Laplace Operator (Laplacian) in Spherical Coordinates – PROOF

mes (67)in DTube • 2 days ago

In this video I derive the Laplace Operator or Laplacian in spherical coordinates by applying the Laplacian in polar coordinates twice: once for the azimuthal (ф) angle and once for the polar angle (θ). Adding the two polar Laplacians, cancelling out terms, and writing the partial derivatives in terms of the spherical coordinates, we obtain the Laplace Operator in spherical coordinates!

#math #polarcoordinates #calculus #sphericalcoordinates #multivariablecalculus

Timestamps:

Converting rectangular coordinates to spherical coordinates – 0:00
Will apply the Laplacian in polar coordinates twice – 9:25
Applying the Laplacian in polar coordinates to the x-y terms – 10:38
Applying the Laplacian in polar coordinates to the z-s terms – 13:43
Adding the two Laplacians and canceling out terms – 17:15
Evaluating the partial derivative in terms of s – 20:20
Combining all the terms, we obtain our Laplace Operator in spherical coordinates – 30:31
Simplifying the Laplacian to match up with Wikipedia – 37:14
Double-checking our Laplacian in Spherical Coordinates with Wikipedia – 39:57

Notes and playlists: