Laplace's Equation in Spherical Coordinates (Axial Symmetry): Separation of Variables!!

Hi! Today we are going to be looking at the fundamental equation in electrostatics, Laplace's Equation. This is a really important Partial DIfferential Equation, and the way we are going to be solving it is by looking for solutions of a certain type. In this case, we will be looking for separable solutions, that is , solutions in which the function which depends on more than one variable is split up into functions that only depend on one variable multiplying each other.

The video is a bit long, but it's not because solving the equation is hard, but rather because there are some previous results that I wanted to explain.

As always, here are the time stamps so you can find what you are looking for faster:

0:00 Introduction
0:16 Laplacian in Spherical Coordinates
1:06 Euler's Differential Equation
2:20 Legendre's Differential Equation
4:15 Rodrigues' Formula for Legendre's Polynomials
6:26 Maxwell's Equations: Derivation of Laplalce's Equation
12:40 1. We propose a Separable Solution
13:28 2. Spherical Coordinates and Axial Symmetry
17:00 3. Laplacian in Sphercial Coordinates
23:05 Mistake that is fixed later: I forgot a sin(theta) dividing
23:20 4. Separation into 2 ODEs
27:50 5. Euler's Differential Equation for R(r)
41:00 Fixing the mistake
41:28 Change of Variables: Reduction to Legendre's Equation
58:50 Solution to the Second ODE
1:05:26 Final result: Expression for the Potential

I hope you enjoyed the video, and if you want to see more problems like this just subscribe to the channel. See you in the next video!

Видео Laplace's Equation in Spherical Coordinates (Axial Symmetry): Separation of Variables!! канала Sandro’s Space