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 ?
Asked
Active
Viewed 568 times
0
-
Have you seen the limit-integral definitions? – May 26 '15 at 21:35
-
No, I have not. – user1234161 May 26 '15 at 21:40
-
1Well look here for the definition of curl. If that's what you're looking for, you might want to find a copy of Schey's Div, Grad, Curl, and All That -- he does of good job of explaining the limit-integral definitions. BTW, you can also define the exterior derivative this way. – May 26 '15 at 21:42
1 Answers
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.
Mark Joshi
- 5,724
-
This is beyond my current math knowledge (I'm a physics undergraduate student) but thanks anyway. – user1234161 May 27 '15 at 07:07