All the Mathematics You Missed: But Need to Know for Graduate School

Contents vii......Page f007.djvu
Preface xiii......Page f013.djvu
On the Structure of Mathematics xix......Page f019.djvu
0.3 Differentiating Vector-Valued Functions xxiii......Page f023.djvu
0.7 Curvature for Curves and Surfaces xxiv......Page f024.djvu
0.9 Complex Analysis xxv......Page f025.djvu
0.13 Fourier Analysis xxvi......Page f026.djvu
0.16 Algorithms xxvii......Page f027.djvu
1.1 Introduction 1......Page p001.djvu
1.2 The Basic Vector Space R^n 2......Page p002.djvu
1.3 Vector Spaces and Linear Transformations 4......Page p004.djvu
1.4 Bases and Dimension 6......Page p006.djvu
1.5 The Determinant 9......Page p009.djvu
1.6 The Key Theorem of Linear Algebra 12......Page p012.djvu
1.7 Similar Matrices 14......Page p014.djvu
1.8 Eigenvalues and Eigenvectors 15......Page p015.djvu
1.9 Dual Vector Spaces 20......Page p020.djvu
1.11 Exercises 21......Page p021.djvu
2.1 Limits 23......Page p023.djvu
2.2 Continuity 25......Page p025.djvu
2.3 Differentiation 26......Page p026.djvu
2.4 Integration 28......Page p028.djvu
2.5 The Fundamental Theorem of Calculus 31......Page p031.djvu
2.6 Pointwise Convergence of Functions 35......Page p035.djvu
2.7 Uniform Convergence 36......Page p036.djvu
2.8 The Weierstrass M-Test 38......Page p038.djvu
2.9 Weierstrass\' Example 40......Page p040.djvu
2.10 Books 43......Page p043.djvu
2.11 Exercises 44......Page p044.djvu
3.1 Vector-Valued Functions 47......Page p047.djvu
3.2 Limits and Continuity 49......Page p049.djvu
3.3 Differentiation and Jacobians 50......Page p050.djvu
3.4 The Inverse Function Theorem 53......Page p053.djvu
3.5 Implicit Function Theorem 56......Page p056.djvu
3.7 Exercises 60......Page p060.djvu
4.1 Basic Definitions 63......Page p063.djvu
4.2 The Standard Topology on R^n 66......Page p066.djvu
4.3 Metric Spaces 72......Page p072.djvu
4.4 Bases for Topologies 73......Page p073.djvu
4.5 Zariski Topology of Commutative Rings 75......Page p075.djvu
4.6 Books 77......Page p077.djvu
4.7 Exercises 78......Page p078.djvu
5 Classical Stokes\' Theorems 81......Page p081.djvu
5.1.1 Vector Fields 82......Page p082.djvu
5.1.2 Manifolds and Boundaries 84......Page p084.djvu
5.1.3 Path Integrals 87......Page p087.djvu
5.1.4 Surface Integrals 91......Page p091.djvu
5.1.6 The Divergence 93......Page p093.djvu
5.1.8 Orientability 94......Page p094.djvu
5.2 The Divergence Theorem and Stokes\' Theorem 95......Page p095.djvu
5.3 Physical Interpretation of Divergence Thm. 97......Page p097.djvu
5.4 A Physical Interpretation of Stokes\' Theorem 98......Page p098.djvu
5.5 Proof of the Divergence Theorem 99......Page p099.djvu
5.6 Sketch of a Proof for Stokes\' Theorem 104......Page p104.djvu
5.8 Exercises 108......Page p108.djvu
6 Differential Forms and Stokes\' Thm. 111......Page p111.djvu
6.1 Volumes of Parallelepipeds 112......Page p112.djvu
6.2.1 Elementary k-forms 115......Page p115.djvu
6.2.2 The Vector Space of k-forms 118......Page p118.djvu
6.2.3 Rules for Manipulating k-forms 119......Page p119.djvu
6.2.4 Differential k-forms and the Exterior Derivative 122......Page p122.djvu
6.3 Differential Forms and Vector Fields 124......Page p124.djvu
6.4 Manifolds 126......Page p126.djvu
6.5.1 Tangent Spaces for Implicit and Parametric Manifolds 132......Page p132.djvu
6.5.2 Tangent Spaces for Abstract Manifolds 133......Page p133.djvu
6.5.3 Orientation of a Vector Space 135......Page p135.djvu
6.5.4 Orientation of a Manifold and its Boundary 136......Page p136.djvu
6.6 Integration on Manifolds 137......Page p137.djvu
6.7 Stokes\' Theorem 139......Page p139.djvu
6.8 Books 142......Page p142.djvu
6.9 Exercises 143......Page p143.djvu
7.1 Plane Curves 145......Page p145.djvu
7.2 Space Curves 148......Page p148.djvu
7.3 Surfaces 152......Page p152.djvu
7.4 The Gauss-Bonnet Theorem 157......Page p157.djvu
7.6 Exercises 158......Page p158.djvu
8 Geometry 161......Page p161.djvu
8.1 Euclidean Geometry 162......Page p162.djvu
8.2 Hyperbolic Geometry 163......Page p163.djvu
8.3 Elliptic Geometry 166......Page p166.djvu
8.4 Curvature 167......Page p167.djvu
8.5 Books 168......Page p168.djvu
8.6 Exercises 169......Page p169.djvu
9 Complex Analysis 171......Page p171.djvu
9.1 Analyticity as a Limit 172......Page p172.djvu
9.2 Cauchy-Riemann Equations 174......Page p174.djvu
9.3 Integral Representations of Functions 179......Page p179.djvu
9.4 Analytic Functions as Power Series 187......Page p187.djvu
9.5 Conformal Maps 191......Page p191.djvu
9.6 The Riemann Mapping Theorem 194......Page p194.djvu
9.7 Several Complex Variables: Hartog\'s Theorem 196......Page p196.djvu
9.8 Books 197......Page p197.djvu
9.9 Exercises 198......Page p198.djvu
10.1 Countability 201......Page p201.djvu
10.2 Naive Set Theory and Paradoxes 205......Page p205.djvu
10.3 The Axiom of Choice 207......Page p207.djvu
10.4 Non-measurable Sets 208......Page p208.djvu
10.5 Gödel and Independence Proofs 210......Page p210.djvu
10.7 Exercises 211......Page p211.djvu
11.1 Groups 213......Page p213.djvu
11.2 Representation Theory 219......Page p219.djvu
11.3 Rings 221......Page p221.djvu
11.4 Fields and Galois Theory 223......Page p223.djvu
11.5 Books 228......Page p228.djvu
11.6 Exercises 229......Page p229.djvu
12.1 Lebesgue Measure 231......Page p231.djvu
12.2 The Cantor Set 234......Page p234.djvu
12.3 Lebesgue Integration 236......Page p236.djvu
12.4 Convergence Theorems 239......Page p239.djvu
12.6 Exercises 241......Page p241.djvu
13.1 Waves, Periodic Functions and Trigonometry 243......Page p243.djvu
13.2 Fourier Series 244......Page p244.djvu
13.3 Convergence Issues 250......Page p250.djvu
13.4 Fourier Integrals and Transforms 252......Page p252.djvu
13.5 Solving Differential Equations 256......Page p256.djvu
13.7 Exercises 258......Page p258.djvu
14.1 Basics 261......Page p261.djvu
14.2 Ordinary Differential Equations 262......Page p262.djvu
14.3.1 Mean Value Principle 266......Page p266.djvu
14.3.2 Separation of Variables 267......Page p267.djvu
14.4 The Heat Equation 270......Page p270.djvu
14.5.1 Derivation 273......Page p273.djvu
14.5.2 Change of Variables 277......Page p277.djvu
14.6 Integrability Conditions 279......Page p279.djvu
14.7 Lewy\'s Example 281......Page p281.djvu
14.9 Exercises 282......Page p282.djvu
15.1 Counting 285......Page p285.djvu
15.2 Basic Probability Theory 287......Page p287.djvu
15.3 Independence 290......Page p290.djvu
15.4 Expected Values and Variance 291......Page p291.djvu
15.5 Central Limit Theorem 294......Page p294.djvu
15.6 Stirling\'s Approximation for n! 300......Page p300.djvu
15.8 Exercises 305......Page p305.djvu
16 Algorithms 307......Page p307.djvu
16.2 Graphs: Euler and Hamiltonian Circuits 308......Page p308.djvu
16.3 Sorting and Trees 313......Page p313.djvu
16.4 P=NP? 316......Page p316.djvu
16.5 Numerical Analysis: Newton\'s Method 317......Page p317.djvu
16.7 Exercises 324......Page p324.djvu
A Equivalence Relations 327......Page p327.djvu
Bibliography 329......Page p329.djvu
Index 338......Page p338.djvu