Laplace Equation in Polar Coordinates

The given equation is Laplace's equation in Cartesian coordinates:

$ \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0 $

We need to transform this equation into polar coordinates ($r, \theta$), where $x = r \cos \theta$ and $y = r \sin \theta$.

Coordinate Transformation Formula

The Laplacian operator ($\nabla^2$) in polar coordinates is given by the formula:

$ \nabla^2 u = \frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} $

Transformed Equation

Setting the Laplacian in polar coordinates equal to zero gives the transformed Laplace equation:

$ \frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} = 0 $

This matches Option A.