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Cambridge University Press 978-1-108-47122-0 — Mathematics for Physicists Alexander Altland , Jan Von Delft Index More Information Cambridge University Press 978-1-108-47122-0 — Mathematics for Physicists Alexander Altland , Jan von Delft Index More Information Index δ-function basis Gaussian representation, 273 definition, 30 qualitative discussion, 272 local, 418 ǫ–δ criterion, 208 orientation, 96 ∃, there exists, 7 orthonormal, 43, 50 ∀, for all,7 bilinear form, 156 ´ , symbol, 217 bin, 218 n-tuple, 34 bit, 8 Abel, Nils Henrik, 8 calculus, 205 absolute value, 17 canonical, 32 action principle, 334 Cantor, Georg Ferdinand Ludwig Philipp, 13 affine space Cartesian product, 13 definition, 26 Cauchy theorem, 344 Euclidean, 26 Cauchy, Augustin-Louis, 339 point, 25 Cauchy–Schwarz inequality, 40 chaos algebra definition, 325 definition, 71 fractal dimension, 325 Grassmann, 161 Lyapunov exponent, 325 alternating form strange attractor, 325 exterior product, 161 characteristic polynomial, 101 maximal degree, 158 closure of degree p, 157 group operation, 7 one-form, 150 interval, 17 pullback, 163 set, 17 top-form, 158 co- and contravariant transformation, 152 volume form, 165 commutator, 183 wedge product, 161 completeness Ampère’s law, 457 relation, 134 analysis, 205 vectors, 30 angle complex conjugate, 15 azimuthal, 423 complex function between vectors, 40 analytic, 338 polar, 423 branch cut, 356 angular momentum, 152 branch point, 356 ansatz, 42 contour integral, 342 anti-derivative, 220 definition, 338 antisymmetric symbol, 12 extended singularity, 346 area element holomorphic, 338 definition, 245 meromorphic, 347 polar coordinates, 246 pole, 347 asymptotic expansion, 267 singularity, 340 augmented matrix, 185 complex numbers 693 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-47122-0 — Mathematics for Physicists Alexander Altland , Jan von Delft Index More Information 694 Index argument, 16 definition definition, 15 trace, 84 imaginary part, 14 Descartes, René, 14 imaginary unit, 14 determinant modulus, 16 definition, 86 real part, 15 invariance under basis change, 93 complex plane, 16 Laplace rule, 88 computer algebra, 226 Leibniz rule, 86 conservation law, 334 multilinearity, 92 continuity equation of metric tensor, 95 differential form, 520 of product of matrices, 93 integral form, 520 difference quotient, 209 relativistically invariant, 529 differentiability contour complex, 265 line, 433 of real function, 209 plot, 433 differentiable manifold surface, 439 atlas, 466 contraction (of indices), 32 chart, 466 convergence-generating factor, 281 cotangent bundle, 478 coordinate definition, 464 basis, 417 flat, 505 coordinates flow, 475 Cartesian, 414 frame, 475 curvilinear, 416, 425 Hodge star, 506 cylindrical, 422 holonomic frame, 475 line, 417 orthonormal basis, 504 polar, 414 pseudo-Riemannian, 504 spherical, 423 Riemannian, 504 stereographic, 491 submanifold, 467 cosine series, 285 transition function, 466 cotangent space, 478 volume of, 506 Coulomb field, 453 differential equation covariant nth order, 301, 321 index, 24 autonomous, 323 notation, 24 boundary condition, 300 cuboid, 241 Cauchy–Riemann, 339 curl (rot) fixed point, 323 Cartesian coordinates, 454 general solution, 300 coordinate invariant, 510 homogeneous first order, 306 in cylindrical coordinates, 458 initial condition, 300, 303 in general coordinates, 457 linear, 306 local basis, 460 logistic, 304 current density ordinary, 301 electrodynamics, 521 partial, 301 magnetostatics, 458 particular solution, 300 curvature quadrature, 303 Gaussian, 505 separation of variables, 302 of surface, 504 system of, 322 curve system of first-order linear equations, 308 acceleration, 408 system of linear, 327 definition, 406 variation of constants, 307 length, 410 differential form, 488 velocity, 407 p-form, 487 cyclic ordering, 58 area form, 488, 492, 494, 506 closed, 494 dagger (†), 69 coordinate representation, 479 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-47122-0 — Mathematics for Physicists Alexander Altland , Jan von Delft Index More Information 695 Index current three-form, 490, 511 electric current two-form, 520 dipole, 270 density, 489 field, 522 exact, 494 elliptical integrals, 374 one-form, 478 equivalence class, 5 pullback, 484 equivalence principle, 512 pushforward, 482 Euclid, 38 top-form, 490 Euclidean twisted, 489 geometry, 40 volume form, 491, 506 space, 47 wedge product, 488 Euler formula, 263 zero-form, 478 Euler, Leonhard, 263 differential geometry, 463 Euler–Lagrange equation, 334 differential quotient, 209 exterior calculus, 493 differentiation exterior derivative chain rule, 213 coordinate representation, 494 graded product rule, 163 definition, 493 higher order, 211 graded Leibniz rule, 493 of inverse function, 213 product rule (Leibniz rule), 212 factorial, 10, 261 vector-valued function, 408 Fermat’s principle, 395, 396 Dirac identity, 352 field Dirac notation, 123 definition, 12 Dirac, Paul Adrien Maurice, 274 Galois, 12 distributions, 274 field (physics) divergence (div) base manifold, 430 coordinate invariant, 509 definition, 430 definition, 451 potential, 442 in Cartesian coordinates, 446 scalar, 430 in general coordinates, 452 target manifold, 430 in spherical coordinates, 438 total differential, 433 local basis, 460 field lines (of vector field), 393 dot product, 39 field-strength tensor, 530 dual force basis, 150 differential form, 480 vector, 150 Lorentz, 523 dual space, 150 Fourier dynamical system momenta, 279 definition, 324 reciprocity, 283 fixed point, 324 Fourier modes flow, 324 completeness, 279 phase space, 324 definition, 278 trajectory, 324 orthonormality relation, 279 Fourier series eigenbasis, 98 basis functions, 132, 279 eigenspace, 105 coefficients, 278 eigenvalue definition, 278 definition, 98 Dirichlet conditions, 282 degenerate, 104 Fourier transform degree of degeneracy, 104 completeness relation, 285 spectrum, 103 convolution theorem, 289 eigenvector, 98 definition, 286 Einstein inverse, 286 equations, 514 multidimensional, 287 summation convention, 32 pair, 286 Einstein, Albert, 512 Parseval’s theorem, 290 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-47122-0 — Mathematics for Physicists Alexander Altland , Jan von Delft Index More Information 696 Index Plancherel theorem, 290 gravitational time-dependent functions, 288 constant, 515 Fourier, Jean Baptiste, 271 redshift, 513 frequency comb, 296 Green function friction, 311 definition, 315 Fubini’s theorem, 240 of harmonic oscillator, 317 function Green, George, 315 continuity, 209 group convolution, 289 Z2, 8 derivative, 209 abelian, 8 differential, 478 associativity, 7 error, 227 axioms, 7 extremum, 235 composition rule, 7 Gaussian, 227 definition, 7 Heaviside step, 276 general linear, 75 hyperbolic, 214 homomorphism, 9 inverse, 213 identity element, 7 local approximation, 448 inverse element, 7 periodic, 284 isomorphism, 10 saddle point, 236 law, 7 smooth, 212 Lorentz, 113 special, 227 neutral element, 7 support, 273 non-abelian, 8 vector-valued, 406 null element, 7 functional orthogonal, 110 action, 331 orthogonal matrix, 112 analysis, 129 pseudo-orthogonal, 113 definition, 330 pseudo-unitary, 113 differentiable, 332 special linear, 95 differential, 332 special orthogonal, 115 extremal function, 332 special unitary, 115 length, 331 symmetric, 10 local, 331 unitary, 110 non-local, 332 unitary matrix, 112 fundamental theorem of algebra, 103 fundemantal theorem of calculus, 219 Hamiltonian, 335 harmonic functions, 278 Galilei transformation, 114 harmonic oscillator Gauss’s critically damped, 312 law, 450 definition, 311 theorem, 450 equation of motion, 313 Gauss, Johann Carl Friedrich, 449 Green function, 317, 352 Gaussian elimination, 78 overdamped, 312 generalized polar coordinates, 552 underdamped, 312 geodesic, 337 Heisenberg uncertainty relation, 127 gradient (grad) Hermitian matrix ∇ (nabla)-operator, 437 diagonalizability, 119 Cartesian coordinates, 437 Hilbert space coordinate invariant, 508 definition, 123, 130 definition, 436 operator, 124 field, 442 Hilbert, David, 131 in spherical coordinates, 439 Hodge star local basis, 460 coordinate-invariant formulation, 168 operator, 437 coordinate representation, 168 Gram–Schmidt orthonormalization, 45 definition, 166 Grassmann identity, 60 hyperbolic coordinates, 552 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-47122-0 — Mathematics for Physicists Alexander Altland , Jan von Delft Index More Information 697 Index imaginary numbers14 definition, 453 index eigenfunctions, spherical, 141 contraction, 32, 155 in general coordinate basis, 453 lowering, 48, 153 in spherical coordinates, 453 raising, 49, 153 local basis, 461 inertia tensor, 156 on unit sphere, 141 inertial coordinate systems, 113 Laplace, Pierre Simon, 453 infinitesimal quantity, 211 Laurent series, 347 inner derivative, 162 law of levers, 57 inner product Lebesque integrable, 221 alternating forms, 167 Legendre complex, 53 functions, 142 definition, 47 polynomials, 138 degenerate, 52 Legendre, Adrien Marie, 138 indefinite, 52 Leibniz, Gottfried Wilhelm, 86 semidefinite, 52 Levi–Civita symbol, 11, 59, 183 integral limit, 208 definite, 220 line, 34 definition, 219 line integral equation, 306 complex, 342 Gaussian, 227 definition, 412 indefinite, 220 linear measure, 245, 251 hull, 28 principal value, 222 span, 28 rational functions, 226 linear independence, 29 tables, 226 linear map two-dimensional, 239 associativity, 70 integral theorems of vector analysis, 459 Hermitian, 117 integration kernel, 73 by parts,
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