stokes proof-help ! for exams coming soon

• Jun 12th 2013, 03:22 AM
n22
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
• Jun 14th 2013, 01:46 PM
Phantasma
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 $d(\langle f\nabla{g}, dr \rangle)$ 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): $\nabla(fg)=f\nabla{g}+g\nabla{f}$.