1. Also known as Gauss’ Theorem
Divergence Theorem
Also known as Gauss’ Theorem

2. Divergence
In calculus, the divergence is used to measure the magnitude of a vector field’s source or sink at a given point
Thus it represents the volume density of the outward flux of a vector field
The air inside the container has to compress and will eventually leak out
Imagine air as it is heated
It expands in all directions
The velocity field points outward from the region
The divergence of the velocity field would have a positive value and the region would be a called a source
If it was being cooled the velocity field would have a negative value and the region would be called a sink

3. Divergence
Thus we can think of the divergence of a vector field as the extent to which it behaves like a sink or a source
We can think of it as a measure of “outgoingness”
To what extent is there more exiting a region than entering it
Thus the expansion of a fluid flowing with a velocity field of is captured by the divergence of
The divergence is a scalar
At a given point, the divergence is a single number that represents how much of the flow is expanding at that point

5. Divergence
Since divergence can be thought of as a measure of the rate at which density exits (or enters a region), the net content leaving (or entering) a region is given by the sum of the partials of the vector field
For a 2D vector field
For a 3D vector field

6. Divergence Theorem Imagine pumping air into a container
The air inside the container has to compress and will eventually leak out
We are making the assumption the container won’t explode
Now if a vector field represents the flow of a fluid, the divergence of represents the expansion (or compression) of the fluid
The divergence theorem says that the total expansion of the fluid inside some 3D region, W, is equal to the flux integral of the vector field over the surface that is the boundary of the of W

7. Divergence Theorem
Given a simple solid region W with a positive orientation and a boundary region S, the divergence theorem says
Essentially this says that the flux can be calculated with a triple integral (sometimes referred to as a volume integral)
This is similar to Green’s theorem which only applies when we have a closed path
The divergence theorem only applies if we have a closed surface
If the surface is not closed, it must be made so to apply this theorem.

8. Divergence Theorem
Calculate the flux of of the surface S which is a hemisphere given by the following
In this case we have a 3D region and with an outward orientation
When we computed this before we got