All the mathematics you missed : but need to know for graduate school / Thomas A. Garrity.

Format:

Book

Author/Creator:

Garrity, Thomas A., 1959-

Language:

English

Subjects (All):

Mathematics.

Physical Description:

xxvii, 347 pages : illustrations ; 24 cm

Place of Publication:

Cambridge, UK ; New York, NY : Cambridge University Press, 2002.

Summary:

Here is an essential resource for advanced undergraduate and beginning graduate students in quantitative subjects who need to quickly learn some serious mathematics.

Contents:

On the Structure of Mathematics xix

0.1 Linear Algebra xxiii

0.2 Real Analysis xxiii

0.3 Differentiating Vector-Valued Functions xxiii

0.4 Point Set Topology xxiv

0.5 Classical Stokes' Theorems xxiv

0.6 Differential Forms and Stokes' Theorem xxiv

0.7 Curvature for Curves and Surfaces xxiv

0.8 Geometry xxv

0.9 Complex Analysis xxv

0.10 Countability and the Axiom of Choice xxvi

0.11 Algebra xxvi

0.12 Lebesgue Integration xxvi

0.13 Fourier Analysis xxvi

0.14 Differential Equations xxvii

0.15 Combinatorics and Probability Theory xxvii

0.16 Algorithms xxvii

1 Linear Algebra 1

1.2 The Basic Vector Space R[superscript n] 2

1.3 Vector Spaces and Linear Transformations 4

1.4 Bases and Dimension 6

1.5 The Determinant 9

1.6 The Key Theorem of Linear Algebra 12

1.7 Similar Matrices 14

1.8 Eigenvalues and Eigenvectors 15

1.9 Dual Vector Spaces 20

2 [epsilon] and [delta] Real Analysis 23

2.1 Limits 23

2.2 Continuity 25

2.3 Differentiation 26

2.4 Integration 28

2.5 The Fundamental Theorem of Calculus 31

2.6 Pointwise Convergence of Functions 35

2.7 Uniform Convergence 36

2.8 The Weierstrass M-Test 38

2.9 Weierstrass' Example 40

3 Calculus for Vector-Valued Functions 47

3.1 Vector-Valued Functions 47

3.2 Limits and Continuity 49

3.3 Differentiation and Jacobians 50

3.4 The Inverse Function Theorem 53

3.5 Implicit Function Theorem 56

4 Point Set Topology 63

4.2 The Standard Topology on R[superscript n] 66

4.3 Metric Spaces 72

4.4 Bases for Topologies 73

4.5 Zariski Topology of Commutative Rings 75

5 Classical Stokes' Theorems 81

5.1 Preliminaries about Vector Calculus 82

5.1.1 Vector Fields 82

5.1.2 Manifolds and Boundaries 84

5.1.3 Path Integrals 87

5.1.4 Surface Integrals 91

5.1.5 The Gradient 93

5.1.6 The Divergence 93

5.1.7 The Curl 94

5.1.8 Orientability 94

5.2 The Divergence Theorem and Stokes' Theorem 95

5.3 Physical Interpretation of Divergence Thm. 97

5.4 A Physical Interpretation of Stokes' Theorem 98

5.5 Proof of the Divergence Theorem 99

5.6 Sketch of a Proof for Stokes' Theorem 104

6 Differential Forms and Stokes' Thm. 111

6.1 Volumes of Parallelepipeds 112

6.2 Diff. Forms and the Exterior Derivative 115

6.2.1 Elementary [kappa]-forms 115

6.2.2 The Vector Space of [kappa]-forms 118

6.2.3 Rules for Manipulating [kappa]-forms 119

6.2.4 Differential [kappa]-forms and the Exterior Derivative 122

6.3 Differential Forms and Vector Fields 124

6.4 Manifolds 126

6.5 Tangent Spaces and Orientations 132

6.5.1 Tangent Spaces for Implicit and Parametric Manifolds 132

6.5.2 Tangent Spaces for Abstract Manifolds 133

6.5.3 Orientation of a Vector Space 135

6.5.4 Orientation of a Manifold and its Boundary 136

6.6 Integration on Manifolds 137

6.7 Stokes' Theorem 139

7 Curvature for Curves and Surfaces 145

7.1 Plane Curves 145

7.2 Space Curves 148

7.3 Surfaces 152

7.4 The Gauss-Bonnet Theorem 157

8 Geometry 161

8.1 Euclidean Geometry 162

8.2 Hyperbolic Geometry 163

8.3 Elliptic Geometry 166

8.4 Curvature 167

9 Complex Analysis 171

9.1 Analyticity as a Limit 172

9.2 Cauchy-Riemann Equations 174

9.3 Integral Representations of Functions 179

9.4 Analytic Functions as Power Series 187

9.5 Conformal Maps 191

9.6 The Riemann Mapping Theorem 194

9.7 Several Complex Variables: Hartog's Theorem 196

10 Countability and the Axiom of Choice 201

10.1 Countability 201

10.2 Naive Set Theory and Paradoxes 205

10.3 The Axiom of Choice 207

10.4 Non-measurable Sets 208

10.5 Godel and Independence Proofs 210

11 Algebra 213

11.1 Groups 213

11.2 Representation Theory 219

11.3 Rings 221

11.4 Fields and Galois Theory 223

12 Lebesgue Integration 231

12.1 Lebesgue Measure 231

12.2 The Cantor Set 234

12.3 Lebesgue Integration 236

12.4 Convergence Theorems 239

13 Fourier Analysis 243

13.1 Waves, Periodic Functions and Trigonometry 243

13.2 Fourier Series 244

13.3 Convergence Issues 250

13.4 Fourier Integrals and Transforms 252

13.5 Solving Differential Equations 256

14 Differential Equations 261

14.2 Ordinary Differential Equations 262

14.3 The Laplacian 266

14.3.1 Mean Value Principle 266

14.3.2 Separation of Variables 267

14.3.3 Applications to Complex Analysis 270

14.4 The Heat Equation 270

14.5 The Wave Equation 273

14.5.1 Derivation 273

14.5.2 Change of Variables 277

14.6 Integrability Conditions 279

14.7 Lewy's Example 281

15 Combinatorics and Probability 285

15.1 Counting 285

15.2 Basic Probability Theory 287

15.3 Independence 290

15.4 Expected Values and Variance 291

15.5 Central Limit Theorem 294

15.6 Stirling's Approximation for n! 300

16 Algorithms 307

16.1 Algorithms and Complexity 308

16.2 Graphs: Euler and Hamiltonian Circuits 308

16.3 Sorting and Trees 313

16.4 P=NP? 316

16.5 Numerical Analysis: Newton's Method 317

A Equivalence Relations 327.

Notes:

Includes bibliographical references (pages [329]-337) and index.

ISBN:

0521797071

OCLC:

47240703