- #1
Dustinsfl
- 2,281
- 5
Consider Laplace's equation on a sphere of unit radius with the boundary condition
Here we will consider a three-term approximation to the solution, i.e., involving the spherical harmonics for and .
Conclude that the form of the solution will be
with (shoudn't there be a 100 in front of this integral?)
Let . We have already solved for the axisymmetric case and know the solution is of the form
Therefore, (I just put this down since I couldn't derive it. Can someone show me how to get to this part?). That is, . So the general solution is
since .
(There has to be a way to show this two more elogantly than just saying this is this except it) Define and also define
Then we have
Lastly, using the boundary condition above, we have that
How do I do this?
From the definition of the spherical harmonics, write down the explicit expressions for for and .
Here we will consider a three-term approximation to the solution, i.e., involving the spherical harmonics
Conclude that the form of the solution will be
with (shoudn't there be a 100 in front of this integral?)
Let
Therefore,
since
(There has to be a way to show this two more elogantly than just saying this is this except it) Define
Then we have
Lastly, using the boundary condition above, we have that
How do I do this?
From the definition of the spherical harmonics, write down the explicit expressions for