Calculus in vector spaces:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Dekker
1979
|
| Series: | Pure and applied mathematics
52 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 782 S. graph. Darst. |
| ISBN: | 0824768329 |
Staff View
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| 100 | 1 | |a Corwin, Lawrence J. |d 1943-1992 |e Verfasser |0 (DE-588)119481731 |4 aut | |
| 245 | 1 | 0 | |a Calculus in vector spaces |c Lawrence J. Corwin ; Robert H. Szczarba |
| 264 | 1 | |a New York [u.a.] |b Dekker |c 1979 | |
| 300 | |a X, 782 S. |b graph. Darst. | ||
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| 490 | 1 | |a Pure and applied mathematics |v 52 | |
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| 650 | 4 | |a Calcul infinitésimal | |
| 650 | 7 | |a Calcul |2 ram | |
| 650 | 4 | |a Espaces vectoriels | |
| 650 | 7 | |a Espaces vectoriels |2 ram | |
| 650 | 7 | |a Exercice analyse |2 Jussieu | |
| 650 | 7 | |a Fonction vectorielle |2 Jussieu | |
| 650 | 7 | |a Théorie spectrale |2 Jussieu | |
| 650 | 7 | |a Théorème Stokes |2 Jussieu | |
| 650 | 7 | |a Vectorruimten |2 gtt | |
| 650 | 4 | |a Calculus | |
| 650 | 4 | |a Vector spaces | |
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Record in the Search Index
| _version_ | 1804116704158023680 |
|---|---|
| adam_text | CONTENTS
PREFACE v
CHAPTER 1. SOME PRELIMINARIES 1
1. The Rudiments of Set Theory 1
2. Some Logic 8
3. Mathematical Induction 12
4. Inequalities and Absolute Value 16
5. Equivalence Relations 20
CHAPTER 2. VECTOR SPACES 23
1. The Cartesian Plane 23
2. The Definition of a Vector Space 29
3. Some Elementary Properties of Vector Spaces 37
4. Subspaces 40
5. Linear Transformations 49
6. Linear Transformations on Euclidean Spaces 58
CHAPTER 3. THE DERIVATIVE 71
1. Normed Vector Spaces 72
2. Open and Closed Sets 76
3. Continuous Functions Between Normed Vector Spaces 83
4. Elementary Properties of Continuous Functions 95
5. The Derivative 99
6. Elementary Properties of the Derivative 107
7. Partial Derivatives and the Jacobian Matrix 110
CHAPTER 4. THE STRUCTURE OF VECTOR SPACES 119
1. Spans and Linear Independence 120
2. Bases 128
3. Bases and Linear Transformations 133
vii
viii CONTENTS
4. The Dimension of a Vector Space 139
5. Inner Product Spaces 147
6. The Norm on an Inner Product Space 153
7. Orthonormal Bases , 157
8. The Cross Product in R 166
CHAPTER 5. COMPACT AND CONNECTED SETS 171
1. Convergent Sequences 171
2. Further Properties of Continuous Functions 179
3. Compact Sets 185
4. Upper and Lower Bounds 189
5. Continuous Functions on Compact Sets 196
6. A Characterization of Compact Sets 200
7. Uniform Continuity 205
8. Connected Sets 209
CHAPTER 6. THE CHAIN RULE, HIGHER DERIVATIVES, AND TAYLOR S
THEOREM 215
1. The Chain Rule 216
2. Proof of the Chain Rule 224
3. Higher Derivatives 229
4. Taylor s Theorem for Functions of One Variable 239
5. Taylor s Theorem for Functions of Two Variables 245
6. Taylor s Theorem for Functions of n Variables 251
7. A Sufficient Condition for Differentiability 257
8. The Equality of Mixed Partial Derivatives 263
CHAPTER 7. LINEAR TRANSFORMATIONS AND MATRICES 267
1. The Matrix of a Linear Transformation 268
2. Isomorphisms and Invertible Matrices 272
3. Change of Basis 278
4. The Rank of a Matrix 285
5. The Trace and Adjoint of a Linear Transformation 290
6. Row and Column Operations 300
7. Gaussian Elimination 307
CHAPTER 8. MAXIMA AND MINIMA 315
1. Maxima and Minima at Interior Points 316
2. Quadratic Forms 324
3. Criteria for Local Maxima and Minima 332
4. Constrained Maxima and Minima: I 341
5. The Method of Lagrange Multipliers 348
6. Constrained Maxima and Minima: II 355
7. The Proof of Proposition 2.3 362
CONTENTS ix
CHAPTER 9. THE INVERSE AND IMPLICIT FUNCTION THEOREMS 367
1. The Inverse Function Theorem 367
2. The Proof of Theorem 1.3 376
3. The Proof of the General Inverse Function Theorem 380
4. The Implicit Function Theorem: I 388
5. The Implicit Function Theorem: II 394
CHAPTER 10. THE SPECTRAL THEOREM 403
1. Complex Numbers 404
2. Complex Vector Spaces 409
3. Determinants 417
4. Properties of the Determinant 424
5. Eigenvectors and Eigenvalues 434
6. The Spectral Theorem 442
7. The Spectral Theorem for Normal Linear Transformations 447
8. Quadratic Forms 451
9. Another Proof of the Spectral Theorem 458
CHAPTER 11. INTEGRATION 463
1. Integration of Functions of One Variable 464
2. Properties of the Integral 475
3. The Integral of a Function of Two Variables 484
4. The Integral of a Function of n Variables 494
5. Properties of the Integral 504
6. Integrable Functions 510
7. The Proof of Theorem 6.2 516
CHAPTER 12. ITERATED INTEGRALS AND THE FUBINI THEOREM 521
1. The Fubini Theorem 521
2. Integrals Over Non Rectangular Regions 530
3. More Examples 537
4. The Proof of Fubini s Theorem 547
5. Differentiating Under the Integral Sign 551
6. The Change of Variable Formula 554
7. The Proof of Theorem 6.2 562
CHAPTER 13. LINE INTEGRALS 573
1. Curves 573
2. Line Integrals of a Function 578
3. Line Integrals of Vector Fields 586
4. Conservative Vector Fields 595
5. Green s Theorem 603
6. The Proof of Green s Theorem 610
x CONTENTS
CHAPTER 14. SURFACE INTEGRALS 617
1. Surfaces 617
2. Surface Area 627
3. Surface Integrals 637
4. Stokes Theorem 645
CHAPTER 15. DIFFERENTIAL FORMS 651
1. The Algebra of Differential Forms 651
2. Basic Properties of the Sum and Product of Forms 658
3. The Exterior Differential 662
4. Basic Properties of the Exterior Differential 669
5. The Action of Differentiable Functions on Forms 672
6. Basic Properties of the Induced Mapping 677
CHAPTER 16. INTEGRATION OF DIFFERENTIAL FORMS 683
1. Integration of Forms 683
2. The General Stokes Theorem 688
3. Green s Theorem and Stokes Theorem 693
4. The Gauss Theorem and Incompressible Fluids 696
5. Proof of the General Stokes Theorem 703
APPENDIX 1. THE EXISTENCE OF DETERMINANTS 707
APPENDIX 2. THE FUNDAMENTAL THEOREM OF ALGEBRA 713
APPENDIX 3. JORDAN CANONICAL FORM 717
1. Generalized Eigenvalues 717
2. The Jordan Canonical Form 721
3. Polynomials and Linear Transformations 730
4. The Proof of Theorem 3.5 737
5. The Proof of Theorem 2.2 742
SOLUTIONS OF SELECTED EXERCISES 747
INDEX 777
|
| any_adam_object | 1 |
| author | Corwin, Lawrence J. 1943-1992 Szczarba, Robert Henry 1932- |
| author_GND | (DE-588)119481731 (DE-588)172411327 |
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| author_sort | Corwin, Lawrence J. 1943-1992 |
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| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV002248111 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:42:44Z |
| institution | BVB |
| isbn | 0824768329 |
| language | English |
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| spelling | Corwin, Lawrence J. 1943-1992 Verfasser (DE-588)119481731 aut Calculus in vector spaces Lawrence J. Corwin ; Robert H. Szczarba New York [u.a.] Dekker 1979 X, 782 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 52 Algèbre linéaire Jussieu Analyse (wiskunde) gtt Calcul différentiel extérieur Jussieu Calcul infinitésimal Calcul ram Espaces vectoriels Espaces vectoriels ram Exercice analyse Jussieu Fonction vectorielle Jussieu Théorie spectrale Jussieu Théorème Stokes Jussieu Vectorruimten gtt Calculus Vector spaces Analysis (DE-588)4001865-9 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s Analysis (DE-588)4001865-9 s DE-604 Szczarba, Robert Henry 1932- Verfasser (DE-588)172411327 aut Pure and applied mathematics 52 (DE-604)BV000001885 52 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001477299&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Corwin, Lawrence J. 1943-1992 Szczarba, Robert Henry 1932- Calculus in vector spaces Pure and applied mathematics Algèbre linéaire Jussieu Analyse (wiskunde) gtt Calcul différentiel extérieur Jussieu Calcul infinitésimal Calcul ram Espaces vectoriels Espaces vectoriels ram Exercice analyse Jussieu Fonction vectorielle Jussieu Théorie spectrale Jussieu Théorème Stokes Jussieu Vectorruimten gtt Calculus Vector spaces Analysis (DE-588)4001865-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4035811-2 |
| 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 | Algèbre linéaire Jussieu Analyse (wiskunde) gtt Calcul différentiel extérieur Jussieu Calcul infinitésimal Calcul ram Espaces vectoriels Espaces vectoriels ram Exercice analyse Jussieu Fonction vectorielle Jussieu Théorie spectrale Jussieu Théorème Stokes Jussieu Vectorruimten gtt Calculus Vector spaces Analysis (DE-588)4001865-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Algèbre linéaire Analyse (wiskunde) Calcul différentiel extérieur Calcul infinitésimal Calcul Espaces vectoriels Exercice analyse Fonction vectorielle Théorie spectrale Théorème Stokes Vectorruimten Calculus Vector spaces Analysis Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001477299&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000001885 |
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