Evaluate integral of second order differential

[billiards]

Homework Statement

Evaluate
$$\int^{2\pi}_{0}\frac{1}{r}\frac{d^{2}f}{d\varphi^{2}}d\varphi$$

Homework Equations

n/a

The Attempt at a Solution

I get the integral as

$$\frac{1}{r}\frac{df}{d\varphi}$$

Not sure how to evaluate this.

 

[jackmell]

Why don't you do a simple one first. Say:

$$f(x)=x^3+2x$$

so that:

$$\frac{df}{dx}=3x^2+2$$

and so if I want to integrate:

$$\int_0^{2\pi}\frac{df}{dx}dx=\int_0^{2\pi} (3x^2+2)dx=(x^3+2x)\biggr|_0^{2\pi}=f(2\pi)-f(0)$$

edit: so I see you got that part. Then it's just the first derivative at the end points right? So just write it as such:

$$\frac{df}{d\phi}\biggr|_{0}^{2\pi}$$

 

[billiards]

Thanks for quick response. (I had some terrible trouble with Latex, I don't know if other users were seeing what I was, but when I tried to preview my post it would not refresh the Latex even though I had changed the coding.)

Is there not any way that I can take this further. This is actually a small part of a larger problem in which I have to show that this integral is negligible.

Hey, I think I get it now! The function is defined on a disk, so it must be the same at 0 and 2 pi. Therefore this term is negligible.

Does that sound right?

 

[jackmell]

I'd say if the derivative is analytic throughout the disk and you're integrating over a closed contour, then the integral is zero.

 

[billiards]

Thanks. This is part of a problem on harmonic functions (I still don't really know what they are) -- I guess that term puts some constraints on the function, hopefully the ones you mentioned above.

Cheers

 

[Mark44]

Thanks for quick response. (I had some terrible trouble with Latex, I don't know if other users were seeing what I was, but when I tried to preview my post it would not refresh the Latex even though I had changed the coding.)

This is a known problem that has yet to be fixed. The problem seems to be that when you preview a post with LaTeX in it, the previewer grabs whatever is in a cache somewhere, so if you have made changes, they won't show up in a preview. The only way around this that I know is to refresh the page after you have clicked Preview Post.

 

[billiards]

$$\sqrt{Thanks Mark}$$

Thant trick works for me!