# Integrate divergence of a vector over an area

1. Oct 22, 2014

### JMoody

Hello, I'm hoping somebody can give me some insight on how to solve this problem. This was a solid mechanics exam question and I wasn't able to finish it because I'm rather weak in math.

1. The problem statement, all variables and given/known data

2. Relevant equations
Recall divergence theorem for part ii. ∫div(V)dA = ∫V⋅ndS where n is normal to the surface.

3. The attempt at a solution
Part i.
Not an issue, I can solve it easily.

Part ii.
I can apply divergence theorem no problem.
For the n vector field I get:
n1 = -e2
n3 = -e1
n2 = .707(e1 + e2)

The problem is that I've been out of school for quite a while, and I can't remember how to successfully integrate over the surface (perimeter) now. I realize I need to break it up in to parts for each line of the surface, but it's been years since I've taken calculus.

Last edited: Oct 22, 2014
2. Oct 22, 2014

### RUber

For c1, you will have $\int_0^1 V\cdot n_1 \, dx_1$.
For c3, you will have $\int_0^1 V\cdot n_2 \, dx_2$.
For c2, it gets a little tougher.
$\int_{c2} V\cdot n_2 dc2$

3. Oct 22, 2014

### JMoody

For c2, should I do a double integral (x1 and x2, both from 0 to 1) of V⋅n2?

4. Oct 22, 2014

### RUber

It might be easier to directly compute the Area integral
$\int_0^1 \int_0^{1-x_1} div(V) dx_2dx_1$

5. Oct 22, 2014

### RUber

I don't think that is right, you would be taking the integral over an area. you want the integral over the line.

6. Oct 22, 2014

### JMoody

Thanks RUber, I think you've provided the help I needed to finish it. It was required to perform divergence theorem for the exam though, I'll give it a shot with both methods and see how the solutions compare.

7. Oct 22, 2014

### RUber

Nevermind this...that doesn't make any sense.
---
Try integrating over x1, and define x2 in terms of x1 along the line. That should do the trick.

8. Oct 22, 2014

### JMoody

I solved it, I get 1/2 using both methods. Thanks again for the help RUber, it helped set me on the write path.

9. Oct 22, 2014

### RUber

When I worked it out, I still had the $\frac 1 {\sqrt{2}}$ in the answer.

10. Oct 22, 2014

### JMoody

I'm not sure where the square root comes from, but I believe the accepted solution was 1/2 when we went over the exam (he just didn't take the time to go over something trivial like integration since most of my peers are coming straight out of undergrad >.<). I calculated it both by using divergence theorem to integrate over the line (more complicated) and by integrating the divergence over the area directly.

11. Oct 22, 2014

### RUber

Aha, I forgot to multiply by the $\sqrt{2}$ in the dS term.
Good work.

12. Oct 23, 2014

### BvU

Why (in post #7) "Nevermind this...that doesn't make any sense." ?
I thought I had div V = 3x2 and that would be a simple integral, so I'm grossly overlooking sonething ?

13. Oct 23, 2014

### RUber

Thanks BvU, I was clearly overthinking things and convinced myself I was wrong.