Please help with this one guys!! i really need solutions tonight if possible.

Verify the Gauss Divergence Theorem:

∫∫∫∇ ⋅F dxdydz= ∫∫ **F .n**ˆ *dA *, where

S

the closed surface S is the sphere *x*2 + *y*2 + *z*2 = 9 and the vector field

*F *= *xz*2*i *+ *x*2 *y ***j **+ *y*2 *z ***k **.
thanks in advance!

I'm confused here. Are we verifying the divergence theorem in general (i.e. prove it is always true) or prove that it is true in this case?

I'll assume we're doing the latter.

\(\displaystyle \iiint_S \nabla \cdot F dV \)

What does the above represent? Well...It represents the entire flux out of our surface. Can we compute this? Of course!

\(\displaystyle \nabla \cdot F = divF \)

Remember that \(\displaystyle divF = \frac{ \partial F1 }{\partial x } + \frac{ \partial F2 }{\partial y } + \frac{ \partial F3 }{\partial z } \)

Assuming you mean

\(\displaystyle F = xz^2 \hat i + x^2 y \hat j + y^2 z \hat k \)

We arrive at

\(\displaystyle \iiint_S \nabla \cdot F dV = \iiint_S (z^2 + x^2 + y^2) dV \)

Of course our surface is defined as \(\displaystyle x^2 + y^2 + z^2 = 3^3 \) which is a sphere! So we have 2 options here, either use cylindrical co-ordinates or spherical. I personally like to use cylindrical, so let's go with that and see if that's easy enough to compute.

\(\displaystyle \iiint_S (z^2 + x^2 + y^2) dV \)

\(\displaystyle \iiint_S r(z^2 + r^2) d \theta dr dz = \int_0^{2 \pi } d \theta \int_0^3 dr \int_0^{ \sqrt{ 3^2 - x^2 - y^2 } } r(z^2 + r^2) dz\)

Compute the above and we will then compare to

\(\displaystyle \iint_D F \cdot \hat N dA \)

Let's remember that the above represents the flux out of the surface. So draw our shape (which is a sphere) and we will note that it is smooth everywhere!!!!

What does that mean? Well...it makes the above integral easy because we only have 1 surface to compute for total flux.

Knowing,

\(\displaystyle \hat N = +- ( - \frac { \partial D}{\partial x } \hat i - \frac { \partial D}{\partial y } \hat j + \hat k ) \)

We can easily compute the above. Do so and compare