Thread: Surface Integration Problem

I'm stuck on this problem,

Consider the domain in the u - v plane bounded by the circle u^2+v^2=1 and the surface S in R^3 defined parametrically by
r(u,v)=(u^2+v^2)i + (uv)j + (u+v)k
where the positive sense around the boundary is determined by the positive sense around the boundary of the region in the u - v plane.

Set F= (yz)i + (xz)j + (xy)k

Compute SS(curlF).n dS. It is a double integral where . is the dot product between the curlF and the normal n.

I've done problems similar to this, but never with different variables as the boundary. How do I set this up as an integral?

Originally Posted by Five Star

I'm stuck on this problem,

Consider the domain in the u - v plane bounded by the circle u^2+v^2=1 and the surface S in R^3 defined parametrically by
r(u,v)=(u^2+v^2)i + (uv)j + (u+v)k

Let us examine this surface.
We have, parametrically,
x=u^2+v^2
y=uv
z=u+v

Note that,
z^2=(u+v)^2=u^2+2uv+v^2=(u^2+v^2)+2(uv)=x+2y.
Thus, the surface is, a plane,
z=x+2y

Is it z^2 = x+2y, not z=x+2y ?
This would not make it a plane right?

Originally Posted by Five Star

Is it z^2 = x+2y, not z=x+2y ?
This would not make it a plane right?

Sorry.
No, that would not be a plane.

Remember when you use a triple intergral and need the upper and lower surface you get the plus/minus when you take the square root.

I was thinking. Is there a name for this figure?

Originally Posted by Five Star

I was thinking. Is there a name for this figure?

It is not on a standard list of solids provided in thy Method of Fluxions (Calculus) scroll.

It is however a general quadric surface, thus maybe it has a name.