Stokes' Theorem: Finding the Flux of a Vector Field on a Surface

[-EquinoX-]

Homework Statement

Let \vec{F} = xy\vec{i} + yz\vec{j} + xz\vec{k} and C is the boundary of S, the surface z = 9 - x2 for 0 ≤ x ≤ 3 and -6 ≤ y ≤ 6, oriented upward. Use Stokes' Theorem to find \int\limits_c \vec{F} \cdot d\vec{r}.<br /> <br /> <h2>Homework Equations</h2><br /> <h2>The Attempt at a Solution</h2><br /> <br /> well I&#039;ve found the curlF at least.. I don&#039;t know what I should do now

 

[gabbagabbahey]

Find an expression for the vector area element of S (it should have an x-component and a z-component)and integrate over the surface.

 

[-EquinoX-]

vectore area element??

 

[gabbagabbahey]

Yes, it's the product of the infinitesimal area element (usually denoted dS or da) with the unit normal to the surface (\vec{da}=\hat{n}da). Have you not heard that term before?

For example, the outward vector area element for a spherical shell of radius R is \vec{da} =R^2 \sin \theta d \theta d \phi \hat{r}, where \theta is the polar angle, \phi is the azimuthal angle, and \hat{r} (sometimes written \hat{e}_r) is the radial unit vector.

Different authors use different notations.

 

[-EquinoX-]

yes, I've heard of it.. I now need to find the normal vector first... how can I do that...

 

[gabbagabbahey]

Start by drawing a sketch of the surface, you should see that the outward normal to the surface is the same as the outward normal of the curve z=9-x^2. Parameterize that curve (I suggest using x=t) and find the tangent and normal in the usual ways.