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Loomis L.H., Sternberg S. — Advanced calculus
Loomis L.H., Sternberg S. — Advanced calculus

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Название: Advanced calculus

Авторы: Loomis L.H., Sternberg S.

Аннотация:

This book is based on an honors course in advanced calculus that we gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis. (from preface)


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second, revised edition

Год издания: 1989

Количество страниц: 589

Добавлена в каталог: 04.06.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute convergence, of an infinite series      221
Absolute value      121 242
Absolutely integrable function      351
Adjoint of T      85
Affine independence      81
Affine span      56
Affine subspace      40 52
Affine transformation      53
Algebra      30 46
Algebra of matrices      92
Algebra, Banach      223
Alternating tensor      310
Amplitude      292
Analytic plane $\mathbb{R}^{2}$      10
angular momentum      523
Annihilator      84
Antisymmetric tensor      310
Associative law, for composition      14
Atlas      364
Ball, closed      124
Ball, open      123 196
Banach algebra      223
Banach space      217
Banach — Tarski paradox      321
Basis      71
Basis for pre-Hilbert space      254
Basis in Hom(V,W)      96
Basis isomorphism      72
Basis, dual      82
Basis, infinite      75
Bessel's inequality      254
Bijective      12
Bilinear      67
Bilinear functionals      305ff
Block decompositions      65ff
Bound variable      2
Boundary      198
Boundary-value problem      294ff
Bounded below      128
Bounded function      122
Bounded linear mapping      127
Bounded set      123
Braces      7
Calculus of variations      182ff
Canonical basis      104
Canonical transformation      558 560
Cartesian axes      38
Cartesian product of two sets      9
Cartesian product, of an indexed collection      14
Cauchy sequence      216
Cellulation      473
Center of mass      529
Center, of gravity      349
Central force      523 564
Centrifugal force      517
Centrifugal potential      517
Chain rule      153
Change of basis      94
Change of variable formula      342 355
Characteristic function of a set      12
Characteristic function, Fourier transform      337
Characteristic polynomial      259
Classical mechanics      509
Closed set      124 197
Closure      198
Codimension      80
Codomain      12
Cofactor      315
Column space (of a matrix)      91
Column vector      93
Commutative diagram      86
Compact transformation      264
Compactness on a manifold      403
Compactness, in general topology      214
Compactness, sequential      206
Compatible atlases      370
Complement      17
Complementary subspaces      57
Completeness      217
Completion      222
Complex conjugation      241
Complex normed linear space      244
complex numbers      240ff
Complex vector space      24 241
Complexification      244
Component      58
Composite truth-functional forms      5
Composite-function rule      143
composition      14
Configuration space      509
Conjugate space      82
Connectives, logical      3
Conservation of energy      521
Conservation, laws      521
Conservation, of angular momentum      523
Constant coefficient equation      278
Constrained maximum      173f
Content      332
Contented function      339
Contented sets      331f
Continuity      126 196 201 203
Contraction mapping      229
Contravariant vector      95
Convergence of infinite series      221
Convergence, sequential      202
Convex set      125
Coordinate      43 72
Coordinate correspondence      36f
Coordinate functional      33 72
Coordinate isomorphism      72
Coordinate n-tuple      72
Coordinate projection      43
Correspondence      9
Coset      52
Cotangent bundle      511
Covariant tensor      307
Covariant vector      95
Cramer's rule      101 315
Critical point      161 188ff
Cylindrical coordinates      345 (Ex. 11.5)
De Morgan's law      17
Definite (form)      115
Degree (of a polynomial)      64
Dense subset      212
density      408
Dependent set of vectors      73
Derivative      146
Derivative in a Banach algebra      226
Derivative over $\mathbb{C}$      242
Derivative, directional      147
Determinant      99ff 108 312ff
Diagonal matrix      262
Diffeomorphism      376
Differentiable manifolds      363 369ff
Differential      142
Differential forms      390f
Differential geometry      459
Differentiation under the integral sign      181
DIMENSION      78
Dimensional identities      78f 84
Direct sum      56
Directed frame      9
Directed line segment      37
Directional derivative      147
Disjoint (family of sets)      18
Distance between vectors      122
Distance, in a metric space      196
Divergence theorem      421
Division algorithm      64
Domain of a variable      8
Domain with regular boundary      419
Domain, of a relation      9
Domain, with almost regular boundary      424
Dual basis      82
Dual space      82
Duality      15 68
Dyad      86
Echelon matrix      104
Eigenvalue      34 258
Eigenvector      34 258
Element (of a set)      6
Elementary bilinear functionals      306
Elementary matrices      105f
Elementary row operations      103
Elliptical coordinates      566
Empty set      7
Equations of variation      513f
Equicontinuity      215
Equivalence relation      19
Equivalent directed line segments      37
Equivalent norms      132 204
Equivalent truth-functional forms      5
Euclidean norm      38 122
Euler angles      548
Euler's equation(s)      184 541 543
Existential quantification      2
Exponential function      226 228
Exterior algebra      316ff
Exterior calculus      429
Exterior differential forms      429
Exterior forms      316
Fibering      19
Finite-dimensional      28
First variation      183
Fixed-point theorem      229ff
Floor of a paving      329
Flow      381
Formal adjoint      295
Fourier coefficient      254
Fourier series      254 301ff
Fourier transform      355 357
Frame      71
Free variables      2
frequency      292
Function      10f
Function space      24
Functional dependence      175ff
Fundamental form on T*(M)      515
Fundamental solution      277
Fundamental theorem of calculus      238
Fundamental Theorem of Ordinary Differential Equations      267
Fundamental theorem, of algebra      243
Geodesic      539
Geodesic coordinates      537
Geodesic curvature      468
Geodesic polar coordinates      470
Germ of functions      138
Global solutions      269
Graph      9 162
Greatest lower bound      119
Group      15
Hamilton — Jacobi equation      563
Hamiltonian mechanics      520
Hamiltonian structure      559
Hamiltonian vector fields      519 559
Heine — Borel property      214
Hermitian symmetry      249
Hilbert space      249
Homogeneity, of a norm      121
Homogeneous equation      277 282ff
Homogeneous function      148
Homogeneous polynomial on V      145
Hyperplane      41
Idempotent      59
If ..., then ...      4
Image      10
Immersion      399
Implicit-function theorem      167 230
Inclusion (set)      7
Increasing norm on $\mathbb{R}^{n}$      134
Increasing sequence      207
Independent set of vectors      71f
Independent subspaces      56
Indexed collection      13
Inertia tensor      546
Infinite series      221 224ff
Infinitesimal      137
Infinitesimal generator      379 381
inhomogeneous equation      277 288ff
Initial condition      267
Injection (product)      47
Injective      12
Inner content      331
Inner paving      331
Integral, of a parametrized arc      236ff
Integration      321 336
Interior      197
Interior point      124
Intermediate-value theorem      120
Intersection      17
Invariant subspace      54 63
Inverse of a relation      9
Inverse, of a function      15
Inverse-mapping theorem      167
Isomorphism      34
Isomorphism between normed linear spaces      132
Iterated integrals      346
Jacobian determinant      159
Jacobian matrix      158f
Kepler's first law      527
Kepler's second law      524
Kepler's third law      527 532
Kernel      34
kinetic energy      521
Kronecker delta function      74
Lagrange multipliers      174
Lagrange's equations      530
Law of Cosines      41 250
Least upper bound property      119
Left inverse      14
Lemma of Du Bois-Reymond      184
Length of a curve, Riemannian manifold      401
Lie bracket      389
Lie derivative of a function      383f
Lie derivative of a linear differential form      392
Lie derivative of a vector field      385 388
Lie derivative of an exterior differential form      431 452
Lie derivative, of a density      417
Limit, of a function      116 126
Line (straight)      38
Line of nodes      548
linear combination      27
Linear combination mapping      31
Linear differential equations      276ff
Linear functionals      32 82
Linear transformation      30
Lipschitz continuity      126
Local solution      269
Locally absolutely integrable density      408
Locally absolutely integrable n-form      435
Locally uniformly Lipschitz      267
Mapping      12
Matrix      32 88
Maximal solution      269
Maximum, relative      161
Mean-value theorem      148f
Member (of a set)      6
Metric space      196
Minimal polynomial      261
Momentum      511
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