Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

by John H. Hubbard, Barbara Burke Hubbard

This book covers most of the standard topics in multivariate calculus, and a
substantial part of a standard first course in linear algebra. The teacher may
find the organization rather less standard.
There are three guiding principles which led to our organizing the material
as we did. One is that at this level linear algebra should be more a convenient
setting and language for multivariate calculus than a subject in its own right.
We begin most chapters with a treatment of a topic in linear algebra and then
show how the methods apply to corresponding nonlinear problems. In each
chapter, enough linear algebra is developed to provide the tools we need in
teaching multivariate calculus (in fact, somewhat the spectral theorem
for symmetric matrices is proved in Section 3.7). We discuss abstract vector
spaces in Section 2.6, but the emphasis is on R°, as we believe that most
students find it easiest to move from the concrete to the abstract.
Another guiding principle is that one should emphasize computationally effective algorithms, and prove theorems by showing that those algorithms really
to marry theory and applications by using practical algorithms as theoretical tools. We feel this better reflects the way this mathematics is used
today, in both applied and in pure mathematics. Moreover, it can be done with
no loss of rigor

Genres: Mathematics, Textbooks, Science, Calculus, Physics, Academic, Nonfiction

1396 pages, Kindle Edition

First published January 1, 1998

Reviews

[Ronald Lett · Rating 5 out of 5]

An excellent companion to Spivak's "Calculus on Manifolds", this text engagingly bridges the gap between the Gibbs formulation of vector calculus and the modern Cartan formulation using differential forms so that one can properly study differential geometry. Gives many excellent exercises and examples of curious behavior. It is also exceptionally reader-friendly; the authors sprinkle historical anecdotes and modern applications throughout the text and have a voice that is very down-to-earth.

[Parsa · Rating 5 out of 5]

I read parts of the last chapter to help me with Spivak and it was really really good. It does in 150 pages what spivak does in about 40, which is mostly a testament of how terse the latter is. I expect to return to this this term for my diff geo class.

[Susmit Islam · Rating 5 out of 5]

Going thru the last part, the part on forms (the book is HUGE), in the fifth edition. Wonderful exposition, every other resource I've consulted on the subject has done little, if anything, to demystify the concrete nature of forms. More importantly, it has the best exposition on exterior calculus (apart maybe from Needham's Visual Differential Geometry). Many texts define the exterior derivative thru its rather opaque properties (especially opaque is the so-called antiderivation property, and the Poincaré lemma). Moreover, even if they satisfied the properties of the more familiar Newtonian derivative (of 0-forms), it wouldn't be entirely clear that satisfying these properties uniquely identifies a map capable of doing differentiation. Haven't gone through the vector calculus and linear algebra parts but given the exposition in the last part it's clear that they too would be chock-full of insights.

[Matt Tokunaga · Rating 5 out of 5]

A classic for a reason. A bit difficult as an introduction to multivariate calculus and proofs though. Goes off the rails a bit when differential forms are introduced but the payoff is worth it. Once you get to generalized Stokes theorem its very satisfying in its simplicity, though its only simple because of all the knowledge you build up beforehand.

[Daniel Zelich · Rating 3 out of 5]

Introduced me to higher math but this book is terribly written. Pedagogically dubious (manifolds after a chapter of multi and lin alg is not conducive to good understanding) and too numerical in its proofs, but otherwise interesting and fun to read.

[Himanshu Singla]

Its good for which class and is it best for 12th standard