Download presentation

Presentation is loading. Please wait.

Published byDominique Hiscox Modified over 3 years ago

1
Lecture 9: Divergence Theorem Consider migrating wildebeest, with velocity vector field 1 m x y Zero influx (s -1 ) into blue triangle: 0 along hypotenuse along this side Zero influx into pink triangle along this side along this side along this side

2
Proof of Divergence Theorem V S Divide into infinite number of small cubes dV i Take definition of div from infinitessimal volume dV to macroscopic volume V Sum Interior surfaces cancel since is same but vector areas oppose 12 DIVERGENCE THEOREM over closed outer surface enclosing V summed over all tiny cubes

3
2D Example “outflux (or influx) from a body = total loss (gain) from the surface” x y Circle, radius R (in m) exactly as predicted by the divergence theorem since There is no ‘Wildebeest Generating Machine’ in the circle!

4
3D Physics Example: Gauss’s Theorem S Where sum is over charges enclosed by S in volume V Consider tiny spheres around each charge: Applying the Divergence Theorem The integral and differential form of Gauss’s Theorem independent of exact location of charges within V

5
Example: Two non-closed surfaces with same rim S2S2 S1S1 Which means integral depends on rim not surface if If then integral depends on rim and surface S is closed surface formed by S 1 + S 2 Special case when shows why vector area

6
Divergence Theorem as aid to doing complicated surface integrals Example y z x Evaluate Directly Tedious integral over and (exercise for student!) gives

7
Using the Divergence Theorem So integral depends only on rim: so do easy integral over circle as always beware of signs

8
Product of Divergences Example free index Kronecker Delta: in 3D space, a special set of 9 numbers also known as the unit matrix Writing all these summation signs gets very tedious! (see summation convention in Lecture 14).

Similar presentations

OK

Lecture 5 R 2R2R Yesterday we introduced electric field lines Today we will cover some extra topics on Electric Fields before going on to Electric Flux.

Lecture 5 R 2R2R Yesterday we introduced electric field lines Today we will cover some extra topics on Electric Fields before going on to Electric Flux.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on forex market Download ppt on design and construction of floating structure Ppt on world book day april Ppt on extranuclear inheritance Ppt online shopping projects Ppt on 3d tv without glasses Ppt on different types of computer softwares for wireline By appt only movie classics Ppt on new zealand culture past Ppt on depth first search adjacency