Can you please critique my reasoning here? I know that there is a leap I am making that is incorrect, or I don't fully understand a critical piece.
I am looking at the Laplacian of $f(x,y)=2yx – xy^4$. I calculated $\Delta f(x,y) = -12xy^2$.
I am trying to interpret what this tells me about the function. I know that the Laplacian is just $\Delta f(x,y)=\nabla \cdot \nabla f(x,y)$ which is the divergence of $\nabla f(x,y)$.
Here is where I am struggling…points where the divergence is positive represent sources in the vector field, and points where the divergence is negative represent sinks in the vector field. Sinks in $\nabla f(x,y)$ correspond to local maximums of $f(x,y)$ while sources corresponds to local minimums.
My question is, there are infinitely many points where $-12xy^2$ is positive and infinitely many where $-12xy^2$ is negative, though I know these points are not all local extrema of $f(x,y)=2yx-xy^4$, so there are some inaccuracies in my logic above.
Am I even remotely interpreting the divergence relating to maximums and minimums correctly here? Any help would be greatly appreciated.
Best Answer
The words "source" and "sink" are somewhat unfortunate in describing a phenomenon spread out over a region. When we have a flow field ${\bf v}$ then ${\rm div}\, {\bf v}({\bf p})>0$ means that for a small disk (in 2D) or ball (in 3D) centered at ${\bf p}$ there is more fluid flowing out per second on one side of this disk than is flowing in on the other side of this disk. If this is true for ${\bf p}$ this is true for all points near ${\bf p}$. The reason could be that it rains in the neighborhood of ${\bf p}$ and that this additional amount of fluid has to flow away somehow in addition to the basic underground flow.
Note that local extrema ${\bf p}$ of some scalar function $f$ are signaled by $\nabla f({\bf p})={\bf 0}$, and not by the divergence of ${\bf v}:=\nabla f$ at ${\bf p}$ being positive, resp., negative.