about $\nabla^{4}\Phi=0$, write down this equation in terms if Cartesian Coordinates

Question

$\nabla^{4}\Phi=0$, write down this equation in terms if Cartesian Coordinates(x,y). I am a bit confused here, the question doesn't tell you if $\Phi$ is scalar or vector, but i think it is a vector, and i am not quite sure about the precise definition of $\nabla^{4}$ either...

Answer 1

I am assuming that $\Phi$ is scalar. I see it as:

$$\nabla^4 \Phi=\nabla \cdot(\nabla( \nabla\cdot (\nabla \Phi)) =\nabla \cdot\left(\nabla\left(\dfrac{\partial\Phi}{\partial x}+\dfrac{\partial\Phi}{\partial y}\right)\right)= \dfrac{\partial^4\Phi}{\partial x^4}+\dfrac{\partial^4\Phi}{\partial y^4}$$