Formal definition of divergence in two dimensions
What we're building to
- In two dimensions, divergence is formally defined as follows:
"Outward flow" at a point doesn't really make sense
- Divergence is a function which takes in individual points in space.
- The idea of outward flow only makes sense with respect to a region in space. You can ask if a fluid flows out of a given region or into it, but it doesn't make sense to talk about fluid flowing out of a single point.
From a region to a point
- is a vector-valued function representing the velocity vector field of some fluid.
- is a closed loop in the -plane.
- is a function that gives the outward unit normal vector at all points on the curve .
- Rather than just writing , write to emphasize that this region contains a specific point .
"Simple" example: Constant divergence
What about higher dimensions?
- Given a fluid flow, divergence tries to capture the idea of "outward flow" at a point. But this doesn't quite make sense, because you can only measure the change in fluid density of a region.
- When talking about a region, the idea of "outward flow" is the same as flux through that region's boundary.
- To adapt the idea of "outward flow in a region" to the idea of "outward flow at a point", start by considering the average outward flow per unit area in a region. This just means dividing the flux integral by the area of the region.
- Next, consider the limit of this outward flow per unit area as the region shrinks around a specific point.
- Putting this all into symbols, we get the following definition of divergence: