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So the question I have is: since to guarantee that the 2D Laplace w/ Neumann BCs have a solution, the net flux through the boundary need to be zero. Why the equation above says the integration should not equal to zero? And basically my question is: why let the net flux through the boundary be zero can guarantee Laplace equation has a solution? kmcrane

@Haloee: the symbol $\overset{!}{=}$ means "should be equal to" or "must be equal to" or "we want this to be equal to." In contrast, the symbol $\ne$ means "not equal to."

To put into words what is being said on the slide: Suppose we want to solve a Laplace equation $\Delta \phi = 0$ subject to Neumann boundary conditions $n \cdot \nabla \phi = g$ for some function $g$ along the boundary. From the divergence theorem, we know that the integral over the entire domain of $\Delta\phi$ must be equal to the integral of the normal derivative of the boundary. And the integral of the right-hand side of our Laplace equation (zero) is just zero. So in order for the left- and right- hand sides of our equation to agree, it must be the case that the integral of the normal derivative over the boundary equals zero. But that won't be true unless we carefully picked our boundary function $g$.