Question

The transformed equation of \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \) in polar coordinates is:

• A. \(\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} = 0\)
• B. incorrect
• C. incorrect
• D. incorrect

Hint:

Memorize Laplace equation in polar coordinates—it is directly used in PDE problems.

Answer 1

The Correct Option is A

Concept: This is Laplace’s equation in Cartesian coordinates. When converted to polar coordinates, it takes a standard known form.

Step 1: Use coordinate transformation.
\[ x = r\cos\theta,\quad y = r\sin\theta \]

Step 2: Apply known Laplacian form.
\[ \nabla^2 u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} \]

Step 3: Substitute into equation.
\[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} = 0 \] Final Answer: \[ \boxed{\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} = 0} \]