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Edwards H. Ч Advanced Calculus: A Differential Forms Approach
Edwards H. Ч Advanced Calculus: A Differential Forms Approach

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Ќазвание: Advanced Calculus: A Differential Forms Approach

јвтор: Edwards H.


An outstanding textbook, complete with examples, exercises, and solutions, for an advanced calculus course in which differential forms can be used to introduce the subject. Enriching reading for its modern viewpoint and techniques. The diverse set of topics from which advanced calculus courses are created are presented here in beautiful unifying generalization.

язык: en

–убрика: ћатематика/јнализ/”чебники по элементарному анализу/

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

ed2k: ed2k stats

√од издани€: 1969

 оличество страниц: 508

ƒобавлена в каталог: 06.04.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
Abel summation      412 426
Abel's theorem      411Ч412
Absolute convergence      409 439n
Addition formulas for $e^x$      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 86Ч87
Affine maps decomposed as a decomposition of simple maps      10Ч11 15 104Ч105 199 200
Algebra of forms, the      90n
Algebraic functions      381Ч383
Almost everywhere convergence      435
Alternating Series Test      377 (Ex. 9)
Analytic functions      299
Approximating sums      25Ч26 29 197Ч198 208Ч213 400 426
Archimedean laws      366 373 377
arcsine function      140 (Ex. 8) 386
arctan      146Ч147
Arithmetic mean      184 (Ex. 6)
Arithmetic modulo n      376 (Ex. 3Ч5)
Arzela's theorem      447 (Ex. 11)
Atlas      203
Banach spaces      448
Basis of a vector space      115Ч117
Betti numbers      327 (Ex. 10)
Binomial coefficients      90n 304 424Ч425
Binomial series      304Ч306
Bolzano Ч 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 65Ч66 72
Canonical form for linear maps      119
Cauchy convergence criterion      456Ч457
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 Ч 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      101Ч102
Chain rule in proof of Stokes' Theorem      62 69Ч70 74
Chain rule in terms of Jacobians      144
Chain rule, principal statement      143
Chain rule, proof      153Ч156
Chain rule, proof for affine maps      87Ч90
Chain rule, reviewed      190
Chain rule, statement for affine maps      87
Closed forms      63 (Ex. 8) 71Ч72 274 320Ч326
Compactness, definition      392
Compactness, related theorems      393Ч398
Completeness of Banach spaces      448
Completeness of Lebesgue integrable functions      442
Completeness of real number system      373Ч375
complex numbers      289Ч291
Computational rules, governing forms and pullbacks      11Ч13 20 42Ч43 86 88Ч89 142Ч143
Computational rules, governing the computation of derived forms      60 68 73 220
Conditional convergence      410n
Conductivity      334
Conformal coordinates      283Ч286
Confusion      26
Conjugate of a complex number      309
Conservative force fields      64 (Ex. 9)
Constraints, examples      164Ч168
Constraints, non-singular      168
Constructive mathematics      463Ч466
Content      197n
Continued fractions      377Ч379 (Ex. 12Ч15)
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. 5Ч7)
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      366Ч367 380
Determinants, definition      101
Determinants, method of evaluation      103Ч104 (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      270Ч276
Differential equations, fundamental existence theorem      245Ч251 320
Differential equations, generalized existence theorem      315
Differential equations, stated in terms of differentials      272 313Ч314
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      407Ч409
Electrical forces      (see Coulomb's Law)
electromagnetic field      343
Elementary functions      384
Elimination of variables in linear equations      76Ч79
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      226Ч232
Elimination Theorem statement, reviewed      191
Envelopes      194 (Ex. 7)
Equality of mixed partials      63 (Ex. 6) 74Ч75 218
Euclidean algorithm      376 (Ex. 6)
Euler      402Ч404
Evaluation of      2-forms 12Ч13
exact differential equations      274
Exact forms      63 (Ex. 8) 71Ч72 273 320Ч326
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      421Ч425 (Ex. 10 11)
Faraday's Law of Induction      71 (Ex. 5) 342
Flows, constant planar flows      2Ч4
Flows, constant spatial flows      5Ч7
Flows, general discussion      328Ч333
Flows, planar flow from unit source      23 (Ex. 2)
Flows, relation between direction of flow and corresponding (nЧ1) 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 454Ч455
Fredholm alternative      126 (Ex. 12)
Functions      (see Algebraic functions Elementary Maps Transcendental Polynomials)
Fundamental theorem of algebra      306Ч307 312
Fundamental Theorem of Calculus, as the case $k=0$ of Stokes' Theorem      73
Fundamental Theorem of Calculus, in vector notation      268
Fundamental Theorem of Calculus, proof      53Ч55
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      52Ч53
Gamma function      423 (Ex. 10(n))
Gauss and the lattice point problem      35 (Ex. 2)
Gauss Ч Seidel iteration      240 (Ex. 2 3) 241 242
Gauss' theorem      (see Divergence Theorem)
Geometric mean      184 (Ex. 6)
Goldbach conjecture      464Ч465
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 Ч Borel theorem      205 393
Holder inequality      174Ч175
Homology, homology basis      323Ч324
Homology, homology theory      320
Homology, simple cases      75 (Ex. 5Ч6)
Hyperbolic functions      253 (Ex. 3)
Image of a map      81
Implicit differentiation, description of the method      144Ч145
Implicit differentiation, examples      145Ч147 150Ч151
Implicit differentiation, proof      156Ч157
Implicit Function Theorem, examples      134Ч139
Implicit Function Theorem, for affine maps      105Ч108
Implicit Function Theorem, for analytic functions      307Ч308
Implicit Function Theorem, for differentiable maps      133Ч134
Implicit Function Theorem, for maps between Banach spaces      451Ч452
Implicit Function Theorem, proof      232Ч234
Implicit Function Theorem, proof reduced to Elimination Theorem      157Ч159
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      208Ч213
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      224Ч225
Integrals, basic properties      49Ч51
Integrals, defined as a limit of sums      30Ч31
Integrals, definition reviewed      400
Integrals, difficulty of defining surface integrals      44Ч48
Integrals, double integrals as iterated integrals      51
Integrals, general properties      219Ч223
Integrals, intuitive description of meaning      24Ч27
Integrals, main theorem      204Ч213
Integrals, of complex†1-forms      294Ч295
Integrating factors      275
Integration by parts      222
Intermediate Value Theorem      159n
Inverse function theorem      454 (Ex. 14 15)
Inverse matrix, formula for      111Ч112
Isoperimetric inequality      188Ч190 (Ex. 19)
Iteration      240 (Ex. 1)
Jacobians      100
Lagrange multipliers      161
Lagrange multipliers, examples      166Ч169 170Ч190
Lagrange multipliers, restatement using differentiable manifolds      192Ч193
Lagrange multipliers, statement of method      169
Laplace's equation      278 288
Laplacian      334
Lebesgue dominated convergence theorem      437Ч438
Lebesgue integration, completeness of space of Lebesgue integrable functions      442Ч446
Lebesgue integration, examples and motivation      426Ч431
Lebesgue integration, main theorem      436
Lebesgue integration, passing to a limit under the integral sign      437Ч441
Lebesgue integration, proof      431Ч435
Leibniz notation      458Ч460
Leibniz's formula      419 (Ex. 1)
Level surfaces      80n
Lexicographic order for k-forms      90Ч91
Light as an electromagnetic phenomenon      348
Line integrals      265Ч266
Linear maps      117Ч122 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      117Ч122 123
Maps, origin of term      8
Mass and energy      351Ч354
Matrices, exterior powers      97Ч100
Matrices, formula for inverse      111Ч112
Matrices, minors of      101
Matrices, of coefficients of an affine map      94
Matrices, products      95Ч97
Matrices, transposes      98
Maxima and minima      160Ч183
Maxwell's equations of electrodynamics      340Ч348
Mesh size (of a subdivision)      30
Microscope, describing meaning of differentiability      151Ч153
Microscope, in proof of independence of parameter      211Ч212
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      357Ч359
Natural numbers, decimal notation      358
Newton on Уaction at a distanceФ      340n
Newton's Law of Gravity      23 (Ex. 1) 335
Newton's method, error estimates      243Ч244
Newton's method, in computation of nih roots      262Ч263
Newton's method, statement      242Ч243
Newton's theorems on the gravitational field of a spherical shell      336Ч337
Norms on vector spaces      447Ч448
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      221Ч222
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      219Ч220
Orientations of the boundary of an oriented solid      66Ч67
Oriented area      6 20
Oriented area, formula for      14 (Ex. 5)
1 2
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