Solving Poisson's Equation with Curvilinear Coordinates

[Benny]

Hi, I've just read through a section on curvilinear coordinates and I would like to know how I could go about doing the following question.

Q. Solve Poisson's equation $\nabla ^2 f = c$ where c is a constant and assuming that f = f(r) depends only on the radius $r = \sqrt {x^2 + y^2 + z^2 }$.

Note: This question shouldn't require any knowledge of how to solve PDEs. Knowledge of the content of the curvilinear coordinates topic should be enough.

If someone could point me in the right direction that would be good thanks. I don't need a full solution, just the essential ingredients which will allow me to complete this question. Any help would be good thanks.

 

[HallsofIvy]

Hi, I've just read through a section on curvilinear coordinates and I would like to know how I could go about doing the following question.

Q. Solve Poisson's equation $\nabla ^2 f = c$ where c is a constant and assuming that f = f(r) depends only on the radius $r = \sqrt {x^2 + y^2 + z^2 }$.

Note: This question shouldn't require any knowledge of how to solve PDEs. Knowledge of the content of the curvilinear coordinates topic should be enough.

If someone could point me in the right direction that would be good thanks. I don't need a full solution, just the essential ingredients which will allow me to complete this question. Any help would be good thanks.

Do you know what $\nabla^2 f$ is in spherical coordinates? The fact that f= f(r) reduces the problem to one dimension only.

 

[Astronuc]

One could write the Laplacian in spherical coordinates, in which case the operator is simply in terms of $$\frac{\partial}{\partial{r}}$$ since the other partials are zero.

 

[Kurdt]

Seeing as no knowledge of how to solve differential equations is required, you can write down the curvilinear expression for the poisson equation, simplify it and hand that in as your solution. I assume you know nabla squared for curvilinear coordinate systems?

Edit: Beaten to it never mind.

 

[Benny]

Thanks for the pointers. I don't know what the Laplacian in spherical coordinates is, off the top of my head, but I should be able to derive it.

BTW: I don't need to 'hand this in', it's just a problem from my exercise book.