Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds Theodore Shifrin
I began writing this text as I taught a brand-new course combining linear algebra and a rigorous approach to multivariable calculus. My goal was to include all the standard computational material found in the usual linear algebra and multivariable calculus courses and more, interweaving the material as effectively as possible, and include complete proofs. I wanted to integrate the material so as to emphasize the recurring theme of implicit versus explicit that persists in linear algebra and analysis. In every linear algebra course we should learn how to go back and forth between a system of equations and a parametrization of its solution set.
The prerequisites for this book are a solid background in single-variable calculus and, if not some experience writing proofs, a strong interest in grappling with them. I have included plenty of examples, clear proofs, and significant motivation for the crucial concepts. I have provided numerous exercises of varying levels of difficulty. The exercises are arranged in order of increasing difficulty… Comments on contents The linear algebraic material with which we begin the course in Chapter 1 is concrete, establishes the link with geometry, and is good self-contained setting for working on proofs. We introduce vectors, dot products, subspaces, and linear transformations and matrix computation. In Chapter 2 we begin to make the transition to calculus, introducing scalar functions of a vector variable - their graphs and their level sets - and vector-valued functions. We come to concepts of differential calculus in Chapter 3. In the first four sections of Chapter 4 we give an accelerated treatment of Guassian elimination and the theory of linear systems, the standard material on linear independence and dimension, and the four fundamental subspaces associated to a matrix. Chapter 5 is a blend of topology, calculus, and linear algebra - quadratic forms and projections. Chapter 6 is a brief, but sophisticated, introduction to the inverse and implicit function theorems. In Chapter 7 we study the multidimensional (Riemann) integral. In Chapter 8 we start by laying the groundwork for the analogous multidimensional result. In Chapter 9 we complete our study of linear algebra.