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

by Thomas A. Garrity , Illustrated by Lori Pedersen

• ISBN13: 9780521792851
• ISBN10: 0521792851
• Format: Hardcover
• Copyright: 2001-11-26
• Publisher: Cambridge University Press

Summary

Few beginning graduate students in mathematics and other quantitative subjects possess the daunting breadth of mathematical knowledge expected of them when they begin their studies. This book will offer students a broad outline of essential mathematics and will help to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential and analytical geometry, real analysis, point-set topology, probability, complex analysis, set theory, algorithms, and more. An annotated bibliography offers a guide to further reading and to more rigorous foundations.

Author Biography

Thomas A. Garrity is Professor of Mathematics at Williams College in Williamstown, Massachusetts. He was an undergraduate at the University of Texas, Austin, and a graduate student at Brown University, receiving his Ph.D. in 1986. From 1986 to 1989, he was G.C. Evans Instructor at Rice University. In 1989, he moved to Williams College, where he has been ever since except in 1992-3, when he spent the year at the University of Washington, and in 2000-1, when he spent the year at the University of Michigan, Ann Arbor

Table of Contents

Preface	xiii
On the Structure of Mathematics	xix
Brief Summaries of Topics	xxiii
Linear Algebra	xxiii
Real Analysis	xxiii
Differentiating Vector-Valued Functions	xxiii
Point Set Topology	xxiv
Classical Stokes' Theorems	xxiv
Differential Forms and Stokes' Theorem	xxiv
Curvature for Curves and Surfaces	xxiv
Geometry	xxv
Complex Analysis	xxv
Countability and the Axiom of Choice	xxvi
Algebra	xxvi
Lebesgue Integration	xxvi
Fourier Analysis	xxvi
Differential Equations	xxvii
Combinatorics and Probability Theory	xxvii
Algorithms	xxvii
Linear Algebra	1	(22)
Introduction	1	(1)
The Basic Vector Space Rn	2	(2)
Vector Spaces and Linear Transformations	4	(2)
Bases and Dimension	6	(3)
The Determinant	9	(3)
The Key Theorem of Linear Algebra	12	(2)
Similar Matrices	14	(1)
Eigenvalues and Eigenvectors	15	(5)
Dual Vector Spaces	20	(1)
Books	21	(1)
Exercises	21	(2)
&epsis; and δ Real Analysis	23	(24)
Limits	23	(2)
Continuity	25	(1)
Differentiation	26	(2)
Integration	28	(3)
The Fundamental Theorem of Calculus	31	(4)
Pointwise Convergence of Functions	35	(1)
Uniform Convergence	36	(2)
The Weierstrass M-Test	38	(2)
Weierstrass' Example	40	(3)
Books	43	(1)
Exercises	44	(3)
Calculus for Vector-Valued Functions	47	(16)
Vector-Valued Functions	47	(2)
Limits and Continuity	49	(1)
Differentiation and Jacobians	50	(3)
The Inverse Function Theorem	53	(3)
Implicit Function Theorem	56	(4)
Books	60	(1)
Exercises	60	(3)
Point Set Topology	63	(18)
Basic Definitions	63	(3)
The Standard Topology on Rn	66	(6)
Metric Spaces	72	(1)
Bases for Topologies	73	(2)
Zariski Topology of Commutative Rings	75	(2)
Books	77	(1)
Exercises	78	(3)
Classical Stokes' Theorems	81	(30)
Preliminaries about Vector Calculus	82	(13)
Vector Fields	82	(2)
Manifolds and Boundaries	84	(3)
Path Integrals	87	(4)
Surface Integrals	91	(2)
The Gradient	93	(1)
The Divergence	93	(1)
The Curl	94	(1)
Orientability	94	(1)
The Divergence Theorem and Stokes' Theorem	95	(2)
Physical Interpretation of Divergence Thm.	97	(1)
A Physical Interpretation of Stokes' Theorem	98	(1)
Proof of the Divergence Theorem	99	(5)
Sketch of a Proof for Stokes' Theorem	104	(4)
Books	108	(1)
Exercises	108	(3)
Differential Forms and Stokes' Thm.	111	(34)
Volumes of Parallelepipeds	112	(3)
Diff. Forms and the Exterior Derivative	115	(9)
Elementary k-forms	115	(3)
The Vector Space of k-forms	118	(1)
Rules for Manipulating k-forms	119	(3)
Differential k-forms and the Exterior Derivative	122	(2)
Differential Forms and Vector Fields	124	(2)
Manifolds	126	(6)
Tangent Spaces and Orientations	132	(5)
Tangent Spaces for Implicit and Parametric Manifolds	132	(1)
Tangent Spaces for Abstract Manifolds	133	(2)
Orientation of a Vector Space	135	(1)
Orientation of a Manifold and its Boundary	136	(1)
Integration on Manifolds	137	(2)
Stokes' Theorem	139	(3)
Books	142	(1)
Exercises	143	(2)
Curvature for Curves and Surfaces	145	(16)
Plane Curves	145	(3)
Space Curves	148	(4)
Surfaces	152	(5)
The Gauss-Bonnet Theorem	157	(1)
Books	158	(1)
Exercises	158	(3)
Geometry	161	(10)
Euclidean Geometry	162	(1)
Hyperbolic Geometry	163	(3)
Elliptic Geometry	166	(1)
Curvature	167	(1)
Books	168	(1)
Exercises	169	(2)
Complex Analysis	171	(30)
Analyticity as a Limit	172	(2)
Cauchy-Riemann Equations	174	(5)
Integral Representations of Functions	179	(8)
Analytic Functions as Power Series	187	(4)
Conformal Maps	191	(3)
The Riemann Mapping Theorem	194	(2)
Several Complex Variables: Hartog's Theorem	196	(1)
Books	197	(1)
Exercises	198	(3)
Countability and the Axiom of Choice	201	(12)
Countability	201	(4)
Naive Set Theory and Paradoxes	205	(2)
The Axiom of Choice	207	(1)
Non-measureable Sets	208	(2)
Godel and Independence Proofs	210	(1)
Books	211	(1)
Exercises	211	(2)
Algebra	213	(18)
Groups	213	(6)
Representation Theory	219	(2)
Rings	221	(2)
Fields and Galois Theory	223	(5)
Books	228	(1)
Exercises	229	(2)
Lebesgue Integration	231	(12)
Lebesgue Measure	231	(3)
The Cantor Set	234	(2)
Lebesgue Integration	236	(3)
Convergence Theorems	239	(2)
Books	241	(1)
Exercises	241	(2)
Fourier Analysis	243	(18)
Waves, Periodic Functions and Trigonometry	243	(1)
Fourier Series	244	(6)
Convergence Issues	250	(2)
Fourier Integrals and Transforms	252	(4)
Solving Differential Equations	256	(2)
Books	258	(1)
Exercises	258	(3)
Differential Equations	261	(24)
Basics	261	(1)
Ordinary Differential Equations	262	(4)
The Laplacian	266	(4)
Mean Value Principle	266	(1)
Separation of Variables	267	(3)
Applications to Complex Analysis	270	(1)
The Heat Equation	270	(3)
The Wave Equation	273	(6)
Derivation	273	(4)
Change of Variables	277	(2)
Integrability Conditions	279	(2)
Lewy's Example	281	(1)
Books	282	(1)
Exercises	282	(3)
Combinatorics and Probability	285	(22)
Counting	285	(2)
Basic Probability Theory	287	(3)
Independence	290	(1)
Expected Values and Variance	291	(3)
Central Limit Theorem	294	(6)
Stirling's Approximation for n!	300	(5)
Books	305	(1)
Exercises	305	(2)
Algorithms	307	(20)
Algorithms and Complexity	308	(1)
Graphs: Euler and Hamiltonian Circuits	308	(5)
Sorting and Trees	313	(3)
P = NP?	316	(1)
Numerical Analysis: Newton's Method	317	(7)
Books	324	(1)
Exercises	324	(3)
A Equivalence Relations	327