Let me return to my model of the eye. With the functions as the solution to the angular equation, we find that the solutions to the radial equation are
(11.36) |
The singular part is not acceptable, so once again we find that the solution takes the form
(11.37) |
We now need to impose the boundary condition that the temperature is C in an opening angle of , and elsewhere. This leads to the equation
This leads to the integral, after once again changing to ,
(11.39) |
These integrals can easily be evaluated, and a sketch for the temperature can be found in figure 11.3 .
Notice that we need to integrate over to obtain the coefficients . The integration over in spherical coordinates is , and so automatically implies that is the right variable to use, as also follows from the orthogonality of .