|
|
 |
| Àâòîðèçàöèÿ |
|
|
|
| Ïîèñê ïî óêàçàòåëÿì |
|
|
|
|
|
|
|
 |
|
|
|
| Edwards H. — Advanced Calculus: A Differential Forms Approach |
|
|
|
| Ïðåäìåòíûé óêàçàòåëü |
Abel summation 412 426
Abel's theorem 411—412
Absolute convergence 409 439n
Addition formulas for  253 (Ex. 2(b))
Addition formulas for general exponentials 256 (Ex. 10)
Addition formulas for trigonometric functions 252 (Ex. 1(e))
Affine manifolds 129
Affine maps 8 86—87
Affine maps decomposed as a decomposition of simple maps 10—11 15 104—105 199 200
Algebra of forms, the 90n
Algebraic functions 381—383
Almost everywhere convergence 435
Alternating Series Test 377 (Ex. 9)
Analytic functions 299
Approximating sums 25—26 29 197—198 208—213 400 426
Archimedean laws 366 373 377
arcsine function 140 (Ex. 8) 386
arctan 146—147
Arithmetic mean 184 (Ex. 6)
Arithmetic modulo n 376 (Ex. 3—5)
Arzela's theorem 447 (Ex. 11)
Atlas 203
Banach spaces 448
Basis of a vector space 115—117
Betti numbers 327 (Ex. 10)
Binomial coefficients 90n 304 424—425
Binomial series 304—306
Bolzano — Weierstrass theorem 393
Boundary of a surface-with-boundary 215 223
Boundedness of domains 31
Boundedness of functions 31
Cancellation on interior boundaries as the underlying idea of the Fundamental Theorem 59 65—66 72
Canonical form for linear maps 119
Cauchy convergence criterion 456—457
Cauchy Convergence Criterion for complex numbers 292 310
Cauchy Convergence Criterion in definition of fc-dimensional volume 197
Cauchy Convergence Criterion in definition of integrals 30
Cauchy Convergence Criterion in definition of real numbers 368
Cauchy Convergence Criterion in successive approximations 230
Cauchy — Riemann equations 283 296
Cauchy's integral formula 298
Cauchy's polygon 254 (Ex. 5)
Cauchy's theorem 296
Cesaro summation 426 (Ex. 14)
Chain rule 143
Chain rule for complex functions 311 (Ex. 10)
Chain rule in Jacobian notation 102
Chain rule in matrix notation 101—102
Chain rule in proof of Stokes' Theorem 62 69—70 74
Chain rule in terms of Jacobians 144
Chain rule, principal statement 143
Chain rule, proof 153—156
Chain rule, proof for affine maps 87—90
Chain rule, reviewed 190
Chain rule, statement for affine maps 87
Closed forms 63 (Ex. 8) 71—72 274 320—326
Compactness, definition 392
Compactness, related theorems 393—398
Completeness of Banach spaces 448
Completeness of Lebesgue integrable functions 442
Completeness of real number system 373—375
complex numbers 289—291
Computational rules, governing forms and pullbacks 11—13 20 42—43 86 88—89 142—143
Computational rules, governing the computation of derived forms 60 68 73 220
Conditional convergence 410n
Conductivity 334
Conformal coordinates 283—286
Confusion 26
Conjugate of a complex number 309
Conservative force fields 64 (Ex. 9)
Constraints, examples 164—168
Constraints, non-singular 168
Constructive mathematics 463—466
Content 197n
Continued fractions 377—379 (Ex. 12—15)
Continuity equation 332
Continuity of a force field 24
Continuity of a function of two variables 31
Continuity of complex functions 293
Continuity of k-forms 143
Continuity of maps between Banach spaces 451
Continuity, main definition 390
Convergence (see Cauchy Convergence Criterion)
Coulomb's law 335
Cramer's rule 109
Critical points 170
Cross products of vectors 266 (Ex. 5—7)
Curl 267
CURVES (see also Manifolds)
Curves, defined by equations 39
Curves, defined by parameters 39
d'Alembertian 347
Decimal approximation 27 (Ex. 2) 380
Decimal fractions 366—367 380
Determinants, definition 101
Determinants, method of evaluation 103—104 (Ex. 2)
Determinants, reason for name 101n
dielectric constant 241
Differentiability of complex functions 293
Differentiability of k-forms 73
Differentiability of maps 143
Differentiability of maps between Banach spaces 451
Differentiability, main definition 390
Differentiability, reviewed 190
Differentiable manifolds 192
Differential equations, describing families of curves 277 (Ex. 2)
Differential equations, elementary techniques of solution 270—276
Differential equations, fundamental existence theorem 245—251 320
Differential equations, generalized existence theorem 315
Differential equations, stated in terms of differentials 272 313—314
Differential forms 73 (see also Forms)
Differentiation under the integral sign 392 (Ex. 16) 399
Directional derivatives 149 (Ex. 7)
Dirichlet problem 288
Div (divergence) 268
Divergence of a flow 61
Divergence theorem 73 268
Dot product 172 269
Double series 407—409
Electrical forces (see Coulomb's Law)
electromagnetic field 343
Elementary functions 384
Elimination of variables in linear equations 76—79
Elimination Theorem statement 158
Elimination Theorem statement for rational numbers 365
Elimination Theorem statement in proof of Implicit Function Theorem 158
Elimination Theorem statement, proof 226—232
Elimination Theorem statement, reviewed 191
Envelopes 194 (Ex. 7)
Equality of mixed partials 63 (Ex. 6) 74—75 218
Euclidean algorithm 376 (Ex. 6)
Euler 402—404
Evaluation of 2-forms 12—13
exact differential equations 274
Exact forms 63 (Ex. 8) 71—72 273 320—326
Exact sequences of affine maps 131 (Ex. 4)
Exponential function 253 (Ex. 2) 311
Exponentials of matrices 253 (Ex. 4) 254
Exterior derivatives 73
Exterior powers of matrices (see Matrices)
Factorial function 421—425 (Ex. 10 11)
Faraday's Law of Induction 71 (Ex. 5) 342
Flows, constant planar flows 2—4
Flows, constant spatial flows 5—7
Flows, general discussion 328—333
Flows, planar flow from unit source 23 (Ex. 2)
Flows, relation between direction of flow and corresponding (n—1) form 21 (Ex. 5 6)
Flows, spatial flow from unit source 24 (Ex. 3) 288
Folium of Descartes 141 (Ex. 9) 193
force (see Work)
| Forms, basic definitions 88
Forms, non-constant forms 142
Forms, on Banach spaces 452 454—455
Fredholm alternative 126 (Ex. 12)
Functions (see Algebraic functions Elementary Maps Transcendental Polynomials)
Fundamental theorem of algebra 306—307 312
Fundamental Theorem of Calculus, as the case  of Stokes' Theorem 73
Fundamental Theorem of Calculus, in vector notation 268
Fundamental Theorem of Calculus, proof 53—55
Fundamental Theorem of Calculus, related to independence of parameter 58 (Ex. 13)
Fundamental Theorem of Calculus, related to Poincare's Lemma 327 (Ex. 9)
Fundamental Theorem of Calculus, reviewed 391 (Ex. 4)
Fundamental Theorem of Calculus, statement 52
Fundamental Theorem of Calculus, uses 52—53
Gamma function 423 (Ex. 10(n))
Gauss and the lattice point problem 35 (Ex. 2)
Gauss — Seidel iteration 240 (Ex. 2 3) 241 242
Gauss' theorem (see Divergence Theorem)
Geometric mean 184 (Ex. 6)
Goldbach conjecture 464—465
Golden section 379n
Grad (gradient) 267
Gravity (see Newton's Law of Gravity)
Green's theorem in the plane 73
Harmonic functions, analyticity of 308
Harmonic functions, as solutions of Laplace's equation 278
Harmonic functions, definition 278
heat capacity 334
Heat equation 333
Heine — Borel theorem 205 393
Holder inequality 174—175
Homology, homology basis 323—324
Homology, homology theory 320
Homology, simple cases 75 (Ex. 5—6)
Hyperbolic functions 253 (Ex. 3)
Image of a map 81
Implicit differentiation, description of the method 144—145
Implicit differentiation, examples 145—147 150—151
Implicit differentiation, proof 156—157
Implicit Function Theorem, examples 134—139
Implicit Function Theorem, for affine maps 105—108
Implicit Function Theorem, for analytic functions 307—308
Implicit Function Theorem, for differentiable maps 133—134
Implicit Function Theorem, for maps between Banach spaces 451—452
Implicit Function Theorem, proof 232—234
Implicit Function Theorem, proof reduced to Elimination Theorem 157—159
Implicit Function Theorem, reviewed 191
Improper integrals 401
Independence of parameter, for surface integrals 47
Independence of parameter, general case 222
Independence of parameter, proof 208—213
Independence of parameter, statement 207
Integers 366
Integrability conditions 315
Integrals (see also Improper integrals Lebesque
Integrals, as area under a curve 56 (Ex. 4) 65
Integrals, as functions of S 224—225
Integrals, basic properties 49—51
Integrals, defined as a limit of sums 30—31
Integrals, definition reviewed 400
Integrals, difficulty of defining surface integrals 44—48
Integrals, double integrals as iterated integrals 51
Integrals, general properties 219—223
Integrals, intuitive description of meaning 24—27
Integrals, main theorem 204—213
Integrals, of complex 1-forms 294—295
Integrating factors 275
Integration by parts 222
Intermediate Value Theorem 159n
Inverse function theorem 454 (Ex. 14 15)
Inverse matrix, formula for 111—112
Isoperimetric inequality 188—190 (Ex. 19)
Iteration 240 (Ex. 1)
Jacobians 100
Lagrange multipliers 161
Lagrange multipliers, examples 166—169 170—190
Lagrange multipliers, restatement using differentiable manifolds 192—193
Lagrange multipliers, statement of method 169
Laplace's equation 278 288
Laplacian 334
Lebesgue dominated convergence theorem 437—438
Lebesgue integration, completeness of space of Lebesgue integrable functions 442—446
Lebesgue integration, examples and motivation 426—431
Lebesgue integration, main theorem 436
Lebesgue integration, passing to a limit under the integral sign 437—441
Lebesgue integration, proof 431—435
Leibniz notation 458—460
Leibniz's formula 419 (Ex. 1)
Level surfaces 80n
Lexicographic order for k-forms 90—91
Light as an electromagnetic phenomenon 348
Line integrals 265—266
Linear maps 117—122 123
Lines of force 71 (Ex. 6)
Liouville's theorem 311 (Ex. 9)
Logarithm function 386 (Ex. 2)
Lorentz transformations 350
Magnetic forces 341
Magnetic permeability 345
Manifolds, affine 129
Manifolds, compact, oriented, differentiable manifolds-with-boundary 219
Manifolds, compact, oriented, differentiable surfaces 203
Manifolds, compact, oriented, differentiable surfaces-with-boundary 214
Manifolds, differentiable 192
Manifolds, solving differential equations 314
Manifolds, usefulness of term 129n
Mappings (see Maps)
Maps, affine maps 8
Maps, differentiable maps 143
Maps, linear maps 117—122 123
Maps, origin of term 8
Mass and energy 351—354
Matrices, exterior powers 97—100
Matrices, formula for inverse 111—112
Matrices, minors of 101
Matrices, of coefficients of an affine map 94
Matrices, products 95—97
Matrices, transposes 98
Maxima and minima 160—183
Maxwell's equations of electrodynamics 340—348
Mesh size (of a subdivision) 30
Microscope, describing meaning of differentiability 151—153
Microscope, in proof of independence of parameter 211—212
Microscope, in rate of convergence of successive approximations 234 (Ex. 2)
Microscope, related to integrals 222
Minkowski's inequality 185 (Ex. 9) 448
MODULO (see Arithmetic modulo n)
Modulus of a complex number 291
Natural numbers 357—359
Natural numbers, decimal notation 358
Newton on “action at a distance” 340n
Newton's Law of Gravity 23 (Ex. 1) 335
Newton's method, error estimates 243—244
Newton's method, in computation of nih roots 262—263
Newton's method, statement 242—243
Newton's theorems on the gravitational field of a spherical shell 336—337
Norms on vector spaces 447—448
One-to-one 81n
Onto as an adjective 79n
Orientations of a surface by a non-zero 2-form 45
Orientations of integrals 29 29n
Orientations of integrals over manifolds 221—222
Orientations of n-space 131 (Ex. 3)
Orientations of planar flows 3
Orientations of space (right- or left-handed) 17
Orientations of spatial flows 5
Orientations of the boundary of an oriented manifold-with-boundary 219—220
Orientations of the boundary of an oriented solid 66—67
Oriented area 6 20
Oriented area, formula for 14 (Ex. 5)
|
|
|
| Ðåêëàìà |
|
|
|
|
|
Î ïðîåêòå