1 / 8

Divergence 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

bjorn
Download Presentation

Divergence Theorem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  4. 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

  5. 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

  6. 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.

  7. 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

More Related