,91
simple arc,242
simple character, 685
simple Lie algehra, 844
simultaneousdiagonalizahility, 123
singleton, 2
singularpoint, 233
singularity,233
isolated, 270
classification, 273
rational
function,
274
skew-symmetric, 738
SL(V)
as aLie group, 816
smooth arc, 242
space
Banach, 148
cornplexcoorilinate,21
dual, 34
inner
product,
25
metric,
8
real
coordinate,
21
square-integrahlefunctions,
lSI
spacelikevector,837
spacetime
sphericallysymmetric,923
static, 923
stationary,923
span,22
speciallinear group, 656
special orthogonalgroup, 657, 824
Lie algebraof, 824
specialrelativity,752,837,868,872,986
specialunitary group, 657
Lie algebraof, 824
spectraldecompusition,117-125
orthogunaloperator, 135
real case, 131
spectral
theorem
INDEX
1023
compact
hermitian,
476
compactnormal, 477
spectrum
closure of, 461
spherical Bessel functions,428, 542
expansion of planewave,543
spherical
coordinates
in TIl dimensions, 593-595
sphericalharmonics,338-346
additiontheorem, 345, 867
definition,341
expansion
interms of, 344
expansionuf plane wave, 544, 646
firstfew,343
Spinoza, 738
square-integrablefunctions,
lSI
stabilizer,663, 818
static
spacetime,
923
stationary spacerime,923
steepest descentmethod, 312-315
step function, 163,288
stereographic
projection
n-sphere,769
two-sphere,766
Stirling approximation,316
Stone-Weierstrass
theorem,
152
generalized, 196
stressenergy
tensor,
920
structure constants, 826, 900, 905
structure
equations,888
Sturm, biography,512
Sturm-Liouville
operator,510
problem, 174,622
system,510, 637
completeness, 524
eigensolutions, 511
eigenvalues, 513
large argrunent,521
largeeigenvalues, 518
regular,510
singular,516
subalgebra,44
subgroup, 656-663
conjugate,657
generated
by a
subset,
657
normal, 659
1024
INDEX
trivial, 656
submanifold,767
open,767
subset, 2
bouoded,459
proper,2
sobspace, 22
invariant,
112
surjective
map.
6
Sylvester's
theorem,
753
symmetricalgebra, 737
symmetricbilinear form,749
classification,
751
inner product, 750
nondegenerate.vsn
symmetricgroup, 655,
664-<i69
characters
graphicalconstruction,714
cycle, 666
representation,
707-723
analyticconstruction, 707
antisynnnetric,679
graphicalcnnstruction,710
products,721
Youngoperators,717
symmetricbomomorphism,655
symmetricprodnct, 737
symmetrizer,
736
symmetrygroup, 944
definingeqnations,957
of a DE, 941
of a snbset,937
projectable,944
transformof a function,943
variational,988
symplecticalgebra, 836
symplecticcbarts, 802
symplecticform, 746, 801
rank of, 747
symplectic geometry,
801-808
conservation
of
energy,
805
symplectic group, 657, 749, 835
symplecticmanifold,801
symplecticmap, 746, 802
symplecticmatrix,749
symplecticstructnre,801
symplectictransformation,746
symplectic vector space, 746-749
canonical basis,747
Hamiltoniandynamics, 748
tangent buodle,780
tangent
space,773
tangent
vector,
772
tangents
toacurve
components,
778
target space, 5
Taylorseries,
252-262
construction,
254
tensor, 731
components
of, 732
contravariant, 731
contravariant-symmetric,736
covariant,731
covariant-symmetric, 736
Levi-Civita,745
skew-symmetric,738
symmetric,736
transformation law,733
types of, 731
tensor
algebra,
731
tensor
bundle,
786
tensor field,
785-790
crucial property of, 786, 787
tensor product, 730, 731
of vectorspaces,
41
tensors
symmetric
product,
737
theta function, 288
timelike vector,837
tntal derivative,954
total divergence,987
trace,
101-103
and determinant, 102
definition, 101
log
of
determinant, 103
relation to determinant, 101
transformation group, 655
translation, 819
translation operator,138
transpose
of a
matrix,
87
transposition, 667
traveling
waves,533
triangle inequality, 31
trivialhomomorphism, 655
trivial
representation,
679
trivialsuhgroup,656
twinparadox
as a variatioualprohlem,986
uncertainty
principle,
78
uncountable
set, 12
union,
2
unitarygroup,656
Lie algebraof, 824
unitaryoperator,
63-<i7
unitary
representation,
678
universal
set,2
Vaudermonde, biography, 95
variation
of
constants,
360
variational
derivative,
977
variationalproblem,980
twinparadox,986
variationalsymmetrygroup,988
vectorfield,780
flowof a, 784
integralcurveof, 783
left-invariaut, 820
Lie algebraof, 782
vector space, 19-45
automorphism,
36
basis
components
in a,22
basisof a,22
complete,146
complex,20
definition, 19
dual,34
endomorphism
of
a, 33
finite-dimension
criterion
for,
462
finite-dimensional,
22
isomorphism.
36
linear
operator
ona,33
normed,31
operator
ona, 33
oriented,746
real, 20
symplectic, 746
vectors,
19
normof, 30
INDEX
1025
uormal,26
orthogoual,26
Volterraequatious,488
Volterra,
hiography, 490
volumeelement, 746
vouDyck, 390
vouHmuholdt,177,614,738
von Neumann,
874
hiography,478
waveequatiou,328,532
symmetrygroup,
961-1J63
waveguide,533
cylindrical,537
rectangular,534
wedgeproduct,740
VVeierstrass,
11,30,296,388,588,739,
841
biography, 462
VVeyl,744,
842,942,996
biography, 849
VVeyl
basis, 835, 844
Weyloperator,847
VViguer,
168,943
biography, 874
VViguer
formula,865
VViguer-Eckart
theorem,707
Wigner-Seitzcell, 208
\VKB method,380-383
connection
formulas,
381
VVordsworth,
806
Wronski,biography, 357
VVrouskiau,355-363,511
comparison
theorem,
362
separationtheorem,361
Young,
850
Youngframe,712, 718
negativeapplication, 715
positive
application,
715
regularapplication, 714
Young
operator,856
Young
pattern,712
Youngtableanx,713, 718, 857
Yukawapotential,214