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

In summary, the conversation discusses a function V(x,y,z) that follows Laplace's equation over a path in space parametrized by \vec{r}(t). The question is whether the extreme values of V along that path are located at the end points, V(\vec{r}(a)) and V(\vec{r}(b)).
  • #1
quasar987
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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 [itex]\vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}[/itex], [itex]a\leq t \leq b[/itex], [itex]\nabla ^2 V(\vec{r}(t))=0[/itex].

Is it true that the extreme values of V along that path are located at the end? (I.e. at [itex]V(\vec{r}(a))[/itex] and [itex]V(\vec{r}(b))[/itex])?
 
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  • #2
I don't know too much math,but i'll say that your problem is 1D,so that V should satisfy some ODE...


Daniel.
 
  • #3


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.
 

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

What is Laplace's equation and what does it mean for a solution to be harmonic?

Laplace's equation is a partial differential equation that describes the behavior of a scalar field in space. A solution is considered harmonic if it satisfies Laplace's equation and has a smooth, well-behaved behavior.

What does it mean for a path in space to be harmonic?

A path in space is considered harmonic if the solution to Laplace's equation remains smooth and well-behaved along the path. This means that the values of the solution do not suddenly change or become discontinuous.

Why is it important to determine if a solution to Laplace's equation is harmonic over a path in space?

Determining if a solution is harmonic over a path in space is important because it ensures that the solution is well-behaved and reliable. This is crucial in many scientific and engineering applications where accurate predictions and calculations are necessary.

How is the harmonic condition checked for a solution to Laplace's equation over a path in space?

The harmonic condition can be checked by evaluating the solution at different points along the path and ensuring that it remains smooth and satisfies Laplace's equation. This can be done analytically or numerically using computational methods.

What are some real-world examples of Laplace's equation and its solution being harmonic over a path in space?

Laplace's equation can be used to model various physical phenomena such as heat flow, fluid dynamics, and electrical potential. In these cases, the solution must be harmonic over a path in space to accurately describe and predict the behavior of the system.

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