Answer to Question #143064 in Calculus for Jay Kumar

Question

If u = x2 + y2
then the value of

Answer 1

$\displaystyle u(x, y) = x^2 + y^2\\ \frac{\partial u}{\partial x} = 2x, \frac{\partial^2 u}{\partial x^2} = 2\\ \frac{\partial u}{\partial y} = 2y, \frac{\partial^2 u}{\partial y^2} = 2\\ \textsf{Since}\, \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 4\,,u\, \textsf{is not harmonic.}\\ u\, \textsf{is not regular and has singularities}\\ \textsf{at every point except at}\,x = y = 0.$