It begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms.
The subject is approached with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. The book contains excellent motivation, numerous illustrations and solutions to selected problems. Skip Navigation and go to main content Bestsellers Books. Print this page. Used from other sellers Check for new and used marketplace copies. An Introduction to Homological Calculus Frank Ayres author , In an introductory "Guide to the Reader," he explains the basic approach: to use low-dimensional cases that can be visualized to support the general ideas.
The emphasis is on understanding rather than on detailed derivations and proofs.
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This is definitely the right approach in a course at this level. One can always keep a copy of a more formal book at hand for those students who want to see more of the proofs. In order to minimize the pre-requisites, the book opens with a chapter on basic multivariable calculus: vectors, functions of several variables, partial derivatives, multiple integrals. This is too short to serve well if students have never seen this material, but will work as a review.
In particular, students who have never had linear algebra will have a hard time learning enough from section 1. A chapter on parametrizations follows. It is, I think, much too brief. For many students, learning to parametrize and to understand parametrizations is the toughest part of a vector calculus course.
A little more help should have been provided here. It clearly sounds the fundamental message: what is inside an integral is not a function, but rather a differential form. Then we get down to fairly serious business. The treatment of forms is done well, but I think these chapters will be heavy going for most students.
Geometric Approach to Differential Forms: David Bachman - Book | Rahva Raamat
We get statements like. Every 2-form projects the parallelogram spanned by V 1 and V 2 onto each of the 2-dimensional coordinate planes, computes the resulting signed areas, multiplies each by some constant, and adds the results. Well, ok… Maybe it's the algebraist in me, but this seems like a very roundabout way of describing the basis of the space of 2-forms. I would have a hard time resisting the impulse to go a little heavier on the linear algebra in order to avoid having to say such things.
It may well be, however, that my students would prefer going Bachman's way. When we get to differential forms in chapter 5, things get necessarily harder. A glance at page 55, for example, shows that Bachman's "geometric approach" does not mean we do not get hairy computations. On that page he is justifying the formula for integrating a 2-form over a surface patch not a term he uses by setting up a kind of Riemann sum and checking that it can be translated to a Riemann sum on the plane.
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The result, of course, is that one integrates the form by integrating its pullback, but that idea is not introduced until much later in the book. The reason is that the derivative is nowhere described as a linear transformation; if one can't push vectors forward, one also can't pull forms back.
The treatment here is arguably geometric, but it is nevertheless pretty messy. On the other hand, it allows Bachman to derive the change of variable formula for multiple integrals from the theory of forms, which is kind of nice. Chapter 6 introduces the exterior derivative.
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I've always thought that the gradient is a poor substitute for the differential, so this feels backwards to me. Once the theory of forms is well set up, the rest of the book proceeds in a fairly brisk fashion.