All the Mathematics You Missed but Need to Know for Graduate School

17996 WordsJun 24, 201272 Pages
All the Mathematics You Missed Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge, but few have such a background. This book will help students see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains the basic points and a few key results of the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential geometry, real analysis, point-set topology, differential equations, probability theory, complex analysis, abstract algebra, and more. An annotated bibliography offers a guide to further reading and…show more content…
. . . . 2 and J Real Analysis 2.1 Limits . . . . . 2.2 Continuity... 2.3 Differentiation 2.4 Integration .. 2.5 The Fundamental Theorem of Calculus. 2.6 Pointwise Convergence of Functions 2.7 Uniform Convergence . 2.8 The Weierstrass M-Test 2.9 Weierstrass ' Example. 2.10 Books .. 2.11 Exercises . E 20 21 21 23 23 25 26 28 31 35 36 38 40 43 44 47 47 3 Calculus for Vector-Valued Functions 3.1 Vector-Valued Functions . . . 3.2 Limits and Continuity . . . . . 3.3 Differentiation and Jacobians . 3.4 The Inverse Function Theorem 3.5 Implicit Function Theorem 3.6 Books .. 3.7 Exercises . . . . Point Set Topology 4.1 Basic Definitions . 4.2 The Standard Topology on R n 4.3 Metric Spaces . . . . . . . . . . 4.4 Bases for Topologies . . . . . . 4.5 Zariski Topology of Commutative Rings 4.6 Books .. 4.7 Exercises . Classical Stokes ' Theorems 5.1 Preliminaries about Vector Calculus 5.1.1 Vector Fields . 5.1.2 Manifolds and Boundaries. 5.1.3 Path Integrals .. 5.1.4 Surface Integrals 5.1.5 The Gradient .. 5.1.6 The Divergence. 49 50 53 56 60 60 63 63 66 72 73 75 77 78 81 82 82 4 5 84 87 91 93 93 CONTENTS 5.1.7 The Curl . 5.1.8 Orientability . 5.2 The Divergence Theorem and Stokes ' Theorem 5.3 Physical Interpretation of Divergence Thm. . 5.4 A Physical Interpretation of Stokes ' Theorem 5.5 Proof of the Divergence Theorem . . . 5.6 Sketch of a Proof for Stokes ' Theorem 5.7 Books .. 5.8 Exercises . 6 Differential Forms and Stokes '

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