stokes proof-help ! for exams coming soon

Hi,

Can some one guide me towards the right answer?thanks.

S be a smooth, orientable surface, with unit normal n and boundary

C. Let C be oriented, with respect to n, as in the statement of Stokes’

Theorem. Let f and g be continuous functions defined on S, with

continuous first and second order partial derivatives.

a)show that ∮(f∇g).dr=∫∫s(∇fx∇g).nds

b)show that ∮(f∇g+g∇f).dr=0

Re: stokes proof-help ! for exams coming soon

Quote:

Originally Posted by

**n22** Hi,

Can some one guide me towards the right answer?thanks.

S be a smooth, orientable surface, with unit normal n and boundary

C. Let C be oriented, with respect to n, as in the statement of Stokes’

Theorem. Let f and g be continuous functions defined on S, with

continuous first and second order partial derivatives.

a)show that ∮(f∇g).dr=∫∫s(∇fx∇g).nds

b)show that ∮(f∇g+g∇f).dr=0

Hint for (a): You want to prove that $\displaystyle d(\langle f\nabla{g}, dr \rangle)$ is equivalent to what follows the integral on the RHS, where d denotes the exterior derivative and the brackets denote the inner product.

Hint for (b): $\displaystyle \nabla(fg)=f\nabla{g}+g\nabla{f}$.