Surface Integral of a Vector Field on a Paraboloid Above a Square

[Niles]

Homework Statement

Hi all. Please take a look at the following problem:

Evaluate the surface integral $$\int{F \cdotp d\vec{S}}$$ for the following vector field:

F(x;y;z) = xyi + yzj + zxk, where i, j and k are unit vectors. S is the part of the paraboloid z = 4-x^2-y^2 that lies above the square $$x \in [0;1]$$ and $$y \in [0;1]$$.

The Attempt at a Solution

Ok, I first find the divergence of F(x,y,z), which is y-x^2-y^2+4 (I have substituted z). Then I find dV, which is just dxdy and use the limits for x and y as stated above.

Is this method correct?

Thanks in advance

 

[HallsofIvy]

Apparently you are referring to the "divergence theorem", that
$$\int\int_T\int (\nabla\cdot \vec{v})dV= \int_S\int \vec{v}\cdot d\vec{S}$$

Are we to assume that the surface is oriented by "outward normals" which is part of the condition for the divergence theorem to hold?

You say "S is part of the paraboloid z = 4-x^2-y^2 that lies above the square $$x \in [0;1]$$ and $$y \in [0;1]$$." So you are not incuding the base? If you are not integrating over a closed surface, the divergence theorem does not hold.

I think it would be simplest just to do the problem directly- integrate over the surface itself. Do you know how to do that?

 

[Niles]

Yes I do. I forgot that the divergence theorem only goes when we are talking about closed surfaces.

Thanks again, Mr.!