One theorem about differential forms swallows four famous results of calculus
Stokes' theorem, Green's theorem, the divergence theorem and even the fundamental theorem of calculus look like separate facts in a textbook. Written in the language of differential forms, they turn out to be special cases of one statement, the generalized Stokes theorem, and the coordinates stop mattering entirely.
A differential form is a thing built to be integrated. A 1-form measures a tiny oriented length and is integrated along a curve; a 2-form measures a tiny oriented patch of area and is integrated over a surface; a 3-form acts as a volume element over a region of space. In general a k-form belongs over a k-dimensional manifold. Crucially, these measurements carry direction, so integration only makes sense on manifolds that have been given an orientation.
That requirement gives geometric meaning to a familiar convention of one-variable calculus: swapping the limits of an integral flips its sign. An interval from a to b is positively oriented when a is less than b, and running the same form over the reversed interval produces the negative result. Measure theory handles integrals differently, treating the integrand as a function against a measure over a set with no orientation at all, a gap that matters little on a line but grows subtler in higher dimensions.
The key operation is the exterior or wedge product, which resembles the cross product of vector calculus because it alternates. The little square with sides dx1 then dx2 has the opposite orientation to the one with sides in reverse order, so their products cancel with a minus sign. Just as a cross product produces the area vector of a parallelogram from two edges, the wedge product builds higher forms out of lower ones. An exterior derivative extends the ordinary derivative to forms, and that is what lets the four classic theorems merge. Forms also survive being carried between spaces by smooth maps, so the change of variables formula reduces to the statement that an integral is unchanged by this transfer.
The ideas have deep roots. Hermann Grassmann's 1844 book on the theory of linear extension contains pieces of the underlying algebra, while Élie Cartan is usually credited with the first algebraic organisation of differential forms, in a paper of 1899. Today they are central to geometry, topology and physics.
Source: Differential form
