Intrinsic definition of divergence and curl

Question

Are the intrinsic definitions of divergence and curl the theorems of Green-Ostrogradski and Stokes-Ampere respectively ?

What is a rigorous derivation of their expression in a coordinate system ?

Answer 1

I don't know these theorems. However, you can define div grad curl using the Riemannian metric's pairing, the hodge * operator, and the exterior derivative. Let $T_1$ denote the map from one-forms to vector fields induced by the pairing.

$T_1 df$ is the gradient.

$T_1 * d T_{1}^{-1}v$ is the curl.

$*d*T_{1}^{-1}v$ is the divergence.