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

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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…

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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 .
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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.

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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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