The Divergence Theorem

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Wilfred Hampton
5 years ago
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1 13.8 The ivegence Theoem Back in 13.5 we ewote Geen s Theoem in vecto fom as C F n ds= div F x, y da ( ) whee C is the positively-oiented bounday cuve of the plane egion (in the xy-plane). Notice this went fom a single integal to a double integal. Notice it went fom C & ds to & da. Now extend this idea to vecto fields on F n d = div F x, y, z dv ( ) whee is the bounday suface of solid egion. Notice this goes fom a double integal to a tiple integal. Notice this goes fom & d to & dv. 3 R and you ve got the ivegence Theoem. *ee section 13.9 page 973 in the back* Notice once again what is on this page: on the left side it elates the integal of a deivative of a function ( div F ) side, the integal of the oiginal function F ove the bounday of the egion (). ove a egion () to, on the ight The ivegence Theoem: Let be a simple solid egion (egions bounded by ectangula boxes, sphees o ellipsoids) and let be the bounday suface of given with positive (outwad) oientation. Let F be a vecto field whose component functions have continuous patial deivatives on an open egion that contains. Then: F d = div F dv The flux of F acoss the suface (the bounday of ) is equal to the tiple integal of the divegence of F ove solid. The net flux emeging fom the suface of equals total divegence within. You did suface integals, ( ( )) ( ) F d, in section 13.6 and used F n d, F u, v X da, o Now we can bing it up a level and find u v div F dv. g g P Q + R da x y if z g ( x, y ) =.

2 It s just anothe way to find suface integals It equies less wok! WAY less wok! F d (the flux of F u acoss ). As befoe this is used to solve poblems involving electic flux, fluid flow, and electic field poblems in physics (flux integals). Let s ty some poblems: xample 1: Find the flux of F ove if 2 2 F = 2 x z, x y, xz and is the unit cube in 1 st octant. How would we have done this poblem back in 13.6? cube div F dv F

3 xample 2: Use the ivegence Theoem to find the flux of the vecto field F ( xyz) x + y + z = 1 with outwad oientation.,, = zyx,, ove the unit sphee How would we have done this poblem back in 13.6? F d = div F dv sphee solid need F

4 How would you have done this poblem befoe I showed you the ivegence Theoem? The poblem was to find the flux of the vecto field F ( xyz) with outwad oientation.,, = zyx,, ove the unit sphee x + y + z = 1 You would have eithe used: CA 1 F d = F ( ( u, v) ) ( u X v) da Hee, you could paametize o epesent suface as ( θ, φ) = sinφcos θ,sinφsin θ,cosφ because in spheical, x= ρ sinφcos θ, y = ρsinφsin θ, z = ρcosφ and ou ρ = 1. o we would use F ( ( θφ, )) ( θ X φ) da How would you find F( ( θ, φ ))? How would you find ( θ X φ )? How would you epesent da? o CA 2 F d = g g P Q + R da x y if you solve z fo z g ( x, y ) =. o solve x + y + z = 1 fo z. Hee, you could paametize o epesent suface as What s the P, Q, and R? What ae the patial deivatives? How would you epesent da? xy (, ) xy,, 1 x y 2 2 =. You can see CA 1 woked out on page 955 o 956 xample 4 if you ae inteested; they don t show all of thei wok.

5 Anothe Poblem you did in ection 13.6 was #25. Find you wok please. Let s take a minute to look back at you HW on this one (o I can show you mine; I did it both ways). valuate the suface integal ( ) F d, F x, y, z = x, z, y and oiented suface : pat of sphee x + y + z = 4 in the 1 st octant, with oientation towads the oigin. This is a closed simple solid so let s ty the ivegence Theoem. xample 3: valuate the suface integal ( ) F d, F x, y, z = x, z, y and oiented suface : pat of sphee x + y + z = 4 in the 1 st octant, with oientation towads the oigin. It is negatively oiented, so don t foget to attach a negative: F d = div F dv

6 Let s ty a diffeent type. Hw #7: Use the ivegence Theoem to calculate the suface integal of F acoss.) (,, ) 3 z z F xyz xy i xe j zk 3 xy, xe, z the cylinde y F d. (In othe wods, calculate the flux = + + =, is the suface of the solid bounded by + z = 1 and the planes x= 1 and x= How many suface integals would have to do if you used F g g ( ( u, v) ) ( u X v) da o P Q + R da x y?

7 When the diections say to VRIFY that the ivegence Theoem is tue, you must demonstate both sides of the ivegence Thm. (and the solutions bette match). This would be a geat final exam question. HW #2: Veify that the ivegence Theoem is tue fo the vecto field ( ) 2 by the paaboloid div F dv 2 2 z = 4 x y and the xy-plane. F xyz,, = x, xyz,, is the solid bounded F d We would need to F d = F d + F d 1 2

8 13.9 No HW in this section. Links them all togethe. Relates them all to the Fundamental Theoem of Calculus whee in each case on the left they have an integal of a deivative ove a egion, and the ight side involves the values of the oiginal function only on the bounday of the egion. TH N

So, if we are finding the amount of work done over a non-conservative vector field F r, we do that long ur r b ur =

So, if we are finding the amount of work done over a non-conservative vector field F r, we do that long ur r b ur = 3.4 Geen s Theoem Geoge Geen: self-taught English scientist, 793-84 So, if we ae finding the amount of wok done ove a non-consevative vecto field F, we do that long u b u 3. method Wok = F d F( () t )