This book has two audiences. Its primary audience is students in a third-semester (multivariable) calculus course, who are studying the material usually known as "vector calculus". Its secondary audience is more advanced students who are seeking a very concrete introduction to (or explanation of) differential forms.
First things first: We address the first part of this introduction to the primary audience.
Our treatment of vector calculus, from the viewpoint of differential forms, is designed to show the mathematical unity behind the subject (but is not intended to be a rigorous treatment, which would go far beyond the bounds of such a course). It brings the ideas involved to the fore, and makes the material clearer and easier to understand.
The subject matter of this course is traditionally taught in the language of elementary physics and engineering. Of course, there is nothing wrong with this language (and many students taking this course are physicists or engineers), but, while being fine for applications, this language makes it look like several different things are going on, whereas from a mathematical viewpoint these "different things" can all be seen to be special cases of a single idea.
Let us consider this subject matter. Let / be a function and F a vector field, both in IR3 (3-space). Then we have:
1) grad(/), a vector field,
2) curl(F), a vector field,
3) div(F), a function.
We will first see how these are all "really" special cases of the following:
