Mathematical analysis: Functions of several real variables and applications
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cham
Springer
[2022]
|
| Schriftenreihe: | Unitext - Matematica per il 3 + 2
137 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035078873&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | X, 657 Seiten Illustrationen, Diagramme |
| ISBN: | 9783031041501 |
Internformat
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Datensatz im Suchindex
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| adam_text |
Contents 1 Sequences and Series of Functions . 1.1 Sequences of Functions: Pointwise and Uniform Convergence. 1.2 First Theorems on Uniform Convergence. 1.3 Theorems on Interchanging Limits and Integrals or Derivatives. 1.4 Uniform Convergence and Monotonicity. 1.5 Series of Functions. 1.6 Power Series. 1.7 Taylor Series. 1.8 Fourier Series. 1.9 The Convergence of Fourier Series. Appendix to Chap. 1. 1.10 The Ascoli-Arzelà Theorem. 1.11 The Weierstrass Approximation Theorem. 1.12 Abel’s Theorem on Power Series. 2 Metric Spaces and Banach Spaces. 2.1 Introduction. 2.2 MetricSpaces.
2.3 Sequences in a Metric Space: Continuous Functions. 2.4 Vector Spaces: Linear Maps. 2.5 The Vector Space R" and Its Dual. 2.6 Normed Vector Spaces. 2.7 The Normed Vector Space R". 2.8 Complete Metric Spaces: Banach Spaces. 2.9 Lipschitz Functions: The Contraction Theorem. 2.10 Compact Sets: Continuous Functions on Compact Sets. 2.11 Connected Open Subsets of R". Appendix to Chap. 2. 2.12 Further Compactness Theorems: Generalised Weierstrass Theorem. 94 1 1 4 7 14 17 22 28 36 42 48 48 50 52 59 59 59 65 69 72 76 78 83 86 89 92 94 vii
viii 3 Contents Functions of Several Variables. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 Round-Upof Topology in R". Limits and Continuity. Partial Derivatives. Higher Derivatives. Schwarz’s Theorem. Gradient. Differentiability . Composite Functions. Directional Derivatives. Functions with Vanishing Gradient on Connected Sets. Homogeneous Functions. Functions Defined by Integrals. Taylor Formula and Higher-Order Differentials. Quadratic Forms. Definite, Semi-definite and Indefinite Matrices. 140 3.13 Local Maxima and Minima. 3.14 Vector-Valued Functions. Appendix to Chap. 3. 3.15 Convex
Functions. 3.16 Complements on Quadratic Forms. 3.17 The Maximum Principle for Harmonic Functions. 4 Ordinary Differential Equations. Introduction : The Initial Value Problem. Cauchy’s Local Existence and Uniqueness Theorem. First Consequences of Cauchy’s Theorem. The Global Existence and Uniqueness Theorem: Extension of Solutions. 210 4.5 Solving First-Order ODEs in Normal Form . 4.6 Solving First-Order ODEs Not in Normal Form. 4.7 Solving Higher-Order Equations. 4.8 Qualitative Study of Solutions. Appendix to Chap. 4. 4.9 Peano’s Theorem. 4.1 4.2 4.3 4.4 5 101 101 103 105 109 113 118 122 127 129 131 135 144 150 158 158 173 181 187 187 196 206 216 221 224 226 232 232 Linear Differential Equations. 237 General
Properties. General Integral of Linear ODEs. The Method of Variation of Parameters. Bernoulli Equations. Homogeneous Equations with Constant Coefficients . Equations with Constant Coefficients and Special Right-Hand Side. 5.7 Linear Euler Equations. Appendix to Chap. 5. 5.8 Boundary Value Problems. 5.9 Linear Systems. 5.1 5.2 5.3 5.4 5.5 5.6 237 241 247 250 252 257 260 263 263 268
Contents 6 7 8 Curves and Integrals Along Curves. 273 6.1 Regular Curves. 6.2 Oriented Curves. 6.3 The Length of a Curve. 6.4 The Integral of a Function Along a Curve. 6.5 The Curvature of a Plane Curve. 6.6 The Cross Product in R3 . 6.7 Biregular Curves in R3: Curvature. Appendix to Chap. 6. 6.8 Curves in R3 : Torsion, Frenet Frame. 273 279 281 286 290 294 297 300 300 Differential One-Forms. 7.1 Vector Fields. Work. Conservative Fields. 7.2 Differential 1-Forms. Line Integrals. 7.3 Exact 1-Forms. 7.4 Exact 1-Forms on the Plane. Simply Connected Open Sets in R2 . 315 7.5 One-Forms in Space. Irrotational Vector Fields . Appendix to Chap.
7. 7.6 Simply Connected Open Sets in R" and Exact 1-Forms. 305 305 308 311 320 323 323 Multiple Integrals . 325 Double Integrals on Normal Domains. Reduction Formulas for Double Integrals. Gauss-Green Formulas. The Divergence Theorem. Stokes’s Formula. 342 8.4 Variable Change in Double Integrals. 8.5 Triple Integrals. 8.6 Peano-Jordan Measurable Subsets of R” . 8.7 The Riemann Integral in R" . 8.8 Properties of Riemann Integrals. 8.9 Summable Functions. Appendix to Chap. 8. 8.10 Jensen’s Inequality. 8.11 The Gamma Function. The Measure of the Unit Ball in R“ . 8.1 8.2 8.3 9 ix 325 335 351 356 362 369 377 382 388 388 389 The Lebesgue Integral. 395 9.1
Introduction. 9.2 Pluri-Intervals. Open Sets. Compact Sets. 9.3 Bounded Measurable Sets . 9.4 Unbounded Measurable Sets. 9.5 Measurable Functions. 9.6 The Lebesgue Integral. Interchanging Limits and Integrals. 9.7 Measure and Integration on Product Spaces. 9.8 Changing Variables in Multiple Integrals. Appendix to Chap. 9. 395 396 401 405 412 419 437 457 477
Contents X 9.9 9.10 9.11 9.12 9.13 10 Lp Spaces. Differentiability of Monotone Functions. Functions with Bounded Variation. Absolutely Continuous Functions. The Indefinite Integral in Lebesgue’s Theory. 477 485 495 504 514 525 Regular Surfaces . 525 Local Coordinates and Change of Parameters. 533 The Tangent Plane and the Unit Normal. 539 The Area of a Surface. 543 Orientable Surfaces: Surfaces with Boundary. 550 Surface Integrals. 556 Stokes’s Formula and the Divergence Theorem. 560 Surfaces and Surface Integrals. 10.1 10.2 10.3 10.4 10.5 10.6 10.7 11 567 11.1 The Implicit Function Theorem for Equations. 567 11.2 The Implicit Function Theorem for Systems. 582 11.3 Local and Global Invertibility. 589 11.4 Constrained Maxima and Minima. Lagrange
Multipliers. 596 Appendix to Chap. 11. 606 11.5 Singular Points of a Plane Curve. 606 12 Manifolds in R” and k-Forms. Implicit Functions. 12.1 ^-Dimensional Manifolds in R". 12.2 The Tangent Space and the Normal Space of a Manifold. 12.3 Measure and Integration on k-Submanifolds in R". 12.4 The Divergence Theorem. 12.5 Alternating Forms. 12.6 Differential k-Forms . 12.7 Orientable Manifolds. Integration of k-Forms on Manifolds. 12.8 Manifolds with Boundary. Stokes’s Formula. Appendix to Chap. 12. 12.9 Exact and Closed Differential Forms. 611 611 619 624 632 638 645 650 659 663 663 Index. 669 |
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| discipline | Mathematik |
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| id | DE-604.BV049736805 |
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| indexdate | 2025-01-11T12:22:00Z |
| institution | BVB |
| isbn | 9783031041501 |
| language | English |
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| physical | X, 657 Seiten Illustrationen, Diagramme |
| publishDate | 2022 |
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| publisher | Springer |
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| series | Unitext - Matematica per il 3 + 2 |
| series2 | Unitext - Matematica per il 3 + 2 |
| spelling | Fusco, Nicola 1956- Verfasser (DE-588)124005721 aut Mathematical analysis Functions of several real variables and applications by Nicola Fusco, Paolo Marcellini, Carlo Sbordone Cham Springer [2022] X, 657 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Unitext - Matematica per il 3 + 2 137 Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Marcellini, Paolo 1947- Verfasser (DE-588)1135682283 aut Sbordone, Carlo 1948- Verfasser (DE-588)1251558771 aut Erscheint auch als Online-Ausgabe 978-3-031-04151-8 Unitext - Matematica per il 3 + 2 137 (DE-604)BV047304938 137 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035078873&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fusco, Nicola 1956- Marcellini, Paolo 1947- Sbordone, Carlo 1948- Mathematical analysis Functions of several real variables and applications Unitext - Matematica per il 3 + 2 Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Mathematical analysis Functions of several real variables and applications |
| title_auth | Mathematical analysis Functions of several real variables and applications |
| title_exact_search | Mathematical analysis Functions of several real variables and applications |
| title_full | Mathematical analysis Functions of several real variables and applications by Nicola Fusco, Paolo Marcellini, Carlo Sbordone |
| title_fullStr | Mathematical analysis Functions of several real variables and applications by Nicola Fusco, Paolo Marcellini, Carlo Sbordone |
| title_full_unstemmed | Mathematical analysis Functions of several real variables and applications by Nicola Fusco, Paolo Marcellini, Carlo Sbordone |
| title_short | Mathematical analysis |
| title_sort | mathematical analysis functions of several real variables and applications |
| title_sub | Functions of several real variables and applications |
| topic | Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035078873&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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