Separation of Variables -- a practice exposition
We want to solve Laplace's equation in order to find out what V (as in "electric potential") is. (Well, I do.) That equation is
∇2V(x,y,z) = 0
∇2V(x,y,z) = δ2V/δx2 + δ2V/δy2 + δ2V/δz2
For purposes of this method, we will guess that V(x,y,z) = X(x)*Y(y)*Z(z) for some unknown functions X, Y, Z. Then, going back to Laplace's equation,
∇2(X*Y*Z) = Y*Z*d2X/dx2
+ X*Z*d2Y/dy2 + X*Y*d2Z/dz2 = 0
The partial derivatives are now, of course, full derivatives because we've broken V down into functions each of which relies on just the one variable. Now divide through by X*Y*Z to make it all prettier.
1/X * d2X/dx2
+ 1/Y * d2Y/dy2 + 1/Z * d2Z/dz2 = 0
At this point we have three mutually independent functions (meaning that they each depend on a different independent variable [meaning that you can change the value of any one of the variables without one of the other variables having to change its value as well]). The sum of these functions must always be zero, which is a constant. The only way for three mutually independent functions to always sum to a given constant regardless of the values of their respective variables is for each of the functions to be constants themselves. Therefore, keeping in mind that this is all based on a tenuous assumption, we have
f(x) + g(y) + h(z) = 0
where f, g, and h are all constants and, for example, f = 1/X * d2X/dx2.
This is as far as we can go without looking at the actual constraints ("boundary values") of a given situation.
OK, time for non-boring stuff. Ummm . . . nothing really comes to mind. Still busy with college. Must scramble off to keep a check-up appointment right now. Oh, and a simple way to make soccer more interesting: Enact an "icing" rule so that the game bears less resemblance to an interminable volleyball point.
Edit: Not that there's anything wrong with volleyball, but soccer shouldn't resemble it as closely as soccer sometimes does.
Also, I just saw a license plate reading "MEZZO". I wonder if the driver honestly believes he's executing a subtle tactic every time he gets into his car?