Let a be a vector field on a Riemannian manifold with metric < , > and connection ∇. Then curl a is a antisymmetric 2-form defined by:
curl a (X,Y) = <∇_Y a, X> - <∇_X a, Y>
There even exists a Helmholtz decomposition of a vector field into its divergence-free and curl-free parts. So maybe Burke was wrong, and div, grad, and curl still live!