vector calculus as forms

After meeting the exterior derivative, the punchline is irresistible: the three operators of vector calculus — gradient, curl, and divergence — are not three things but one thing, the exterior derivative d, applied to forms of degree 0, 1, and 2. Once you set up a dictionary between vector fields and forms in R^3, the messy list of formulas for grad, curl, and div collapses into the single rule 'apply d'.

Here is the dictionary in R^3 (it uses the metric to raise and lower indices, so it is special to Cartesian-style geometry). A scalar function f is a 0-form. A vector field (P, Q, R) corresponds to the 1-form P dx + Q dy + R dz, and also, through the volume form, to the 2-form P dy^dz + Q dz^dx + R dx^dy. Now watch: d of the 0-form f is df = f_x dx + f_y dy + f_z dz, the gradient. d of the 1-form for a field F gives a 2-form whose components are exactly curl F. d of the 2-form for a field F gives a 3-form whose single coefficient is exactly div F. Three classical operators, one operator d, distinguished only by the degree it acts on.

This is not merely tidy; it explains the identities you once memorized. Because d squared is zero, curl of grad is zero (that is d of d on a 0-form) and divergence of curl is zero (d of d on a 1-form) — both fall out for free. And the generalized Stokes' theorem becomes the umbrella over the gradient theorem, Green's theorem, the classical Stokes theorem, and the divergence theorem, since each is just 'integrate d omega = integrate omega on the boundary' at the appropriate degree. The form picture is what reveals that the whole subject was one idea seen from several angles.