The old-fashioned derivative you learned about in basic calculus and the gradient, curl, and divergence operators discussed in vector calculus are all special cases of the exterior derivative operator introduced in differential geometry. The exterior derivative maps an n-form ω on a manifold M to an n+1-form dω on that manifold, and in particular maps a simple function (or scalar field, or 0-form) f to a 1-form df. Combined with the rules of exterior algebra and the product rule familiar from differential calculus, these two rules are sufficient to compute the exterior derivative of a form of any order in terms of a set of basis vectors (or basis vector fields) xi and their corresponding forms dxi:

  1. d(dω) = 0
  2. df = Σ fxi dxi

(where fxi is the partial derivative of f with respect to the xi-coordinate).

Stokes' Theorem provides a good geometric rationale for calling this the exterior derivative: it represents the function that, when integrated over a whole submanifold, is equivalent to the original function when integrated over the boundary (or exterior) of that manifold.