Information captured by differential forms

Differential 1-forms are the dual of vector fields on a manifold. Thus, if you are interested in vector fields, you should also be interested in their ``mirror image,'' differential forms. A good intuition regarding their multiplication is that $\mathrm{d}x\wedge\mathrm{d}y$ is an oriented area. Thus, $\mathrm{d}y\wedge\mathrm{d}x = -\mathrm{d}x\wedge\mathrm{d}y$ since its orientation is opposite that of $\mathrm{d}x\wedge\mathrm{d}y$. The exterior derivative unifies and generalizes the gradient, curl, divergence, and Laplacian. Some special cases of the nilpotency of $\mathrm{d}$ (i.e., $\mathrm{d}^2 = 0$) are $\nabla\cdot\nabla\times{\bf F} = 0$ and $\nabla\times\nabla f = 0$.

In terms of integration, notice first that the volume form transforms the right way under coordinate transformations, $$\begin{eqnarray} \omega &=& h(p) \mathrm{d}x^1\wedge\cdots\wedge \mathrm{d}x^n \\ &=& h(p) \frac{\partial x^1}{\partial y^{\mu_1}} \mathrm{d}x^{\mu_1}\wedge\cdots\wedge \frac{\partial x^n}{\partial y^{\mu_n}} \mathrm{d}x^{\mu_n} \\ &=& h(p) \det\left(\frac{\partial x^\mu}{\partial y^\nu}\right) \mathrm{d}y^1\wedge\cdots\wedge \mathrm{d}y^n, \end{eqnarray}$$ where $h(p)$ is positive definite and $\det\left(\frac{\partial x^\mu}{\partial y^\nu}\right)$ is the Jacobian of the coordinate transformation. It also ``knows'' about the orientability of the space. In fact, a space is orientable if and only if it has a volume form.

With differential forms, many important results from multivariable calculus can be unified into a single formula, Stokes' theorem: $$\int_M \mathrm{d}\omega = \int_{\partial M} \omega.$$ For example, some special cases of Stokes' theorem are the curl and divergence theorems in 3-space: $$\begin{eqnarray} \int_S \nabla\times {\bf F}\cdot d{\bf S} &=& \oint_{\partial S} {\bf F}\cdot d{\bf r} \\ \int_V \nabla\cdot {\bf F} dV &=& \oint_{\partial V} {\bf F}\cdot d {\bf S} \end{eqnarray}$$ Of course, Stokes' theorem also generalizes these theorems to higher dimensions.

The utility of differential forms extends far beyond this. For example, de Rham cohomology depends fundamentally on differential forms and can be used to classify topological spaces.