Check whether the potential function V = A log ρ + B in cylindrical co-ordinate is a solution of Laplace's equation. A and B are constants.

Explanation: 

Laplace’s equation states that the sum of the second-order partial derivatives of U (function) with respect to the coordinates, equals zero.

∇2 is called the Laplacian or Laplace operator.

In Cartesian coordinates:

\({∇ ^2}U = \frac{{{\partial ^2}U}}{{\partial {x^2}}} + \frac{{{\partial ^2}U}}{{\partial {y^2}}} + \frac{{{\partial ^2}U}}{{\partial {z^2}}} = 0\)

In Cylindrical coordinates:

\({∇ ^2}U = \frac{1}{ρ}\frac{\partial }{{\partial ρ}}\left( {ρ\frac{{\partial U}}{{\partial ρ}}} \right) + \frac{1}{{{ρ^2}}}\frac{{{\partial ^2}U}}{{\partial {{\rm{\Theta }}^2}}} + \frac{{{\partial ^2}U}}{{\partial {z^2}}} = 0\)

Analysis:

V = A log ρ + B

\({∇ ^2}V = \frac{1}{ρ}\frac{\partial }{{\partial ρ}}\left( {ρ\frac{{\partial V}}{{\partial ρ}}} \right) + \frac{1}{{{ρ^2}}}\frac{{{\partial ^2}V}}{{\partial {{\rm{\Theta }}^2}}} + \frac{{{\partial ^2}V}}{{\partial {z^2}}} \)

\({∇ ^2}V = \frac{A}{ρ}\frac{\partial }{{\partial ρ}}\left( {ρ\frac{{1}}{{ ρ}}} \right) + 0 + 0\)

= 0

∴ It satisfies the Laplace equation.