# Stokes' Theorem

1. May 1, 2005

### mkkrnfoo85

Here is the problem:

S is the ellipsoid $$x^2+y^2+2z^2=10$$

and F is a vector field $$F=(sin(xy),e^x,-yz)$$

Find: $$\int \int_S ( \nabla \mbox {x} F) \cdot dS$$

So, I know that Stokes' Theorem states that:
$$\int \int_S ( \nabla \mbox {x} F) \cdot dS = \int_{\partial S} F \cdot ds$$
where $$\partial S$$ equals the boundary of the ellipsoid. How do you find $$\partial S$$? My professor just told me that any closed surface has no boundary and therefore the answer is 0, but would someone show me how I can show this? And can someone tell me under what conditions does the answer become 0 using Stokes' Theorem? Thanks a lot.

2. May 1, 2005

### jdavel

Stokes' theorem is derived by dividing a bounded surface up into a bunch of small flat surfaces and then noticing that the line integrals around all these surfaces cancel eachother out except at the boundary of the big surface. Think about that and what it means if the surface you're splitting up has no boundary.