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

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, first published in 2002, will help students to 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 all 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, probability, complex analysis, abstract algebra, and more. An annotated bibliography then offers a guide to further reading and to more rigorous foundations. This book will be an essential resource for advanced undergraduate and beginning graduate students in mathematics, the physical sciences, engineering, computer science, statistics, and economics who need to quickly learn some serious mathematics. 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. On The Structure Of Mathematics -- Linear Algebra -- Real Analysis -- Differentiating Vector-valued Functions -- Point Set Topology -- Classical Stokes' Theorems -- Differential Forms And Stokes' Theorem -- Curvature For Curves And Surfaces -- Geometry -- Complex Analysis -- Countability And The Axiom Of Choice -- Algebra -- Lebesgue Integration -- Fourier Analysis -- Differential Equations -- Combinatorics And Probability Theory -- Algorithms -- Linear Algebra -- The Basic Vector Space R[superscript N] -- Vector Spaces And Linear Transformations -- Bases And Dimension -- The Determinant -- The Key Theorem Of Linear Algebra -- Similar Matrices -- Eigenvalues And Eigenvectors -- Dual Vector Spaces -- [epsilon] And [delta] Real Analysis -- Limits -- Continuity -- Differentiation -- Integration -- The Fundamental Theorem Of Calculus -- Pointwise Convergence Of Functions -- Uniform Convergence -- The Weierstrass M-test -- Weierstrass' Example -- Calculus For Vector-valued Functions -- Vector-valued Functions -- Limits And Continuity -- Differentiation And Jacobians -- The Inverse Function Theorem -- Implicit Function Theorem -- Point Set Topology -- The Standard Topology On R[superscript N] -- Metric Spaces -- Bases For Topologies -- Zariski Topology Of Commutative Rings -- Classical Stokes' Theorems -- Preliminaries About Vector Calculus -- Vector Fields -- Manifolds And Boundaries -- Path Integrals -- Surface Integrals -- The Gradient -- The Divergence -- The Curl -- Orientability. Thomas A. Garrity. Includes Bibliographical References (pages 329-337) And Index. Annotation 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 Though a bit of an exaggeration, it can be said that a mathematical problem can be solved only if it can be reduced to a calculation in linear algebra.