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



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Íàçâàíèå: Advanced Calculus: A Differential Forms Approach

Àâòîð: Edwards H.

Àííîòàöèÿ:

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.


ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/Àíàëèç/Ó÷åáíèêè ïî ýëåìåíòàðíîìó àíàëèçó/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1969

Êîëè÷åñòâî ñòðàíèö: 508

Äîáàâëåíà â êàòàëîã: 06.04.2005

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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Ïðåäìåòíûé óêàçàòåëü
Abel summation      412 426
Abel's theorem      411—412
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 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 $k=0$ 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)
1 2
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