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Vector
Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 4th edition
Preface


Chapters 1 through 6 of this book cover
the standard topics in multivariate calculus and a first course in
linear algebra. The book can also be used for a course in analysis,
using the proofs in the appendix.
The organization and selection of material differs from the standard
approach in three ways, reflecting the following guiding
principles.
First,
we believe that at this level linear algebra should be more a
convenient setting
and language for multivariate calculus than a subject in its own right.
The
guiding principle of this unified approach is that locally,
a nonlinear function behaves like its derivative.
Thus whenever we have a question about a nonlinear function we will
answer it by looking carefully at a linear transformation:
its derivative. In this approach, everything learned about
linear algebra pays off twice: first for understanding linear
equations, then as a tool for understanding nonlinear equations.
We discuss
abstract vector spaces in section 2.6, but the emphasis is on R^{n},
as we believe that most students find it easiest to move from the
concrete to the abstract.
Second, we
emphasize computationally
effective algorithms, and prove theorems by showing that these
algorithms work.
We feel this better reflects the way this mathematics is
used today, in both applied and pure mathematics. Moreover, it can be
done with no loss of rigor.
For linear equations, row reduction is the central
tool from which everything else follows; we use row reduction
to prove all the standard results about dimension and rank.
For nonlinear equations, the cornerstone is Newton's method,
the best and most widely used method for solving nonlinear equations;
we use it both as a computational tool and in proving the
inverse and implicit function theorems. We include
a section on numerical methods of integration, and we encourage the use
of computers both to reduce tedious calculations and as an
aid in visualizing curves and surfaces.
Third, we
use differential forms to generalize the fundamental theorem of
calculus to higher
dimensions.
The great conceptual simplifications gained by doing
electromagnetism in the language of forms is a central
motivation for using forms. We apply the language of forms to
electromagnetism and potentials in section 6.11 and 6.12, which are expanded in this fourth edition.
Complete preface (in pdf)
Return
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Forms"
Review of
2nd edition from the Mathematical Association of America Monthly
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