Need help finding flux across a surface

[nprince]

Homework Statement

Find the upward flux across the surface z = x of the vector field F=(y,-x,0).

Homework Equations

∫∫(F$\bullet$n)ds

The Attempt at a Solution

I know what to do once I parametrize z = x but I am stuck on the parametrization. Would it just be: r(x,y) = (zi + j + xk) ?

 

[nprince]

or could I say f(x,y,z) = z - x

so the normal vector would be : $\nabla$f / ||$\nabla$f||

= -i + k / $\sqrt{}$-1²+1²

 

[HallsofIvy]

The first thing you are going to have to do is put bounds on your surface. You don't want to integrate across the entire infinite plane, do you?

As far as the parameterization of the surface is concerned, it is r(x, y)= xi+ yj+ xk. You are missing the "y" but I suspect that was a typo.

The best way to find n, or better, $\vec{n} dS= d\vec{S}$, is to find the "fundamental vector product" of the surface. That is the cross product $\vec{r}_x\times\vec{r}_y$.

$$d\vec{S}= (\vec{r}_x\times\vec{r}_y)dxdy$$

I personally don't like the notation $\vec{n}dS$ because if you take that literally, as you have here, you have to divide by the length of the normal vector to find $\vec{n}$ and then multiply by it to find dS- and, of course, those will cancel!

 

[nprince]

So then it's just the double integral of the vector field dot that cross product and I just need to find the limits of integration. Thanks!