Edwards H. — Advanced Calculus: A Differential Forms Approach :: Электронная библиотека попечительского совета мехмата МГУ

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Edwards H. — Advanced Calculus: A Differential Forms Approach

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Название: Advanced Calculus: A Differential Forms Approach

Автор: Edwards H.

Аннотация:

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:

Рубрика: Математика/Анализ/Учебники по элементарному анализу/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1969

Количество страниц: 508

Добавлена в каталог: 06.04.2005

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Предметный указатель

Oriented length      19 20
Oriented volume      16—18 20
orthogonal matrices      193—194 (Ex. 5—6)
Orthogonal trajectories      277 (Ex. 3)
Parabola of safety      195 (Ex. 7)
Parameterization      44—48
Parameterization of the circle      257—258
Parametric description of domains      38—43
Partial derivatives      41
Partial derivatives, definition      142
Partition of unity, continuous      206
Partition of unity, differentiable      213 (Ex. 1) 217 220
Pi, computation of      256—260
Pi, definition      34 (Ex. 1)
Pi, represented as a simple integral      65 (Ex. 14)
Picard's iteration      246
Poincare's lemma      325
Poisson's equation      339 341 348
Poisson's integral formula      287 308—310
Polar coordinates      47 64—65
Polar coordinates, differentiation of      146—147
Polynomials      381
Potential theory      335—340
Power series, operations with      420—421 (Ex. 6—9)
Product formula for sine      402 415—417
Product of forms      89
Products (infinite)      401
Products of matrices      (see Matrices)
Ptolemy's trigonometric tables      379—380 (Ex. 16)
Pullbacks of      3-forms 15—16
Pullbacks of constant forms under affine maps      20 86
Pullbacks of non-constant forms under non-affine maps      40—43 142—143
Pullbacks of non-constant forms under non-affine maps, related to Leibniz notation      460
Pullbacks of non-constant forms under non-affine maps, reviewed      190
Pullbacks, computation of      12—13
Pullbacks, geometrical significance      8—12 18
Pullbacks, origin of term      11
Quadratic forms, elliptic, hyperbolic, parabolic      177—178
Quadratic forms, origin of term      175n
Quadratic forms, positive definite      183 186—187
Rank of a differentiable map (at a point not a singularity)      139
Rank of a linear map      119 124
Rank of a system of linear equations      81
Rank of an affine map      128
Rank of the exterior powers of a matrix      130 (Ex. 2)
Rank, geometrical significance      82—83
Rank, relation to pullbacks      84
Rank, reviewed      191
Ratio Test      401
Rational numbers      359—367
Real numbers      367—375
Rearrangement of series      405 (Ex. 6—9) 409
Refinement (of a subdivision)      32
Relativity      (see Special relativity)
Relaxations      238
Riemann integration      426—427
Riemann's criterion for the convergence of an integral      37 (Ex. 7)
Schwarz inequality      172
Second Derivative Test      187 (Ex. 18)
Separation of variables      273
Series (infinite)      400
Sets      461—462 467
Sexagesimal fractions      379—380 (Ex. 16 18)
Shears      9

Simple iteration      240 (Ex. 1)
Sine, product formula for      402 415—417
Singularities of a differentiable map, definition      134 141
Singularities of a differentiable map, examples      134—139
Singularities of a differentiable map, reviewed      192
Snell's Law of Refraction      183 (Ex. 2)
Solution manifolds of a differential equation      314
Space as a term not implying three dimensions      80n
Special relativity      351—354
Spherical coordinates      45
Spherical coordinates, differentiation of      150 (Ex. 10)
Square roots      244 (Ex. 1)
Stationary points      170n
Stereographic projection      43—44 (Ex. 3) 47
Stokes' Theorem, in vector notation      266—268
Stokes' Theorem, proved when domain is a rectangle      60 67—68
Stokes' Theorem, statement      72—73
Stokes' Theorem, statement and proof      217—218 220—221
Subscript notation      77n
Successive approximations      226
Successive approximations of the inverse of a matrix      236
Successive approximations of the solution of an ordinary differential equation      246
Successive approximations of the solution of linear equations      236
Surface integrals      266
Surfaces      (see also Manifolds)
Surfaces, defined by equations      41
Surfaces, defined by parameters      41
Tangent line to a curve in the plane      148 (Ex. 4)
Tangent plane to a surface in space      149 (Ex. 8)
Taylor series      234 (Ex. 1) 300
Tensors      23n
topology      321 324
Torus, as a compact, oriented surface      214 (Ex. 3)
Torus, parameterization of      49 (Ex. 7)
Torus, volume of      70 (Ex. 21(c))
Transcendental functions      383—384
Transpose of a matrix      98
Triangle inequality, for complex numbers      310 (Ex. 1)
Triangle inequality, for rational numbers      365
Triangle inequality, in Banach spaces      448
Trigonometric functions      251 (Ex. 1) 311
Uniform continuity      388
Uniform convergence      449
Uniform differentiability      389—390
Vector fields      265
Vector potential      347
Vector products      (see Cross products)
Vector spaces, bases      115—117
Vector spaces, dimension      115—117
Vector spaces, examples      113—114
Vector spaces, formal definition      122—123
Vector spaces, glossary of terms      123—124
Vector spaces, informal definition      113
Vector spaces, linear maps      117—122
Vector spaces, origin of term      113n
Vector spaces, subspaces      114—115
Velocity vectors      147
Volume of an n-dimensional ball      425 (Ex. 12)
Volume, k-dimensional      197
Wallis' formula      405 (Ex. 5)
Wave equation      348
Work as the differential of potential      61 333—340
Work in a central force field      23 (Ex. 1)
Work in a constant force field      1—2
“Uncertainty” of an approximating sum      32—34

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