Module 1 : A Crash Course in Vectors
Lecture 6 : Laplacian
  Laplacian in cylindrical and spherical coordinates:
  In cylindrical :
\begin{displaymath}\nabla^2 = \frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\... ...{\partial^2}{\partial\theta^2}+ \frac{\partial^2}{\partial z^2}\end{displaymath}
  In spherical :
 
\begin{displaymath}\nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\... ...ht)+ \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\end{displaymath}
  Frequently the Laplacian of a vector field is used. It is simply a short hand notation for the componentwise Laplacian
 
\begin{displaymath}\nabla^2\vec F = \hat\imath \nabla^2 F_x + \hat\jmath \nabla^2 F_y +\hat k \nabla^2 F_z\end{displaymath}
 

Exercise 5

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