Stoke's Theorem

**Altair** · February 5th 2009, 08:38 AM

Question asks to evaluate the surface integral for the given $F$ and $S$ with stokes theorem.

$F= [z^2, x^2, y^2], s: z^2 = x^2 + y^2$ for $y >= 0$ , $0<=z<=2$

My question is that in transformation from x,y to u,v shouldn't there be jacobian multiplied ? I dont see it anywhere in the solution.

**mr fantastic** · February 7th 2009, 01:33 AM

Originally Posted by Altair

Question asks to evaluate the surface integral for the given $F$ and $S$ with stokes theorem.

$F= [z^2, x^2, y^2], s: z^2 = x^2 + y^2$ for $y >= 0$ , $0<=z<=2$

My question is that in transformation from x,y to u,v shouldn't there be jacobian multiplied ? I dont see it anywhere in the solution.

There is no trouble or mystery here.

The surface is being represented by the parametrised position vector $\vec{r}(u, v) = x(u, v) \vec{i} + y(u, v) \vec{j} + z(u, v) \vec{k}$ . Using this representation, $\vec{dS} = \frac{\partial \vec{r}}{\partial u} du \times \frac{\partial \vec{r}}{\partial v} dv$ .

This is a formula that should be somewhere in your notes or textbook.

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