Math 2A lecture 22 p. 1/12 Math 2A lecture 22 Integral theorems in 3D T.J. Barnet-Lamb tbl@brandeis.edu Brandeis University
Math 2A lecture 22 p. 2/12 Announcements Homework eleven due Friday. Homework twelve is posted, due Tuesday after Thanksgiving. Office hours are 2 3.3pm today See the website for all sorts of course-related fun http://people.brandeis.edu/ tbl/math2a/ It is your responsibility to log into LATTE and check that the grades are entered correctly. So far, HW 1 6 and HW 8 1 should be posted.
Previously on math 2a We looked at several kinds of integral in 3D. Vector line integrals measure how much the vector field is helping/hindering us as we try to move along a path Flux integrals measure how much vector stuff is going through a boundary which will be a surface in 3D. We looked at div (which measures how much vector stuff is being created or destroyed at a given point in a flow), and curl (which measures how much a little circle/sphere would want to rotate from the momentum the flow gives it). Math 2A lecture 22 p. 3/12
Math 2A lecture 22 p. 4/12 Previously on math 2a We saw two forms of Greens theorem (not counting the crazy one in the book!) One connected 2D flux integrals with (2D region) integrals of div. One connected 2D line integrals with (2D region) integrals of curl. We could see the first one quite clearly as a consequence of the law of conservation of vector stuff The other only became clear after quite a bit of algebra. But after we d found it, we could kinda-sorta see visually why it makes sense.
Math 2A lecture 22 p. 5/12 Example: Gauss s theorem Compute the flux of the vector field F(x,y,z) = y,x,z outward through the boundary of the solid region E enclosed by the paraboloid z = 1 x 2 y 2 and the plane z =.
Math 2A lecture 22 p. 5/12 Example: Gauss s theorem Compute the flux of the vector field F(x,y,z) = y,x,z outward through the boundary of the solid region E enclosed by the paraboloid z = 1 x 2 y 2 and the plane z =. Deja vu? This is the same example we did directly last time. It will be easier using Gauss s theorem (aka the divergence theorem).
Math 2A lecture 22 p. 5/12 Example: Gauss s theorem Compute the flux of the vector field F(x,y,z) = y,x,z outward through the boundary of the solid region E enclosed by the paraboloid z = 1 x 2 y 2 and the plane z =. Deja vu? This is the same example we did directly last time. It will be easier using Gauss s theorem (aka the divergence theorem). We work out F = 1.
Math 2A lecture 22 p. 6/12 Example: Gauss s theorem Thus the divergence theorem tells us we can work out the answer by integrating 1 over the inside of the region. (Let s call this 3D region R.) answer = = R = 2π 1dxdy dz = (1 r 2 )r dθ dr = r 2 1r dz dθ dr r r 3 dθ dr r r 3 dr = 2π[r 2 /2 r 4 /4] 1 = 2π(1/4) = π/2
Math 2A lecture 22 p. 7/12 Example: Gauss s theorem II Compute the flux of the vector field F(x,y,z) = z,y,x outward through the unit sphere. (Remember that the unit sphere can be parameterized by r(u,v) = sin u cos v, sin u sin v, cos u. ) The divergence is 1. Thus we get the answer by integrating 1 over the region (R, say) inside the unit sphere. We just get (4/3)π. We could also do the integral the long way. This integral is easiest in spherical polars, and I forgot to mention how to do integrals in spherical polars before! The only trick is that dxdy dz ρ 2 sin φdρdθ dφ
Math 2A lecture 22 p. 8/12 Example: Gauss s theorem II Doing the integral the long way: answer = = = R π π 1dxdy dz = π 1 3 [ρ3 ] 1 sin φdθ dφ = π 2π 3 sin φdφ = 2π 3 [ cos φ]π = 4π 3 1ρ 2 sin φdρdθ dφ 1 3 sinφdφ
Math 2A lecture 22 p. 9/12 Example: Stokes s theorem Compute the line integral F dr, where C is the circular C path p(t) = cos t, sin t, 1 + sin t + cos t and F is the vector field F(x,y,z) = y,x 2 x,z 2 z.
Math 2A lecture 22 p. 9/12 Example: Stokes s theorem Compute the line integral F dr, where C is the circular C path p(t) = cos t, sin t, 1 + sin t + cos t and F is the vector field F(x,y,z) = y,x 2 x,z 2 z. The curl is simply,, 2x. Thus we just need to integrate the flux of,, 2x through any surface with boundary C. Let s take the elliptical region in the plane z = 1 + x + y. This has parameterization r(u,v) = v cosu,v sin u, 1 + v cosu + v sin u for 2π and v 1.
Math 2A lecture 22 p. 1/12 Example: Stokes s theorem We work out (curl F)(r(u,v)) =,, 2v cosu r u = v sinu,v cosu, v sin u + v cosu r v = cosu, sin u, cos u + sinu r u r v =?,?, v sin 2 u v cos 2 u =?,?, v We want the flux upwards through the surface, so we want the other normal. Then the answer is (curl F)(r(u,v)) ( r u r v )dv du = 2v 2 cos udv du
Math 2A lecture 22 p. 11/12 Example: Stokes s theorem Working out the integral F(r(u,v)) ( r u r v )dv du = = 2 3 v3 1 cosudu = 2 3 cosudu = 2v 2 cosudv du
Math 2A lecture 22 p. 12/12 Exercise: Stokes theorem Compute the line integral F dr, where C is the circular C path p(t) = cost, sin t, 2 cos t and F is the vector field F(x,y,z) = y 2,x,z 2. Also set up the integral you would have to do if you did this directly as a line integral.