Question

prove that the function u = 2x (1-y) is harmonic

Answer 1

$$u=2x(1-y)$$

$$\dfrac{\partial u}{\partial x}=2(1-y)$$

$$\dfrac{\partial^2 u}{\partial x^2}=0$$

$$\dfrac{\partial u}{\partial y}=-2x$$

$$\dfrac{\partial^2 u}{\partial y^2}=0$$

$$\nabla^2u(x,y)=\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}$$

$$=0+0=0$$

Then the function $u = 2x (1-y)$ is harmonic.