Here are two cute checks of the fundamental theorems:
(a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem (v = ∇T , in this case). Show that the result is consistent with what you already knew about second derivatives.
(b) Combine Corollary 2 to Stokes’ theorem with the divergence theorem. Show that the result is consistent with what you already knew.
(a) Corollary 2 says ∮(∇T)·dl = 0. Stokes’ theorem says ∮ (∇T)·dl =∫[∇×(∇T)]·da. So ∫[∇×(∇T)]·da = 0, and since this is true for any surface, the integrand must vanish: ∇×(∇T) = 0, confirming Eq. 1.44.
∇ × (∇T ) = 0. (1.44)
(b) Corollary 2 says ∮(∇×v)·da = 0. Divergence theorem says ∮(∇×v)·da =∫∇·(∇×v) dτ. So ∫∇·(∇×v) dτ = 0, and since this is true for any volume, the integrand must vanish: ∇(∇×v) = 0, confirming Eq. 1.46.
∇ · (∇ × v) = 0. (1.46)