Advanced calculus / Lynn H. Loomis and Shlomo Sternberg.
Series Addison-Wesley series in mathematicsEditor: Reading, Mass. : Addison-Wesley, c1968Descripción: 580 p. : il. ; 25 cmTema(s): CalculusOtra clasificación: 26-01 (57-01)Chapter 0 Introduction 1 Logic: quantifiers [1] 2 The logical connectives [3] 3 Negations of quantifiers [6] 4 Sets [6] 5 Restricted variables [8] 6 Ordered pairs and relations [9] 7 Functions and mappings [10] 8 Product sets; index notation [12] 9 Composition [14] 10 Duality [15] 11 The Boolean operations [17] 12 Partitions and equivalence relations [19] Chapter 1 Vector Spaces 1 Fundamental notions [21] 2 Vector spaces and geometry [36] 3 Product spaces and Hom(V, W) [43] 4 Affine subspaces and quotient spaces [52] 5 Direct sums [56] 6 Bilinearity [67] Chapter 2 Finite-Dimensional Vector Spaces 1 Bases [71] 2 Dimension [77] 3 The dual space [81] 4 Matrices [88] 5 Trace and determinant [99] 6 Matrix computations [102] *7 The diagonalization of a quadratic form [111] Chapter 3 The Differential Calculus 1 Review in R [117] 2 Norms [121] 3 Continuity [126] 4 Equivalent norms [132] 5 Infinitesimals [136] 6 The differential [140] 7 Directional derivatives; the mean-value theorem [146] 8 The differential and product spaces [152] 9 The differential and [153] 10 Elementary applications [161] 11 The implicit-function theorem [164] 12 Submanifolds and Lagrange multipliers [172] *13 Functional dependence [175] *14 Uniform continuity and function-valued mappings [179] *15 The calculus of variations [182] *16 The second differential and the classification of critical points [186] *17 The Taylor formula [191] Chapter 4 Compactness and Completeness 1 Metric spaces; open and closed sets [195] *2 Topology [201] 3 Sequential convergence [202] 4 Sequential compactness [205] • 5 Compactness and uniformity [210] 6 Equicontinuity [215] 7 Completeness [216] 8 A first look at Banach algebras [223] 9 The contraction mapping fixed-point theorem [228] 10 The integral of a parametrized arc [236] 11 The complex number system [240] *12 Weak methods [245] Chapter 5 Scalar Product Spaces 1 Scalar products [248] 2 Orthogonal projection [252] 3 Self-adjoint transformations [257] 4 Orthogonal transformations [262] 5 Compact transformations [264] Chapter 6 Differential Equations The fundamental theorem [266] Differentiable dependence on parameters [274] The linear equation [276] The nth-order linear equation [281] Solving the inhomogeneous equation [288] The boundary-value problem [294] Fourier series [301] Chapter 7 Multilinear Functionals Bilinear functionals [305] Multilinear functionals 306 Permutations [308] The sign of a permutation [309] The subspace An of alternating tensors [310] The determinant [312] The exterior algebra [316] Exterior powers of scalar product spaces [319] The star operator [320] Chapter 8 Integration Introduction [321] Axioms [322] Rectangles and paved sets [324] The minimal theory [327] The minimal theory (continued) [328] Contented sets [331] When is a set contented? [333] Behavior under linear distortions [335] Axioms for integration [335] Integration of contented functions [338] The change of variables formula [342] Successive integration [346] Absolutely integrable functions [351] Problem set: The Fourier transform [355] Chapter 9 Differentiable Manifolds 1 Atlases [364] 2 Functions, convergence [367] 3 Differentiable manifolds [369] 4 The tangent space [373] 5 Flows and vector fields [376] 6 Lie derivatives [383] 7 Linear differential forms [390] 8 Computations with coordinates [393] 9 Riemann metrics [397] Chapter 10 The Integral Calculus on Manifolds 1 Compactness [403] 2 Partitions of unity [405] 3 Densities [408] 4 Volume density of a Riemann metric [411] 5 Pullback and Lie derivatives of densities [416] 6 The divergence theorem [419] 7 More complicated domains [424] Chapter 11 Exterior Calculus 1 Exterior differential forms [429] 2 Oriented manifolds and the integration of exterior differential forms [433] 3 The operator d [438] 4 Stokes’ theorem [442] 5 Some illustrations of Stokes’ theorem [449] 6 The Lie derivative of a differential form [452] Appendix I. “Vector analysis” [457] Appendix II. Elementary differential geometry of surfaces in E3 [459] Chapter 12 Potential Theory in En 1 Solid angle [474] 2 Green’s formulas [476] 3 The maximum principle [477] 4 Green’s functions [479] 5 The Poisson integral formula [482] 6 Consequences of the Poisson integral formula [485] 7 Harnack’s theorem [487] 8 Subharmonic functions [489] 9 Dirichlet’s problem [491] 10 Behavior near the boundary [495] 11 Dirichlet’s principle [499] 12 Physical applications [501] 13 Problem set: The calculus of residues [503] Chapter 13 Classical Mechanics 1 The tangent and cotangent bundles [511] 2 Equations of variation [513] 3 The fundamental linear differential form on T*(M) [515] 4 The fundamental exterior two-form on T*(M) [517] 5 Hamiltonian mechanics [520] 6 The central-force problem [521] 7 The two-body problem [528] 8 Lagrange’s equations [530] 9 Variational principles [532] 10 Geodesic coordinates [537] 11 Euler’s equations [541] 12 Rigid-body motion [544] 13 Small oscillations [551] 14 Small oscillations (continued) [553] 15 Canonical transformations [558] Selected References [569] Notation Index [572]
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 L863 (Browse shelf) | Available | A-2954 |
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| 26 L732 Análise no espaço Rn / | 26 L743 Einführung in die höhere Analysis : | 26 L781-3 The elements of the theory of real functions. | 26 L863 Advanced calculus / | 26 M165 The generalized Riemann integral / | 26 M175 Real analysis / | 26 M175e Análisis real / |
Bibliografía: p. 569-571.
Chapter 0 Introduction --
1 Logic: quantifiers [1] --
2 The logical connectives [3] --
3 Negations of quantifiers [6] --
4 Sets [6] --
5 Restricted variables [8] --
6 Ordered pairs and relations [9] --
7 Functions and mappings [10] --
8 Product sets; index notation [12] --
9 Composition [14] --
10 Duality [15] --
11 The Boolean operations [17] --
12 Partitions and equivalence relations [19] --
Chapter 1 Vector Spaces --
1 Fundamental notions [21] --
2 Vector spaces and geometry [36] --
3 Product spaces and Hom(V, W) [43] --
4 Affine subspaces and quotient spaces [52] --
5 Direct sums [56] --
6 Bilinearity [67] --
Chapter 2 Finite-Dimensional Vector Spaces --
1 Bases [71] --
2 Dimension [77] --
3 The dual space [81] --
4 Matrices [88] --
5 Trace and determinant [99] --
6 Matrix computations [102] --
*7 The diagonalization of a quadratic form [111] --
Chapter 3 The Differential Calculus --
1 Review in R [117] --
2 Norms [121] --
3 Continuity [126] --
4 Equivalent norms [132] --
5 Infinitesimals [136] --
6 The differential [140] --
7 Directional derivatives; the mean-value theorem [146] --
8 The differential and product spaces [152] --
9 The differential and [153] --
10 Elementary applications [161] --
11 The implicit-function theorem [164] --
12 Submanifolds and Lagrange multipliers [172] --
*13 Functional dependence [175] --
*14 Uniform continuity and function-valued mappings [179] --
*15 The calculus of variations [182] --
*16 The second differential and the classification of critical points [186] --
*17 The Taylor formula [191] --
Chapter 4 Compactness and Completeness --
1 Metric spaces; open and closed sets [195] --
*2 Topology [201] --
3 Sequential convergence [202] --
4 Sequential compactness [205] --
• 5 Compactness and uniformity [210] --
6 Equicontinuity [215] --
7 Completeness [216] --
8 A first look at Banach algebras [223] --
9 The contraction mapping fixed-point theorem [228] --
10 The integral of a parametrized arc [236] --
11 The complex number system [240] --
*12 Weak methods [245] --
Chapter 5 Scalar Product Spaces --
1 Scalar products [248] --
2 Orthogonal projection [252] --
3 Self-adjoint transformations [257] --
4 Orthogonal transformations [262] --
5 Compact transformations [264] --
Chapter 6 Differential Equations --
The fundamental theorem [266] --
Differentiable dependence on parameters [274] --
The linear equation [276] --
The nth-order linear equation [281] --
Solving the inhomogeneous equation [288] --
The boundary-value problem [294] --
Fourier series [301] --
Chapter 7 Multilinear Functionals --
Bilinear functionals [305] --
Multilinear functionals 306 Permutations [308] --
The sign of a permutation [309] --
The subspace An of alternating tensors [310] --
The determinant [312] --
The exterior algebra [316] --
Exterior powers of scalar product spaces [319] --
The star operator [320] --
Chapter 8 Integration --
Introduction [321] --
Axioms [322] --
Rectangles and paved sets [324] --
The minimal theory [327] --
The minimal theory (continued) [328] --
Contented sets [331] --
When is a set contented? [333] --
Behavior under linear distortions [335] --
Axioms for integration [335] --
Integration of contented functions [338] --
The change of variables formula [342] --
Successive integration [346] --
Absolutely integrable functions [351] --
Problem set: The Fourier transform [355] --
Chapter 9 Differentiable Manifolds --
1 Atlases [364] --
2 Functions, convergence [367] --
3 Differentiable manifolds [369] --
4 The tangent space [373] --
5 Flows and vector fields [376] --
6 Lie derivatives [383] --
7 Linear differential forms [390] --
8 Computations with coordinates [393] --
9 Riemann metrics [397] --
Chapter 10 The Integral Calculus on Manifolds --
1 Compactness [403] --
2 Partitions of unity [405] --
3 Densities [408] --
4 Volume density of a Riemann metric [411] --
5 Pullback and Lie derivatives of densities [416] --
6 The divergence theorem [419] --
7 More complicated domains [424] --
Chapter 11 Exterior Calculus --
1 Exterior differential forms [429] --
2 Oriented manifolds and the integration of exterior differential forms [433] --
3 The operator d [438] --
4 Stokes’ theorem [442] --
5 Some illustrations of Stokes’ theorem [449] --
6 The Lie derivative of a differential form [452] --
Appendix I. “Vector analysis” [457] --
Appendix II. Elementary differential geometry of surfaces in E3 [459] --
Chapter 12 Potential Theory in En --
1 Solid angle [474] --
2 Green’s formulas [476] --
3 The maximum principle [477] --
4 Green’s functions [479] --
5 The Poisson integral formula [482] --
6 Consequences of the Poisson integral formula [485] --
7 Harnack’s theorem [487] --
8 Subharmonic functions [489] --
9 Dirichlet’s problem [491] --
10 Behavior near the boundary [495] --
11 Dirichlet’s principle [499] --
12 Physical applications [501] --
13 Problem set: The calculus of residues [503] --
Chapter 13 Classical Mechanics --
1 The tangent and cotangent bundles [511] --
2 Equations of variation [513] --
3 The fundamental linear differential form on T*(M) [515] --
4 The fundamental exterior two-form on T*(M) [517] --
5 Hamiltonian mechanics [520] --
6 The central-force problem [521] --
7 The two-body problem [528] --
8 Lagrange’s equations [530] --
9 Variational principles [532] --
10 Geodesic coordinates [537] --
11 Euler’s equations [541] --
12 Rigid-body motion [544] --
13 Small oscillations [551] --
14 Small oscillations (continued) [553] --
15 Canonical transformations [558] --
Selected References [569] --
Notation Index [572] --
MR, 37 #2912
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