Homework Help: Double derivatives of F as a function of F and harmonicity

1. Mar 17, 2013

Verdict

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

2. Relevant equations
I can't think of much. u(x,y) is harmonic, so its double derivatives with respect to x and y add up to zero. I'm not 100% sure, but does being harmonic also imply that u satisfies the cauchy riemann equations? That might come in handy in the denominator.

3. The attempt at a solution
Where to begin, where to begin. To me, the question sounds very vague. The second derivative of a constant, isn't that just zero? What should I do? I know this is a lousy way to post a question, but I honestly don't even know where to begin. If you have any tips (just to get me started!) that would be great.

Kind regards

Last edited: Mar 17, 2013
2. Mar 17, 2013

SteamKing

Staff Emeritus
I always thought that if u(x,y) is harmonic, then DEL^2 u = 0.

3. Mar 17, 2013

Verdict

That was a terrible typo on my part, in my case it means that the second derivative of u with respect to x plus the second derivative of u with respect to y add up to 0, which is the 2D version of what you are saying

4. Mar 18, 2013

HallsofIvy

I really don't understand what you mean by "the equation F(x,y)= constant" can be expressed as u(x,y)= constant with u harmonic" unless you mean that f(x, y) is a harmonic function plus a constant: F(x,y)= u(x,y)+ C.
Then $\partial^2 F/\partial x^2= \partial^2 u/\partial x^2$ and $\partial^2F/\partial y^2= \partial^2 u/\partial y^2$ and, as you say, the sum of those is 0.

5. Mar 18, 2013

Verdict

I'm not sure, I'll email my teacher to ask that, because it doesn't make sense indeed.

6. Mar 18, 2013

Verdict

consider the set defined by u(x,y)=constant. This is not the only way to describe this set. For example, if u(x,y)=constant also (u(x,y))^2=constant, or take any function F applied to u(x,y) give F(u(x,y))=constant. Now can you choose F such that F(u(x,y)) is harmonic?

7. Mar 19, 2013

Verdict

i've been trying to figure this out for a day or so now, but I really don't seem to be making a lot of progress. I tried plugging in F(u(x, y)) with u harmonic, but not a whole lot cancels. Could anyone provide a hint?

8. Mar 20, 2013

Verdict

I was thinking, maybe I should use implicit differentiation, as the function is equal to a constant? I don't really see where that gets me, but I am pretty desperate at this point.