The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls we studied in Setion 3.6. We will first look t Green s theorem for retngles, nd then generlize to more omplex urves nd regions in R. Green s theorem for retngles Suppose F : R R is C 1 on n open set ontining the losed retngle [, ] [, d], nd let F 1 nd F e the oordinte funtions of F. If C denotes the oundry of, oriented in the lokwise diretion, then we my deompose C into the four urves C 1, C, C 3, nd C 4 shown in Figure 4.4.1. Then d C 3 C 4 C C 1 Figure 4.4.1 The oundry of retngle deomposed into four smooth urves α(t) (t, ), t, is smooth prmetriztion of C 1, β(t) (, t), 1 Copyright y n Sloughter 001
Green s Theorem Setion 4.4 t d, is smooth prmetriztion of C, γ(t) (t, d), t, is smooth prmetriztion of C 3, nd δ(t) (, t), t d, is smooth prmetriztion of C 4. Now F ds F ds + F ds + F ds + F ds C C 1 C C 3 C 4 F ds + F ds F ds F ds, (4.4.1) C 1 C C 3 C 4 nd F ds C 1 F ds C F ds C 3 nd F ds C 4 d d ((F 1 (t, ), F (t, )) (1, 0)dt ((F 1 (, t), F (, t)) (0, 1)dt ((F 1 (t, d), F (t, d)) (1, 0)dt ((F 1 (, t), F (, t)) (0, 1)dt Hene, inserting (4.4.) through (4.4.5) into (4.4.1), C F ds d F 1 (t, )dt + d F (, t)dt (F (, t) F (, t))dt F 1 (t, d)dt Now, y the Fundmentl Theorem of Clulus, for fixed vlue of t, F 1 (t, )dt, (4.4.) F (, t)dt, (4.4.3) F 1 (t, d)dt, (4.4.4) F (, t)dt, (4.4.5) d F (, t)dt (F 1 (t, d) F 1 (t, ))dt. (4.4.6) x F (x, t)dx F (, t) F (, t) (4.4.7) nd d y F 1(t, y)dy F 1 (t, d) F 1 (t, ). (4.4.8)
Setion 4.4 Green s Theorem 3 Thus, omining (4.4.7) nd (4.4.8) with (4.4.6), we hve C F ds d d d x F (x, t)dxdt d d x F (x, y)dxdy ( x F (x, y) y F 1(x, y) y F 1(t, y)dydt y F 1(x, y)dydx ) dxdy. (4.4.9) If we let p F 1 (x, y), q F (x, y), nd C ( ommon nottion for the oundry of ), then we my rewrite (4.4.9) s pdx + qdy This is Green s theorem for retngle. Exmple If [1, 3] [, 5], then xydx + xdy 3 5 1 3 1 ( q x p ) dxdy. (4.4.10) y ( x x ) y xy dxdy (1 x)dydx 3(1 x)dx 3x 3 1 3 x 3 6. Clerly, this is simpler thn evluting the line integrl diretly. Green s theorem for regions of Type III Green s theorem holds for more generl regions thn retngles. We will onfine ourselves here to disussing regions known s regions of Type III, ut it is not hrd to generlize to regions whih my e sudivided into regions of this type (for n exmple, see Prolem 1). Rell from Setion 3.6 tht we sy region in R is of Type I if there exist rel numers < nd ontinuous funtions α : R R nd β : R R suh tht 1 {(x, y) : x, α(x) y β(x)}. (4.4.11) We sy region in R is of Type II if there exist rel numers nd d nd ontinuous funtions γ : R R nd δ : R R suh tht {(x, y) : y d, γ(y) x δ(y)}. (4.4.1)
4 Green s Theorem Setion 4.4 C 3 C 4 C C 1 Figure 4.4. eomposing the oundry of region of Type I efinition Type III. We ll region in R whih is oth of Type I nd of Type II region of Exmple In Setion 3.6, we sw tht the tringle T with verties t (0, 0), (1, 0), nd (1, 1) nd the losed disk B ((0, 0), 1) {(x, y) : x + y 1} re of oth Type I nd Type II. Thus T nd re regions of Type III. We lso sw tht the region E eneth the grph of y x nd ove the intervl [ 1, 1] is of Type I, ut not of Type II. Hene E is not of Type III. Exmple Any losed retngle in R is region of Type III, s is ny losed region ounded y n ellipse. Now suppose is region of Type III nd is the oundry of, tht is, the urve enlosing, oriented ounterlokwise. Let F : R R e C 1 vetor field, with oordinte funtions p F 1 (x, y) nd q F (x, y). We will first prove tht pdx p dxdy. (4.4.13) y Sine is, in prtiulr, region of Type I, there exist ontinuous funtions α nd β suh tht {(x, y) : x, α(x) y β(x)}. (4.4.14) In ddition, we will ssume tht α nd β re oth differentile (without this ssumption the line integrl of F long would not e defined). As with the retngle in the previous proof, we my deompose into four urves, C 1, C, C 3, nd C 4, s shown in Figure 4.4.. Then ϕ 1 (t) (t, α(t)),
Setion 4.4 Green s Theorem 5 t, is smooth prmetriztion of C 1, ϕ (t) (, t), α() t β(), is smooth prmetriztion of C, ϕ 3 (t) (t, β(t)), t, is smooth prmetriztion of C 3, nd ϕ 4 (t) (, t), α() t β(), is smooth prmetriztion of C 4. Now pdx pdx + pdx pdx pdx, (4.4.15) C 1 C C 3 C 4 where pdx C 1 pdx C 3 nd C pdx (F 1 (t, α(t)), 0) (1, α (t))dt β() α() pdx C 4 Hene pdx (F 1 (, t), 0) (0, 1)dt (F 1 (t, β(t)), 0) (1, β (t))dt β() α() F 1 (t, α(t))dt, (4.4.16) β() (F 1 (, t), 0) (0, 1)dt F 1 (t, α(t))dt Now, y the Fundmentl Theorem of Clulus, α() 0dt 0, (4.4.17) F 1 (t, β(t))dt, (4.4.18) β() α() F 1 (t, β(t))dt 0dt 0. (4.4.19) (F 1 (t, β(t)) F 1 (t, α(t)))dt. (4.4.0) nd so β(t) α(t) y F 1(t, y)dy F 1 (t, β(t)) F 1 (t, α(t)), (4.4.1) pdx β(t) α(t) β(x) α(x) p y y F 1(t, y)dydt y F 1(x, y)dydx dxdy. (4.4.)
6 Green s Theorem Setion 4.4 A similr lultion, treting s region of Type II, shows tht qdy q x dxdy. (4.4.3) (You re sked to verify this in Prolem 7.) Putting (4.4.) nd (4.4.3) together, we hve p F ds pdx + qdy y dxdy + q x dxdy ( q x p ) dxdy. (4.4.4) y Green s Theorem Suppose is region of Type III, is the oundry of with ounterlokwise orienttion, nd the urves desriing re differentile. Let F : R R e C 1 vetor field, with oordinte funtions p F 1 (x, y) nd q F (x, y). Then ( q pdx + qdy x p ) dxdy. (4.4.5) y Exmple Let e the region ounded y the tringle with verties t (0, 0), (, 0), nd (0, 3), s shown in Figure 4.4.3. If we orient in the ounterlokwise diretion, then ( (3x + y)dx + 5xdy x (5x) ) y (3x + y) dxdy (5 1)dxdy 4 dxdy (4)(3) 1, where we hve used the ft tht the re of is 3 to evlute the doule integrl. The line integrl in the previous exmple redued to finding the re of the region. This n e exploited in the reverse diretion to ompute the re of region. For exmple, given region with re A nd oundry, it follows from Green s theorem tht A dxdy pdx + qdy (4.4.6) for ny hoie of p nd q whih hve the property tht q x p y 1. (4.4.7)
Setion 4.4 Green s Theorem 7 3.5 1.5 1 0.5 0.5 1 1.5.5 Figure 4.4.3 A tringle with ounterlokwise orienttion For exmple, letting p 0 nd q x, we hve nd, letting p y nd q 0, we hve A xdy (4.4.8) A ydx. (4.4.9) The next exmple illustrtes using the verge of (4.4.8) nd (4.4.9) to find A: A 1 ( xdy ) ydx 1 xdy ydx. (4.4.30) Exmple Let A e the re of the region ounded y the ellipse with eqution x + y 1, where > 0 nd > 0, s shown in Figure 4.4.4. Sine we my prmetrize, with ounterlokwise orienttion, y ϕ(t) ( os(t), sin(t)),
8 Green s Theorem Setion 4.4 Figure 4.4.4 The ellipse x + y 1 with ounterlokwise orienttion 0 t π, we hve Prolems A 1 1 1 π 0 π ( π. 0 π xdy ydx ( sin(t), os(t)) ( sin(t), os(t)dt ( sin (t) + os (t))dt dt 0 ) (π) 1. Let e the losed retngle in R with verties t (0, 0), (, 0), (, 4), nd (0, 4), with oundry oriented ounterlokwise. Use Green s theorem to evlute the following line integrls. () xydx + 3x dy () ydx + xdy. Let e the tringle in R with verties t (0, 0), (, 0), nd (0, 4), with oundry oriented ounterlokwise. Use Green s theorem to evlute the following line integrls. () xy dx + 4xdy () ydx + xdy () ydx xdy
Setion 4.4 Green s Theorem 9 3. Use Green s theorem to find the re of irle of rdius r. 4. Use Green s theorem to find the re of the region enlosed y the hypoyloid x 3 + y 3 3, where > 0. Note tht we my prmetrize this urve using 0 t π. ϕ(t) ( os 3 (t), sin 3 (t)), 5. Use Green s theorem to find the re of the region enlosed y one petl of the urve prmetrized y ϕ(t) (sin(t) os(t), sin(t) sin(t)). 6. Find the re of the region enlosed y the rdioid prmetrized y 0 t π. ϕ(t) (( + os(t)) os(t), ( + os(t)) sin(t)), 7. Verify (4.4.3), thus ompleting the proof of Green s theorem. 8. Suppose the vetor field F : R R with oordinte funtions p F 1 (x, y) nd q F (x, y) is C 1 on n open set ontining the Type III region. Moreover, suppose F is the grdient of slr funtion f : R R. () Show tht q x p y 0 for ll points (x, y) in. () Use Green s theorem to show tht pdx + qdy 0, where is the oundry of with ounterlokwise orienttion. 9. How mny wys do you know to lulte the re of irle? 10. Who ws George Green? 11. Explin how Green s theorem is generliztion of the Fundmentl Theorem of Integrl Clulus. 1. Let >, let C 1 e the irle of rdius entered t the origin, nd let C e the irle of rdius entered t the origin. If is the nnulr region etween C 1 nd C nd F is C 1 vetor field with oordinte funtions p F 1 (x, y) nd q F (x, y), show tht ( q x p ) dxdy pdx + qdy + pdx + qdy, y C 1 C where C 1 is oriented in the ounterlokwise diretion nd C is oriented in the lokwise diretion. (Hint: eompose into Type III regions 1,, 3, nd 4, eh with oundry oriented ounterlokwise, s shown in Figure 4.4.5.)
10 Green s Theorem Setion 4.4 1 3 4 Figure 4.4.5 eomposition of n nnulus into regions of Type III