Munkres J.R. — Analysis on manifolds :: Электронная библиотека попечительского совета мехмата МГУ

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Munkres J.R. — Analysis on manifolds

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Название: Analysis on manifolds

Автор: Munkres J.R.

Аннотация:

This book is intended as a text for a course in analysis, at the senior or first-year graduate.
A year-long course in real analysis is an essential part of the preparation of any potential mathematician. For the first half of such a course, there is substantial agreement as to what the syllabus should be. Standard topics include: sequence and series, the topology of metric spaces, and the derivative and the Riemannian integral for functions of a single variable. There are a number of excellent texts for such a course, including books by Apostol [A], Rudin [Ru], Goldberg [Go], and Roy den [Ro], among others.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 366

Добавлена в каталог: 14.02.2006

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Предметный указатель

$B^{n}(a)$       see “n-ball”
$C^{1}$       see “Class $C^{1}$ ”
$C^{r}$       see “Class $C^{r}$ ”
$dx_{i}$ , elementary 1-form      253
$dx_{I}$ , elementary k-form      254
$d\omega$ , differential      256
$f\otimes g$ , tensor product      223
$f\wedge g$ , wedge product      238
$f^{\sigma}$       229
$I_{k}$ , identity matrix      5
$sgn\sigma$       228
$S^{n-1}(a)$       see “n-1 sphere”
$S_{k}$ , symmetric group      227
$X_{I}$ , submatrix      184
$Y_{\alpha}$       see “Parametrized-manifold”
$\alpha^{*}$ , dual transformation of forms      267
$\alpha_{*}$ , induced transformation of vectors      246
$\epsilon$ -neighborhood of point      26
$\epsilon$ -neighborhood of set      34
$\mathbf{H}^{k}$ , $\mathbf{H}^{k}_{+}$       200
$\mathbf{H}^{k}(A)$ , deRham group      335
$\mathbf{L}^{1}$ left half-line      283
$\mathbf{R}^{n}$ as metric space      25
$\mathbf{R}^{n}$ as vector space      2
$\mathcal{A}^{k}(V)$ , alternating tensors      229
$\mathcal{A}^{k}(V)$ , alternating tensors, basis for      232
$\mathcal{L}^{k}(V)$ , k-tensors on V      220
$\mathcal{L}^{k}(V)$ , k-tensors on V, basis for      221
$\mathcal{O}(n)$ , orthogonal group      209
$\mathcal{T}(M)$       see “Tangent bundle”
$\mathcal{T}_{p}(M)$       see “Tangent space”
$\Omega^{k}$ , linear space of k-forms      255 351
$\partial M$       see “Boundary of manifold”
$\phi_{I}$ , elementary tensor      221
$\psi_{i}$ , elementary alternating tensor      232
$\Sigma_{J}$       222
$\Sigma_{[I]}$       184
$\widetilde{\phi}_{i}$ , elementary 1-form      249
$\widetilde{\psi}_{i}$ , elementary k-form      249
Addition of matrices      4
Addition of vectors      1
Additivity of integral      106 109
Additivity of integral (extended)      125
Additivity of volume      112
Alternating tensor      229
Alternating tensor, elementary      232
Antiderivative      99
Approaches as a limit      28
ARC      306
Area      179
Area of 2-sphere      216—217
Area of parametrized-surface      191
Area of torus      217
Ascending k-tuple      184
Ball, $B^{n}(a)$       see “n-ball”
Ball, open      26
Basis      2 10
Basis for $\mathbf{R}^{n}$       3
Basis, usual, for tangent space      249
Bd A      29
Boundary of manifold      205 346
Boundary of manifold, induced orientation      288 346
Boundary of set      29
Bounded set      32
Cauchy — Schwarz inequality      9
Centroid of $E^{n}_{+}$       218
Centroid of bounded set      168
Centroid of cone      168
Centroid of half-ball      169
Centroid of manifold      218
Centroid of parametrized-manifold      193
Chain rule      56
Change of variables      147
Change of Variables Theorem      148
Change of variables theorem, proof      161
Class $C^{1}$       50
Class $C^{r}$ , form      250 351
Class $C^{r}$ , function      52 144 199
Class $C^{r}$ , manifold      196 200 347
Class $C^{r}$ , manifold-boundary      206
Class $C^{r}$ , tensor field      248
Class $C^{r}$ , vector field      247
Class $C^{\infty}$       52
Closed cube      30
Closed form      259
Closed form, not exact      261 308 343
Closed set      26
Closure      26
Cofactors      19
Cofactors, expansion by      23
Column index      4
Column matrix      6
Column rank      7
Column space      7
Common refinement      82
Compact      32
Compact support      139
Compact vs. closed and bounded      33 38
Compactness of interval      32
Compactness of rectangle      37
Comparison property of integral      106
Comparison property of integral (extended)      125
Component function      28
Component interval      81
Components of alternating tensor      233
Components of form      249
Composite function, class $C^{r}$       58
Composite function, differentiability      56
Cone      168
Connected      38
Connectedness of convex set      39
Connectedness of interval      38
Conservative vector field      323
Content      113
Continuity of algebraic operations      28
Continuity of composites      27
Continuity of projection      28
Continuity of restriction      27
continuous      27
Continuously differentiable      50
Convex      39
Coordinate patch      196 201 346
Coset      334
Covering      32
Cramer’s Rule      21
Cross product      183 313
Cross-section      121
cube      30
Cube, open      26
Curl      264
Cylindrical coordinates      151
d(x,C)      34
Darboux integral      89
deRham group      335
deRham group of $\mathbf{R}^{n}-0$       341
deRham group of $\mathbf{R}^{n}-p-q$       344
deRham’s theorem      354
Derivative      41 43
Derivative of composite      56
Derivative of inverse      60
Derivative vs. directional derivative      44
Determinant of product      18
Determinant of transpose      19
Determinant vs. rank      16
Determinant, axioms      15
Determinant, definition      234
Determinant, formula      234
Determinant, geometric interpretation      169
Determinant, properties      16
Df, derivative      43

df, differential      253 255
Diagonal      36
Diffeomorphism      147
Diffeomorphism of manifolds      347
Diffeomorphism, preserves rectifiability      154
Diffeomorphism, primitive      156
Differentiable      41—43
Differentiable homotopy      325
Differentiable manifold      346
Differentiable vs. continuous      45
Differentiably homotopic      325
Differential form of order 0      251
Differential form on manifold      351
Differential form on open set in $\mathbf{R}^{n}$       248
Differential of 0-form      253
Differential of k-form      256
Differential operator      256
Differential operator as directional derivative      262
Differential operator in manifold      352
Dimension of vector space      2
Directional derivative      42
Directional derivative in manifold      349
Directional derivative vs. continuity      44
Directional derivative vs. derivative      44
Distance from point to set      34
Divergence      263
Divergence theorem      319
Dominated by      139
Dot product      3
Dual basis      222
Dual space V*      220
Dual transformation of forms      267
Dual transformation of forms, calculation      269 273
Dual transformation of forms, properties      268
Dual transformation of tensors      224
Echelon form      8
Elementary 1-form      249 253
Elementary alternating tensor      232
Elementary alternating tensor as wedge product      237
Elementary k-form      249 254
Elementary k-tensor      221
Elementary matrix      11
Elementary permutation      227
Elementary row operation      8
Entry of matrix      4
Euclidean metric      25
Euclidean norm      4
Euclidean space      25
Even parametrization      228
Exact form      259
Expansion by cofactors      23
Ext A      29
Extended integral      121
Extended integral as limit of integrals      123 130
Extended integral as limit of series      141
Extended integral vs. ordinary integral      127 129 140
Extended integral, properties      125
Exterior      29
Extreme-value theorem      34
Face of rectangle      92
Final point of arc      306
FORM      see “Differential form”
Frame      171
Fubini’s theorem for rectangles      100
Fubini’s theorem for simple regions      116
Fundamental theorem of calculus      98
Gauss — Jordan reduction      7
Gauss’ Theorem      319
Gradient      48 263
Gradient theorem      312
Gram — Schmidt process      180
Graph      97 114
Green’s Theorem      308
Half-ball      169
hemisphere      192
Homeomorphism      345
Homologically trivial      259
Homotopy equivalence      336
Homotopy equivalence theorem      336
Homotopy, differentiable      325
Homotopy, straight-line      331
Identity matrix      5
Implicit differentiation      71 73
Implicit function theorem      74
Improper integral      121
Increasing function      90
Independent      2 10
Induced orientation of boundary      288 307 346
Induced transformation of deRham group      335
Induced transformation of quotient space      335
Induced transformation of tangent vectors      246
Initial point of arc      306
Inner product      3
Inner product space      3
Int A      29
Integrable      85
Integrable, extended sense      121
Integral of constant      87
Integral of form on 0-manifold      307
Integral of form on differentiable manifold      353
Integral of form on manifold in $\mathbf{R}^{n}$       293—294
Integral of form on open set in $\mathbf{R}^{k}$       276
Integral of form on parametrized-manifold      276
Integral of max, min      105
Integral of scalar function over manifold      210 212
Integral of scalar function over parametrized-manifold      189
Integral of scalar function over Riemannian manifold      355
Integral of scalar function vs. integral of form      299
Integral over bounded set      104
Integral over bounded set, existence      109 111
Integral over bounded set, properties      106
Integral over interval      89
Integral over rectangle      85
Integral over rectangle, evaluation      102
Integral over rectangle, existence      93
Integral over rectifiable set      112
Integral over simple region      116
Integral, extended      see “Extended integral”
Interior of manifold      205 346
Interior of set      29
Intermediate-value theorem      38
Invariance of domain      67
Inverse function theorem      69
Inverse function, derivative      60
Inverse function, differentiability      65
Inverse matrix      13
Inverse matrix, formula      22
Inversion in a permutation      228
Invertible matrix      13
Inward normal      318
Isolated point      27
Isometry      120 174
Isometry, preserves volume      176
Isomorphism, linear      6
Iterated integrals      103
Jacobian matrix      47
Jordan content      113
Jordan-measurable      113
k-form      see “Form”
Klein bottle      285
Left half-line      283
Left inverse      12
Left-handed      171
Leibnitz notation      60
Leibnitz’s rule      324
Length      179
Length of interval      81
Length of parametrized-curve      191
Length of vector      4
Lie group      209

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