Difficult Surface Integrals

1. May 23, 2004

Theelectricchild

I expected Stokes theorem to make my life easier but these problems are even harder than the normal ones ive been doing.

Use Stokes' Theorem to evaluate $$\int\int_ScurlFdS$$

where F(x, y, z) = < x^2*y^3*z, sin(xyz) ,xyz >

S: Part of cone $$y^2 = x^2 + z^2$$ that lies between the planes y = 0 and y = 3 oriented in the direction of the positive y - axis.

not sure about this one... I parameterized the curve C as r(t) = < 3cost, 3, -3sint > to get a curve with positive orientation induced by S having normal vector in pos-y direction...a circle in plane y = 3,

Use Stokes' Theorem to evaluate $$\int\int_ScurlFdS$$

where $$F(x, y, z)=< e^{xy}*cos(z) ,x^2*z, xy >$$

S: Hemisphere $$x = sqrt{1-y^2-z^2}$$ oriented in the direction of the positive x - axis.

curve C : r(t) = <0, cost, sint> ... a circle again in plane x=0, hmmm, wonder if I'm messing these up...?
answer: 0 (this seemed way too easy...everything was zero!)

Last edited: May 23, 2004
2. May 23, 2004

arildno

$$\iint_{S}\nabla\times\vec{F}\cdot{d}\vec{S}=\oint_{C}\vec{F}\cdot{d}\vec{r}$$

In that case, your last calculation is correct; even if it is indecently easy..

3. May 23, 2004

Theelectricchild

Even for the first one? The answer just seemed completely off considering its size.

4. May 23, 2004

arildno

At a glance, the first one ought to be zero as well..

5. May 23, 2004

arildno

Sorry, first glance wrong..

Last edited: May 23, 2004
6. May 23, 2004

arildno

If 3^{7}=2187, then the first should be correct as well.

7. May 23, 2004

thank you :)