Chapter 3: Calculating Potentials Separation of Variables
Separation of variables is a powerful technique that can be used to solve partial differential equations. In Physics 217 we will be using this technique to solve Laplace’s equation.
Separation of variables relies on the first uniqueness theorem.
In three dimensions, the potential V will in general depend on three variables: for example V(x, y, z) or V(r,q,f).
To solve Laplace’s equation we try the following solutions:
-
- V(x, y, z) = X(x) Y(y) Z(z)
-
- V(r,q,f) = V(r) a(q) ?(f)
-
If you can solve Laplace’s equation using these assumption, you have found THE solution.