Differential Forms MA 305

We’ve hit the big three theorems of vector calculus: Green’s, Stokes’, and the Divergence Theorem (although Green’s theorem is really a pretty obvious special case of Stokes’ theorem). But it’s an amazing fact that all of these theorems are really special cases of an even larger theorem that unifies all that we’ve studied so far. This new theory also extends what we’ve done to spaces of arbitrary (finite) dimension. It rests on the idea of differential forms. Differential forms are hard to motivate right off the bat. It won’t be immediately clear how differential forms are related to what we’ve done, but be patient. For now treat differential forms as a new kind of mathematical object; we’re going to learn how to manipulate them—add, multiply, differentiate, and integrate. It will be quite abstract and “formal” at first, but you’ll soon see how they connect with what we’ve already done.