MTH234 Chapter 16 - Vector Calculus Michigan State University

Transcription

1 MTH234 hapter 6 - Vector alculus Michigan State Universit 4 Green s Theorem Green s Theorem gives a relationship between double integrals and line integrals around simple closed curves. (Start and end at the same point. Are not self-intersecting ecept at endpoints.) Draw picture of region D with boundar. Positive and negative orientation. Definition(s) 4... A simple closed curve has positive orientation if its parametrization traverses the curve eactl once in a counterclockwise direction. 2. A simple closed curve has negative orientation if its parametrization traverses the curve eactl once in a clockwise direction. Theorem 4.2. Let be a positivel oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded b. If F P, Q have continuous partial derivatives on an open region that contains D then, P d + Q d D ( Q P ) da or equivalentl F T ds D ( Q P ) da The idea of the proof is important because it will come up again in Stokes Theorem. The idea is circulation. Because we have a closed simple curve the integral F T ds counts how the particles on the curve are circulating. Green s Theorem sas that instead of counting how picture here. breaking up and canceling ecept at edges the particles are circulating on the curve we can count how the particles are circulating inside the curve.

2 MTH234 hapter 6 - Vector alculus Michigan State Universit That is (irculation of points on curve) (irculation of points inside curve) Idea of Proof So now we need to determine circulation at a point. consider circulation around small rectangles. Along the 4 boundaries of the rectangle we get: First lets Top: F(, + ) ( i) P (, + ) (, + ) F ( i) ( +, + ) Bottom: F(, ) (i) P (, ) F ( j) F (j) Right: F( +, ) j Q( +, ) (, ) F (i) ( +, ) Left: F(, ) ( j) Q(, ) Grouping favorabl we get: irculation of Top + Bottom + Right + Left irculation of P (, + ) + P (, ) + Q( +, ) + Q(, ) irculation of ( P (, + ) + P (, )) + Q( +, ) Q(, ) irculation of ( P + Q ) Now we need to scale from circulation on a rectangle to circulation at a point irculation at irculation of Area of irculation at ( P + Q ) Q P Q P And so now we are read to see wh we love Green s Theorem 2

3 MTH234 hapter 6 - Vector alculus Michigan State Universit Eample 4.3. Find the work done b F 4 2, 2 4 once counterclockwise around the curve given b the picture: Solution. Let s pretend we forgot Green s Theorem on the eam. 3 To parametrize this curve correctl I need to break it into 4 pieces F T ds B F T ds + F T ds + F T ds + F T ds T LL RL Parametrizing the four pieces we see that (in a counterclockwise direction) B : r(t) cos t, sin t t [π, 0] r (t) sin t, cos t T : r(t) 3 cos t, 3 sin t t [0, π] r (t) sin t, 3 cos t LL : r(t) t, 0 t [, 3] r (t), 0 RL : r(t) t, 0 t [, ] r (t), 0 Let s calculate these individual integrals 0 B F T ds π 4 2, 2 4 r (t) dt 0 4(cos t) 2(sin t), 2(cos t) 4(sin t) sin t, cos t dt π 0 π 4 sin t cos t + 2 sin2 t + 2 cos 2 t 4 sin t cos t dt 0 8 sin t cos t + 2 dt π 2(0 π) 2π π T F T ds 0 4 2, 2 4 r (t) dt π 4(3 cos t) 2(3 sin t), 2(3 cos t) 4(3 sin t) sin t, 3 cos t dt 0 π 0 6 sin t cos t + 8 sin2 t + 8 cos 2 t 36 sin t cos t dt π 72 sin t cos t + 8 dt 0 8(π 0) 8π LL F T ds 4 2, 2 4 r (t) dt 4(t) 2(0), 2(t) 4(0), 0 dt 4t dt [ 2t 2] 2( 9) 6 3 RL F T ds 4 2, 2 4 r (t) dt 3 4(t) 2(0), 2(t) 4(0), 0 dt 3 4t dt [ 2t 2] 3 2(9 ) 6 Giving us our final answer of F T ds 2π + 8π π Now let s imagine ou remember Green s Theorem. 3

4 MTH234 hapter 6 - Vector alculus Michigan State Universit Eample 4.3. Find the work done b F 4 2, 2 4 once counterclockwise around the curve given b the picture: 3 Work F T ds (4 2) d + (2 4) d d d 4 d d (Area) 4 2 (π(3)2 π() 2 ) 2(9π π) 6π Notation The notation P d + Q d Is sometimes used to indicate that the line integral is calculated using the positive orientation of the closed curve. 2. Another notation for the positivel oriented boundar curve of a region D is D. Fun Reads There is additional material in 6.4 that is covered in the book that MSU will not currentl be testing on. Those wishing to gain a greater understanding of the power of Green s Theorem ma wish to read the section on finding area using line integrals (top of page ) and the section on Etended Versions of Green s Theorem (starting on page ). 4

5 MTH234 hapter 6 - Vector alculus Michigan State Universit Group Work. (WW#3) Use Green s Theorem to evaluate the line integral 4 cos( ) d sin( ) d. Where is the rectangle with vertices (0, 0), (2, 0), (0, 4), and (2, 4). 2. alculate ( 4 + 2)d + (5 + sin )d where is the boundar of region shown to the right: - - 5