Laplace Operator (Laplacian) in Polar Coordinates – PROOF

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!

#math #polarcoordinates #calculus #partialdifferentialequation #multivariablecalculus