Deriving the Spherical Laplacian (Shortcut Method) w/ Bonus Polar, Cylindrical

Here we derive the spherical laplacian using complex variables to eliminate the mountains of work that would normally be required.

Useful on examinations.

Also useful for those who can never remember the spherical laplacian and want a short method to derive it.

[pi]

You, sir, are a gift from God. You're amazing. I'm in high school, I've never taken a single calculus class, and I've understood all of this. The only calculus I've done was mainly multivariable calculus (Maxwell's equations, specifically), and I wanted a spherical Schrödinger equation. You have helped so much.

[maxwibert]

2:00 Isn't it c = r e^i(pi/2 - phi)? what am I missing here

[cucumcumber]

why are 12:07 and 13:30 completely different?

[scitwi]

Ha! I knew that it could be done easier with complex numbers! Can you put some links in the description to some resources where I can read some more about these methods? They seem very useful.

[bonbonpony]

Hmm… At 21:16 this seems a bit suspicious to me: if you didn't divide the second equation by ρ, you would get a different formula for the polar form of the Laplacian :q And since the right hand is 0, you could multiply or divide the second equation by pretty much everything and this would still work, giving all sorts of wrong formulas of course :P I think you might be tricking us here (or yourself) in a similar way as those people who prove that 1=2 because of a missed/hidden division by 0.