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Multivariable
Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds Theodore Shifrin |
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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. |
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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. |