1. ## Multivariable Calculus Parametrization

I am trying to find the surface area.

So I know that once I have the parameters, I can find Tu, Tv and cross them to find the normal vector.

However, I seem to have trouble whenever the function isn't given in terms of u and v.

This is the problem I am looking at:-

Calculate [Double Integral] f(x,y,z)dS for the given surface and function

y=9-z^2, 0<=x<=3, 0<=z<=3, f(x,y,z) = z

2. ## Re: Multivariable Calculus Parametrization

The surface is defined by $y= 9- z^2$ with no condition on x. So use x and z as parameters. x= u, $y= 9- v^2$, z= v.

3. ## Re: Multivariable Calculus Parametrization

Thanks!

That works.

I've got a new problem.

Verify Stokes Theorem for the given vector field and surface, oriented with an upward pointed normal.

F = <2xy, x,y+z>, the surface z = 1 - x^2 -y^2 for x^2 +y^2<=1

For using line integrals, what should I use my c(t) as?

For the Stokes theorem part, I can calculate the curl, but again I don't know how to go from there (as in find the boundaries and dS).

Thanks!

4. ## Re: Multivariable Calculus Parametrization

Thanks!

That works.

I've got a new problem.

Verify Stokes Theorem for the given vector field and surface, oriented with an upward pointed normal.

F = <2xy, x,y+z>, the surface z = 1 - x^2 -y^2 for x^2 +y^2<=1

For using line integrals, what should I use my c(t) as?

For the Stokes theorem part, I can calculate the curl, but again I don't know how to go from there (as in find the boundaries and dS).

Thanks!