Question

Show that potential function U= x²+y²+2z² satisfies Laplace equation

Answer 1

$\nabla^2U=\frac{\partial^2U}{\partial x^2}+\frac{\partial^2U}{\partial y^2}+\frac{\partial^2U}{\partial z^2}=0,$

$\frac{\partial U}{\partial x}=2x,$ $\frac{\partial U}{\partial y}=2y,$ $\frac{\partial U}{\partial z}=4z,$

$\frac{\partial^2U}{\partial x^2}=2,$ $\frac{\partial^2U}{\partial y^2}=2,$ $\frac{\partial^2U}{\partial z^2}=4,$

$\nabla^2U=2+2+4=8 \not =0,$

function U does not satisfy Laplace equation.