# Thread: Stokes Theorem = 0???

1. ## Stokes Theorem = 0???

(1)

using stokes theorem and cutting the surface into 2 parts how can we prove that

$\int$ curl A.n dS = 0

assume the surface "S" to be smooth and closed, and "n" is the unit outward normal as usual.

(2)
How can you prove
$\int$ curl A.n dS = 0
using the divergence theorem?

2. Maybe if you cut the surface horizontally down the middle then $\iint_K \mbox{curl}\bold{A}\cdot \bold{N} dS = \int_C \bold{F}\cdot d\bold{R}$ by Stokes' Theorem. But there are two peices the upper half and the lower half. Both are described by the same RHS expression except that they differ in signs because when we take the lower half we consider negative orientation around the boundary. And hence together these two line integrals add up to zero.

3. i understand that. but what are the parameters that i work with. and what are the limits of integral. i have not been given this in the original question

4. i think i have managed to work out part 2:

if F = curl A
the div theorem says:
$\int_v div(curlA)dV = \oint Curl A.ndS$

div(curl A) = 0
So: $\oint CurlA.ndS = 0$

i understand that the identity div(curl A)=0 but how is this proved please?