Laplace Operator (Laplacian) in Polar Coordinates – PROOF

mes (67)in DTube • 4 days ago

In this video I derive the Laplace Operator or Laplacian in polar coordinates, which will come in handy when I derive the Laplacian in spherical coordinates in the next video. Since polar coordinates are in two dimensions (2D), the corresponding Laplacian is also in 2D, and is the sum of the second partial derivatives of a function in terms of x and y. I use the tree structure for remembering the order of chain rule for partial derivatives to first write the first partial derivative of x and y in terms of their corresponding polar coordinates r and θ. The second partial derivative is simply the partial derivative of the first partial derivative. After a lot of algebra and cancellations, I obtain the Laplacian in Polar Coordinates! Note that in the derivation I used the symmetry of partial derivatives, but after watching this video again, I realized it's not actually needed as the terms cancel without it — let me know if you spot which terms these are!

#math #polarcoordinates #calculus #partialdifferentialequation #multivariablecalculus

Timestamps:

Derivation based on Michael Dabkowski's Mike, the Mathematician YouTube channel @mikethemathematician – 0:00
Laplace operator in 2D rectangular coordinates – 1:20
Converting rectangular coordinates to polar coordinates – 2:22
Writing the first partial derivative using the chain rule and associated Tree Structure – 5:15
Obtaining first partial derivative in terms of x by plugging in our polar coordinates conversion to get rid of x and y terms – 10:08
Obtaining first partial derivative in terms of y using similar steps – 20:33
Obtaining the second partial derivative in terms of x – 25:13
Obtaining the second partial derivative in terms of y – 33:21
Symmetry of second derivatives (Schwarz integrability condition or Clairaut's theorem): Partial derivative of a partial derivative can switch order – 39:50
Adding second partial derivatives to obtain our Laplacian in polar coordinates – 41:30
- Note that I could have canceled terms without even using the symmetry of second derivatives!
Double-checking our equation with Wikipedia – 54:08