How Do You Solve a Complex Divergence Problem Using Spherical Coordinates?

[[Quadratic]]

Homework Statement

Apologies for the attachment.

Homework Equations

Limit definition of the divergence as seen in attachment
Volume of a sphere: $\frac{4}{3}\pi r^{3}$

The Attempt at a Solution

The first thing I did was parameterize the vector function F(x,y,z) = <xy,x,y+z>
My parameterization is as follows:

$x = a+rcos\vartheta sin\varphi \\ y = b+rsin\vartheta sin\varphi \\ z = c+rcos\varphi \\ dS = S_{\varphi} X S_{\vartheta} d\varphi d\vartheta$

Setting up the limit and integral:
$lim_{r\rightarrow0}\frac{1}{\frac{4}{3}\pi r^3} \int^{2\pi}_{\vartheta=0} \int^{\pi}_{\varphi=0} <(a + rcos\vartheta sin\varphi)(b + rsin\vartheta sin\varphi),a + rcos\vartheta sin\varphi,rsin\vartheta sin\varphi + c + rcos\varphi> \bullet S_{\varphi} X S_{\vartheta} d\varphi d\vartheta \\$

I apologize for the large attachment and my messy latex. Any suggestions to clean it up are welcome. Am I on the right track so far, before I continue? I tried using the Jacobian thinking it would clean up the integrand but I didn't really get anywhere, and my professor told me we are not doing a change of variables here. Thanks in advance.

 

[haruspex]

You somehow have to perform the dot product. You have F in Cartesian but not dS. I assume you're allowed to put dS in Cartesian? If not, I don't know how you're expected to continue.

 

[[Quadratic]]

You somehow have to perform the dot product. You have F in Cartesian but not dS. I assume you're allowed to put dS in Cartesian? If not, I don't know how you're expected to continue.

I really don't know. My teacher is rather "disorganized" to put it lightly and frequently makes mistakes on our exams. For example our last exam had 12 problems, and 5 of them had pretty critical errors and were not corrected until around an hour into the exam. He also assigns us problems that require methods he skips over in his lectures. Look at me, now I'm just complaining :)

Almost everyone in the class is having trouble with this, and our professor assured us he worked this one out and it is solvable.

 

[haruspex]

I really don't know. My teacher is rather "disorganized" to put it lightly and frequently makes mistakes on our exams. For example our last exam had 12 problems, and 5 of them had pretty critical errors and were not corrected until around an hour into the exam. He also assigns us problems that require methods he skips over in his lectures. Look at me, now I'm just complaining :)

Almost everyone in the class is having trouble with this, and our professor assured us he worked this one out and it is solvable.

OK, so assume you're allowed to put dS in Cartesian. Have you tried that?