Calculus in vector spaces:
Addressing linear algebra from the basics to the spectral theorem and examining a host of topics in multivariable calculus, including differentiation, integration, maxima and minima, the inverse and implicit function theorems, and differential forms, this thoroughly revised Second Edition of an inva...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Dekker
1995
|
| Edition: | 2. ed. |
| Series: | Pure and applied mathematics
189 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | Addressing linear algebra from the basics to the spectral theorem and examining a host of topics in multivariable calculus, including differentiation, integration, maxima and minima, the inverse and implicit function theorems, and differential forms, this thoroughly revised Second Edition of an invaluable reference/text - widely successful through five printings - continues to provide unified, integrated coverage of the two fields Demonstrating that mathematics is a noncompartmentalized discipline of interrelated subjects, Calculus in Vector Spaces, Second Edition introduces the derivative as a linear transformation ... presents linear algebra in a concrete context based on complementary ideas in calculus ... explains differential forms on Euclidean space permitting Green's theorem, Gauss's theorem, and Stokes's theorem to be understood in a natural setting ... gives a new clarification of compactness as defined in terms of coverings and in terms of sequences ... supplies a novel treatment of eigenvalues and eigenvectors ... and more |
| Physical Description: | XI, 583 S. graph. Darst. |
| ISBN: | 0824792793 |
Staff View
MARC
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| 520 | |a Demonstrating that mathematics is a noncompartmentalized discipline of interrelated subjects, Calculus in Vector Spaces, Second Edition introduces the derivative as a linear transformation ... presents linear algebra in a concrete context based on complementary ideas in calculus ... explains differential forms on Euclidean space permitting Green's theorem, Gauss's theorem, and Stokes's theorem to be understood in a natural setting ... gives a new clarification of compactness as defined in terms of coverings and in terms of sequences ... supplies a novel treatment of eigenvalues and eigenvectors ... and more | ||
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Record in the Search Index
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| adam_text | Contents
Preface to the Second Edition iii
Preface to the First Edition v
Chapter 1. Some Preliminaries 1
1. The Rudiments of Set Theory 1
2. Some Logic 6
3. Mathematical Induction 9
4. Inequalities and Absolute Value 13
5. Equivalence Relations 16
Chapter 2. Vector Spaces 19
1. The Cartesian Plane 19
2. The Definition of a Vector Space 24
3. Some Elementary Properties of Vector Spaces 30
4. Subspaces 33
5. Linear Transformations 39
6. Linear Transformations on Euclidean Spaces 46
vii
viii Contents
Chapter 3. The Derivative 55
1. Normed Vector Spaces 56
2. Open and Closed Sets 60
3. Continuous Functions Between Normed Vector Spaces 65
4. Elementary Properties of Continuous Functions 75
5. The Derivative 80
6. Elementary Properties of the Derivative 85
7. Partial Derivatives and the Jacobian Matrix 87
Chapter 4. The Structure of Vector Spaces 95
1. Spans and Linear Independence 96
2. Bases 101
3. Bases and Linear Transformations 105
4. The Dimension of a Vector Space 110
5. Inner Product Spaces 116
6. The Norm on an Inner Product Space 121
7. Orthonormal Bases 123
8. The Cross Product in R3 130
Chapter 5. Compact and Connected Sets 133
1. Convergent Sequences 133
2. Compact Sets 139
3. Upper and Lower Bounds 143
4. Continuous Functions on Compact Sets 147
5. A Characterization of Compact Sets 151
6. Uniform Continuity 155
7. Connected Sets 157
Chapter 6. The Chain Rule, Higher Derivatives, and Taylor s
Theorem 163
1. The Chain Rule 164
2. Proof of the Chain Rule 170
3. Higher Derivatives 173
4. Taylor s Theorem for Functions of One Variable 181
5. Taylor s Theorem for Functions of Two Variables 186
6. Taylor s Theorem for Functions of n Variables 191
7. A Sufficient Condition for Differentiability 195
8. The Equality of Mixed Partial Derivatives 200
Contents ix
Chapter 7. Linear Transformations and Matrices 203
1. The Matrix of a Linear Transformation 203
2. Isomorphisms and Invertible Matrices 207
3. Change of Basis 211
4. The Rank of a Matrix 216
5. The Trace and Adjoint of a Linear Transformation 219
6. Row and Column Operations 227
7. Gaussian Elimination 232
Chapter 8. Maxima and Minima 239
1. Maxima and Minima at Interior Points 240
2. Quadratic Forms 246
3. Criteria for Local Maxima and Minima 252
4. Constrained Maxima and Minima: I 257
5. The Method of Lagrange Multipliers 263
6. Constrained Maxima and Minima: II 268
7. The Proof of Proposition 2.3 273
Chapter 9. The Inverse and Implicit Function Theorems 277
1. The Inverse Function Theorem 277
2. The Proof of Theorem 1.3 283
3. The Proof of the General Inverse Function Theorem 285
4. The Implicit Function Theorem: I 292
5. The Implicit Function Theorem: II 296
Chapter 10. The Spectral Theorem 303
1. Complex Numbers 304
2. Complex Vector Spaces 308
3. Eigenvectors and Eigenvalues 314
4. The Spectral Theorem 318
5. Determinants 324
6. Properties of the Determinant 329
7. More on Determinants 336
8. Quadratic Forms 340
Chapter 11. Integration 345
1. Integration of Functions of One Variable 345
2. Properties of the Integral 353
3. The Integral of a Function of Two Variables 359
4. The Integral of a Function of n Variables 366
x Contents
5. Properties of the Integral 375
6. Integrable Functions 379
7. The Proof of Theorem 6.2 384
Chapter 12. Iterated Integrals and the Fubini Theorem 387
1. The Fubini Theorem 387
2. Integrals Over Nonrectangular Regions 393
3. More Examples 399
4. The Proof of Fubini s Theorem 407
5. Differentiating Under the Integral Sign 409
6. The Change of Variable Formula 412
7. The Proof of Theorem 6.2 418
Chapter 13. Line Integrals 427
1. Curves 427
2. Line Integrals of Functions 431
3. Line Integrals of Vector Fields 436
4. Conservative Vector Fields 443
5. Green s Theorem 450
6. The Proof of Green s Theorem 455
Chapter 14. Surface Integrals 461
1. Surfaces 461
2. Surface Area 468
3. Surface Integrals 475
4. Stokes Theorem 481
Chapter 15. Differential Forms 487
1. The Algebra of Differential Forms 487
2. Basic Properties of the Sum and Product of Forms 492
3. The Exterior Differential 494
4. Basic Properties of the Exterior Differential 500
5. The Action of Differentiable Functions on Forms 502
6. Further Properties of the Induced Mapping 505
Chapter 16. Integration of Differential Forms 509
1. Integration of Forms 509
2. The General Stokes Theorem 513
3. Green s Theorem and Stokes Theorem 517
4. The Gauss Theorem and Incompressible Fluids 519
5. Proof of the General Stokes Theorem 525
Contents xi
Appendix 1. The Existence of Determinants 527
Appendix 2. Jordan Canonical Form 531
1. Generalized Eigenvalues 531
2. The Jordan Canonical Form 534
3. Polynomials and Linear Transformations 540
4. The Proof of Theorem 3.5 545
5. The Proof of Theorem 2.2 549
Solutions of Selected Exercises 553
Index 577
|
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| id | DE-604.BV010118831 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:46:50Z |
| institution | BVB |
| isbn | 0824792793 |
| language | English |
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| physical | XI, 583 S. graph. Darst. |
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| publisher | Dekker |
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| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Corwin, Lawrence J. Verfasser aut Calculus in vector spaces Lawrence J. Corwin ; Robert H. Szczarba 2. ed. New York u.a. Dekker 1995 XI, 583 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 189 Addressing linear algebra from the basics to the spectral theorem and examining a host of topics in multivariable calculus, including differentiation, integration, maxima and minima, the inverse and implicit function theorems, and differential forms, this thoroughly revised Second Edition of an invaluable reference/text - widely successful through five printings - continues to provide unified, integrated coverage of the two fields Demonstrating that mathematics is a noncompartmentalized discipline of interrelated subjects, Calculus in Vector Spaces, Second Edition introduces the derivative as a linear transformation ... presents linear algebra in a concrete context based on complementary ideas in calculus ... explains differential forms on Euclidean space permitting Green's theorem, Gauss's theorem, and Stokes's theorem to be understood in a natural setting ... gives a new clarification of compactness as defined in terms of coverings and in terms of sequences ... supplies a novel treatment of eigenvalues and eigenvectors ... and more Cálculo Espaces vectoriels ram Calculus Vector spaces Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s Analysis (DE-588)4001865-9 s DE-604 Szczarba, Robert H. Verfasser aut Pure and applied mathematics 189 (DE-604)BV000001885 189 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006719144&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Corwin, Lawrence J. Szczarba, Robert H. Calculus in vector spaces Pure and applied mathematics Cálculo Espaces vectoriels ram Calculus Vector spaces Lineare Algebra (DE-588)4035811-2 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4035811-2 (DE-588)4001865-9 |
| title | Calculus in vector spaces |
| title_auth | Calculus in vector spaces |
| title_exact_search | Calculus in vector spaces |
| title_full | Calculus in vector spaces Lawrence J. Corwin ; Robert H. Szczarba |
| title_fullStr | Calculus in vector spaces Lawrence J. Corwin ; Robert H. Szczarba |
| title_full_unstemmed | Calculus in vector spaces Lawrence J. Corwin ; Robert H. Szczarba |
| title_short | Calculus in vector spaces |
| title_sort | calculus in vector spaces |
| topic | Cálculo Espaces vectoriels ram Calculus Vector spaces Lineare Algebra (DE-588)4035811-2 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Cálculo Espaces vectoriels Calculus Vector spaces Lineare Algebra Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006719144&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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