18.02A Topic 41: Extensions and applications of Green’s theorem.Author: Jeremy OrloffRead: SN: V5, V6 (pp.1,2)Review of Green’s Theorem:Tangential (work) form:orICF · dr =ICF · T ds =ZZRcurlF · k dAICM dx + N dy =ZZRNx− MydA.Normal (flux) form:orICF · n ds =ZZRdivF dAICM dy − N dx =ZZRMx+ NydA.xyRCnTBoth forms require (for now) C is a positively oriented (interior on left), simple closedcurve and R is its interior.T = unit tangent vector, n = unit normal = CW rotation of T, it points out from R.Application:Definition: A region D in the plane is simply connected if it has “no holes”.Examples:D1D2D3xyD4xyD5= whole planeD1-D5 are simply connected. For any simple closed curve inside them its interior isentirely inside the region. (Sometimes we say any curve can be shrunk to a pointwithout leaving the region.)The regions at right are not simply connected. That is, the interior ofthe curve C is not entirely in the region.CAnnulusxyCPuntured planeTheorem: If D is a simply connected region and My= Nxin D the F = hM, Niis conservative in D, i.e. F = ∇f for some f defined in D.Proof: Since D is simply connected if C is a simple closed curve in D its interiorR is also in D. Therefore, using Green’s Theorem, Nx= My⇔ curlF = 0 ⇒ICF · dr =ZZRcurlF · k dA = 0.xyDCR118.02A topic 41 2Summary: F = Mi + Nj = hM, Ni on simply connected region D.The following statements are equivalent.1)ZQPF · dr is path independent. 1’)ZQPM dx + N dy is path independent.2)ICF · dr = 0 for any closed path C. 2’)ICM dx + N dy = 0.3) F = ∇f for some f in D 3’) M dx + N dy = df.If F is continuously differentiable then 1,2,3 ⇒ 4.4) curlF = 0 in D4’) My= Nxin D.Why we need D simply connected in the theorem: If there is a holethen F might not be defined on the interior of C. (See example 1 below.)DCExtended Green’s Theorem.If R has multiple boundary curves we can extend Green’s Theorem.Suppose R = region between C1and C2(note R is always to the leftas you traverse either curve.)⇒IC1F · dr +IC2F · dr =ZZRcurlF · k dA.C1C2RC1C2C3C4RLikewise for more than two curves:IC1F · dr +IC2F · dr +IC3F · dr +IC4F · dr =ZZRcurlF · k dA.Proof. Make a cut so the new curve is simple.Green’s Thm. ⇒IC1+C3+C2−C3F · dr =ZZRcurlF · k dA⇔IC1+C2F · dr =ZZRcurlF · k dA. QEDC1C2C3−C3Example 1: Let F =−yi + xjr2(“tangential field”)F is defined on D = plane - (0,0) = punctured planexyIt’s easy to compute (we’ve done it before) that curlF = 0 in D.Question: For the tangential field F what values canICF · dr takefor C a simple closed curve (positively oriented)?We have two cases i) C1not around 0 ii) C2around 0xyC1RC2i) Green’s Theorem ⇒IC1F · dr =ZZRcurlF · k dA = 0.18.02A topic 41 3ii) We show thatIC2F · dr = 2π.Let C3be a small circle of radius a, entirely inside C2.By extended Green’s TheoremIC2F · dr −IC3F · dr =ZZRcurlF · k dA = 0⇒IC2F · dr =IC3F · dr.On the circle C3we can easily compute the line integral:F · T = 1/a ⇒IC3F · T ds =ZC31ads =2πaa= 2π. QEDxyC2C3RAnswer to question: 0 or 2π.If C is not simple,ICF · dr = 2πn, where n is the number of timesC goes (CCW) around (0,0).(Not for class) n is called the winding number of C around 0. n alsoequals the number of times C crosses the positive x-axis, counting +1from below and −1 from above.xyCExample 2: Isy dx − x dyy2exact? If so find a potential function.M =1y, N = −xy2are continuosly differentiable whenever y 6= 0i.e, in the two half-planes R1and R2–both simply connectedMy= −1/y2= Nx⇒ in each half-plane M dx + N dy = df.We use method 2: fx= 1/y ⇒ f = x/y + g(y).fy= −x/y2+ g0(y) = −x/y2⇒ g0(y) = 0 ⇒ g(y) = c.⇒ f(x, y) = x/y + c.xyR1R2Example 3: Let F = rn(xi + yj). Use extended Green’s Theorem to show that Fis conservative for all integers n. Find a potential function.First note, M = rnx, N = rny ⇒ My= nrn−2xy = Nx⇔ curlF = 0.We show F is conservative by showingRCF · dr = 0 for all simple closed curves C.If C1is a simple closed curve not around 0 then Green’s Theorem impliesRC1F·dr = 0.If C3is a circle centered on (0,0) then, since F is radialIC3F · dr =IC3F · T ds = 0.18.02A topic 41 4If C3completely surrounds C2then extended Green’s Theoremimplies (since curlF = 0 between the curves)IC2F · dr =IC3F · dr = 0.ThusICF · dr = 0 for all closed loops ⇒ F is conservative.xyC1RC2C3To find the potential function we use method 1 over the curve C shown.The calculation works for n 6= −2. For n = −2 everything is the same except weget natural logs instead of powers. (We also ignore the fact that if (x1, y1) is on thenegative x-axis we shoud use a different path that doesn’t go through the origin. Thisisn’t really an issue since we already know a potential function exists, so continuitywould handle these points without using an integral.)f(x1, y1) =ZCrnx dx + rny dy=Zy11(1 + y2)n/2y dy +Zx11(x2+ y21)n/2x dx=(1 + y2)(n+2)/2n + 2y11+(x2+ y21)(n+2)/2n + 2x11=(1 + y21)(n+2)/2− 2(n+2)/2n + 2+(x21+ y21)(n+2)/2− (1 + y21)(n + 2)/2=(x21+ y21)(n+2)/2− 2(n+1)/2n + 2⇒ f(x, y) =rn+2n + 2+ C.If n = −2 we get f(x, y) = ln r + C.xyC1C2(x1, y1)(1, 1)End of topic 41
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