Let's say I have the Laplace equation: $\nabla^2\phi=0$. Does it make any sense to integral both sides? Or is it only solvable writing it as $\frac{\partial^2\phi}{\partial x^2} +\frac{\partial^2\phi}{\partial y^2} +\frac{\partial^2\phi}{\partial z^2}=0$ and then solving a PDE? Is there any meaning to doing a single integral on a gradient of some scalar function which is dependent upon many variables? How can it be done?
I think my confusion stems from a line integral, where $\int_C \vec{\nabla} \phi \cdot \vec{ds} =\phi(B)-\phi(A)$ considering A is the starting position of the curve C, and B the end. But in any other context, is it applicable somehow?
I will be very glad for someone to put light on this, and maybe end my confusion regarding the connection between integrals and gradients. I'm a physics undergraduate, so I haven't covered this material with mathematical rigor.