Is the solution to Laplace's equation harmonic over a path in space

[quasar987]

I was hesistant wheter to post this in the physics of math section but it's much of math problem I think.

Suppose I have a function V(x,y,z) which obeys Laplace's equation over some path in space. That is to say, for some path parametrized by $\vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}$, $a\leq t \leq b$, $\nabla ^2 V(\vec{r}(t))=0$.

Is it true that the extreme values of V along that path are located at the end? (I.e. at $V(\vec{r}(a))$ and $V(\vec{r}(b))$)?

 

[dextercioby]

I don't know too much math,but i'll say that your problem is 1D,so that V should satisfy some ODE...


Daniel.