Also known as Gauss’ Theorem Divergence Theorem Also known as Gauss’ Theorem

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

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

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

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

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.

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