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.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1991

Количество страниц: 366

Добавлена в каталог: 14.02.2006

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Предметный указатель

LIMIT      28
Limit of composite      30
Limit point      26
Limit vs. continuity      29
Line integral      278
Line segment      39
linear combination      2
Linear in $i^{th}$ variable      220
Linear isomorphism      6
Linear space      1
Linear space of k-forms      255
Linear subspace      2
Linear transformation      6
Linearity of integral of form      295
Linearity of integral of scalar function      213
Linearity of integral, extended      125
Linearity of integral, ordinary      106
Lipschitz condition      160
Locally bounded      133
Locally of class $C^{r}$       199
Lower integral      85
Lower sum      82
Manifold      200
Manifold of dimension 0      201
Manifold without boundary      196
Matrix      4
Matrix addition      4
Matrix cofactors      22
Matrix multiplication      5
Matrix, column      6
Matrix, elementary      11
Matrix, invertible      13
Matrix, non-singular      14
Matrix, row      6
Matrix, singular      14
Mayer — Vietoris theorem      337
Mean-value theorem in $\mathbf{R}$       49
Mean-value theorem in $\mathbf{R}^{m}$       59
Mean-value theorem, second-order      52
Measure zero in $\mathbf{R}^{n}$       91
Measure zero in manifold      213
mesh      82
Metric      25
Metric space      25
Metric, Euclidean      25
Metric, Riemannian      354
Metric, sup      25
Minor      19
Mixed partials      52 103
Moebius band      285
Monotonicity of integral      106
Monotonicity of integral (extended)      125
Monotonicity of volume      112
Multilinear      220
Multiplication by scalar      1 4
Multiplication of matrices      5
n-1 sphere      207
n-1 sphere as manifold      208
n-1 sphere, volume      218
n-ball, $B^{n}(a)$       207
n-ball, $B^{n}(a)$ , as manifold      208
n-ball, $B^{n}(a)$ , volume      168
n-manifold      see “Manifold”
Natural orientation of n-manifold      286
Natural orientation of tangent space      298
Neighborhood      26 (see also “ $\epsilon$ -neighborhood”)
Non-orientable manifold      281
Non-singular matrix      14
Norm      4
Normal field to n-1 manifold vs. orientation      285 312
Normal field to n-1 manifold, formula      314
Odd permutation      228
Open ball      26
Open covering      32
Open cube      26
Open rectangle      30
Open set      26
Opposite orientation of manifold      286 346
Opposite orientation of vector space      171
Order (of a form)      248
Orientable      281 346
Orientation for 0-manifold      307
Orientation for 1-manifold      282
Orientation for boundary      288
Orientation for manifold      281 346
Orientation for n-1 manifold      285 312
Orientation for n-manifold      286
Orientation for vector space      171 282
Orientation-preserving, diffeomorphism      281
Orientation-preserving, linear transformation      172
Orientation-reversing, diffeomorphism      281
Orientation-reversing, linear transformation      172
Oriented manifold      281 346
Orthogonal group      209
Orthogonal matrix      173
Orthogonal set      173
Orthogonal transformation      174
Orthonormal set      173
Oscillation      95
Outward normal      318
Overlap positively      281 346
Parallelopiped      170
Parallelopiped, volume      170 182
Parametrized-curve      48 191
Parametrized-manifold      188
Parametrized-manifold, volume      188
Parametrized-surface      191
Partial derivatives      46
Partial derivatives, equality of mixed      52 103
Partial derivatives, second-order      52
Partition of interval      81
Partition of rectangle      82
Partition of unity      139
Partition of unity on manifold      211 352
Peano curve      154
Permutation      227

Permutation group      227
Poincare lemma      331
Polar coordinate transformation      54 148
Potential function      323
Preserves $i^{th}$ coordinate      156
Primitive diffeomorphism      156
Product, matrix      5
Product, tensor      see “Tensor product”
Product, wedge      see “Wedge product”
Projection map      167
Pythagorean theorem for volume      184
Quotient space V/W      334
Rank of matrix      7
rectangle      29
Rectangle, open      30
Rectifiable set      112
Reduced echelon form      8
Refinement of partition      82
Restriction of coordinate patch      207
Restriction of form      337
Reverse orientation      see “Opposite orientation”
Riemann condition      86
Riemann integral      89
Riemannian manifold      355
Riemannian metric      354
Right inverse      12
Right-hand rule      172
Right-handed      171
Row index      4
Row matrix      6
Row operations      8
Row rank      7
Row space      7
Scalar field      48 251
Sign of permutation      228
Simple region      114
Singular matrix      14
Size of matrix      4
Skew-symmetric      265
Solid torus      151
Solid torus as manifold      208
Solid torus, volume      151
span      2 10
Sphere      see “n-1 sphere”
Spherical coordinate transformation      55 150
Stairstep form      8
Standard basis      3
Star-convex      330
Stokes’ theorem for 1-manifold      308
Stokes’ theorem for arc      306
Stokes’ theorem for differentiable manifold      353
Stokes’ theorem for k-manifold in $\mathbf{R}^{n}$       303
Stokes’ theorem for surface in $\mathbf{R}^{3}$       319
Straight-line homotopy      331
Subinterval determined by partition      82
Subrectangle determined by partition      82
Subspace of metric space      25
Subspace, linear      2
Substitution rule      144
Sup metric      25
Sup norm for matrices      5
Sup norm for vectors      4
Support      139
Symmetric group      227
Symmetric set      168
Symmetric tensor      229
T*      see “Dual transformation of tensors”
Tangent bundle      248
Tangent space to $\mathbf{R}^{n}$       245
Tangent space to manifold      247 349
Tangent vector field to $\mathbf{R}^{n}$       247
Tangent vector field to manifold      248
Tangent vector to $\mathbf{R}^{n}$       245
Tangent vector to manifold      247 348 351
Tensor      220
Tensor field on manifold      249
Tensor field to $\mathbf{R}^{n}$       248
Tensor product      223
Tensor product, properties      224
Topological property      27
Torus      151
Torus as manifold      208
Torus, area      217
Total volume of rectangles      91
Transition function      203 346
TRANSPOSE      9
triangle      193
Triangle inequality      4
Uniform continuity      36
Upper half-space      200
Upper integral      85
Upper sum      82
Usual basis for tangent space      249
V(X), volume function      181
V*, dual space      220
V/W, quotient space      334
Vector      1
Vector addition      1
Vector space      1
Velocity vector      48 245 349
Volume form      300
Volume form for Riemannian manifold      355
Volume of $M\times N$       218
Volume of bounded set      112
Volume of cone      168
Volume of manifold      212
Volume of n-bail      168
Volume of n-sphere      218
Volume of parallelopiped      182
Volume of parametrized-manifold      188
Volume of rectangle      81
Volume of Riemannian manifold      355
Volume of solid torus      151
Wedge product, definition      238
Wedge product, properties      237
width      81

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