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.

 

[tacman]

The solution to Laplace's equation is indeed harmonic over a path in space. This means that the function V(x,y,z) satisfies the Laplace's equation at every point along the path, as given by the equation \nabla ^2 V(\vec{r}(t))=0. This is a fundamental property of Laplace's equation and is true regardless of the specific path chosen.

As for the question of whether the extreme values of V along the path are located at the end points, it depends on the specific path and the boundary conditions. If the boundary conditions dictate that the function must have a maximum or minimum value at the end points, then yes, the extreme values of V will be located at the end points. However, if the boundary conditions allow for extreme values at other points along the path, then it is possible for the extreme values to occur at points other than the end points.

In summary, the solution to Laplace's equation is harmonic over a path in space, but the location of the extreme values of the function V along the path depends on the specific path and boundary conditions.