# Homework Help: Problem with vector field proof

1. Aug 15, 2012

### Roidin

Suppose that F and G are vector fields and that F-G = ▽μ for some real-valued function μ(x,y). Prove that

∫ F.dx = ∫ G.dx for all piecewise smooth curves C in the xy-plane

I just need some help in getting started really. Thanks

2. Aug 15, 2012

### LCKurtz

If $dx$ is a scalar, what does $\vec F \cdot dx$ mean? Have you stated the problem correctly? Is C a closed curve?

3. Aug 15, 2012

### Roidin

Thanks LCKurtz,

F.dx is the line integral of the vector field and yes C is closed.

4. Aug 15, 2012

### LCKurtz

That's unusual notation. And since it matters that C is closed, don't you think it would have been good to mention that?

Let's call $\vec H = \vec F -\vec G$. So say $\vec H = \langle h_1,h_2\rangle$ and you are talking about$$\oint_C \vec H \cdot d\vec r =\oint_C h_1\, dx + h_2\, dy$$So now if you use $\vec H(x,y) =\nabla \mu(x,y)$ what happens?

Last edited: Aug 15, 2012
5. Aug 20, 2012

### gtfitzpatrick

Hi im doing a similar problem so sooner than start a new thread...
$\oint_C \vec H \cdot d\vec r =\oint_C \frac{dμ}{dx}\, dx + \frac{dμ}{dy}\, dy$?

6. Aug 20, 2012

### LCKurtz

Yes. Now do you see how to argue that it is zero?

7. Aug 20, 2012

### gtfitzpatrick

No. in a word.
I thought i might be able to work something out using greens theorem(as the curve is closed) but its not working out...

8. Aug 20, 2012

### LCKurtz

Green's theorem is what you want. What do you get and why don't you think it works?

I have to go for a couple of hours. I'll check back and see if you have it by then.

Last edited: Aug 20, 2012
9. Aug 20, 2012

### gtfitzpatrick

$\oint_C \vec H \cdot d\vec r =\oint_C \frac{dμ}{dx}\, dx + \frac{dμ}{dy}\, dy$ = $\int\int\frac{d^2μ}{dxdy}-\frac{d^2μ}{dxdy} dxdy = 0$ but this can only be true if $\oint F.dr = \oint G.dr$ qed?

10. Aug 20, 2012

### LCKurtz

Yes. It would be more clear if the left side had started with$$\oint_C \vec F\cdot d\vec R -\oint_C\vec G\cdot d\vec R =\oint_C (\vec F-\vec G)\cdot d\vec R=\oint_C\vec H\cdot d\vec R =\ ...$$

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