# 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

Staff Emeritus
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.