Math S21a: Multivariable calculus Oliver Knill, Summer 2011Lecture 21: Greens theoremGreen’s theorem is the second integral theorem in the plane. This entire section deals withmultivariable calculus in the plane, where we have two integral theorems, the fundamental theoremof line int egr als and Greens theorem. First two reminders:If~F (x, y) = hP (x, y), Q(x, y)i is a vector field and C : ~r (t) = h x(t), y(t)i, t ∈ [a, b] isa curve, the line integralZC~F~dr =Zba~F (x(t), y(t)) · ~r′(t) dtmeasures the work done by the field~F along the path.The curl of a two dimensional vector field~F (x, y) = hP (x, y), Q(x, y)i is defined asthe scalar fieldcurl(F )(x, y) = Qx(x, y) − Py(x, y) .The curl(F ) measures the vorticity of the vector field.One can write ∇ ×~F = curl(~F ) because the two dimensional cross product of (∂x, ∂y) with~F = hP, Qi is the scalar Qx− Py.1 For~F (x, y) = h−y, xi we have curl(F )(x, y) = 2.2 If~F (x, y) = ∇f is a gradient field then the curl is zero because if P (x, y) = fx(x, y), Q(x, y) =fy(x, y) and curl(F ) = Qx− Py= fyx− fxy= 0 by Clairo t ’s theorem.Green’s theorem: If~F (x, y) = hP (x, y), Q(x, y)i is a smooth vector field and Ris a region for which the boundary C is a curve parametrized so that R is ”to theleft”. ThenZC~F ·~dr =Z ZGcurl(F ) dxdy .Pro of. Look first at a small square G = [x, x + ǫ] × [y, y + ǫ]. The line integral of~F = hP, Qi alongthe boundary isRǫ0P (x+t, y)d t+Rǫ0Q(x+ǫ, y+t) dt−Rǫ0P (x+t, y+ǫ) dt−Rǫ0Q(x, y+ t) dt. This lineintegr al measures the ”circulation” at the place (x, y). Because Q(x + ǫ, y) − Q(x, y) ∼ Qx(x, y)ǫand P (x, y + ǫ) − P (x, y) ∼ Py(x, y)ǫ, the line integral is (Qx− Py)ǫ2∼Rǫ0Rǫ0curl(F ) dxdy. Allidentities hold in t he limit ǫ → 0. To prove the statement for a general region G, chop it intosmall squares of size ǫ. Summing up all the line integrals around the boundaries gives the lineintegr al around the boundary b ecause in the interior, the line integrals cancel. Summing up thevort icities on the squares is a Riemann sum approximation of the double integral.George Green lived from 1793 to 1841. He was a physicist a self-taught mathematician andmiller. His work greatly contributed to modern physics.3 If~F is a gradient field then both sides of Green’s theorem are zero:RC~F ·~dr is zero bythe fundamental theorem for line int egr als. andR RGcurl(F ) · dA is zero because curl ( F ) =curl(grad(f)) = 0.The already established Clairot identitycurl(grad(f)) = 0can also checked by writing it as ∇ × ∇f and using that the cross product of two identical vectorsis 0. Treating ∇ as a vector is called nabla calculus.4 Find the line integral of~F (x, y) = hx2−y2, 2xyi = hP, Qi along the boundar y of the rectangle[0, 2]×[0, 1 ]. Solution: curl(~F ) = Qx−Py= 2y−2y = −4y so thatRC~F~dr =R20R104y dydx =2y2|10x|20= 4.5 Find the a rea of the regio n enclosed by~r(t) = hsin(πt)2t, t2− 1ifor −1 ≤ t ≤ 1 . To do so, use Greens theorem with the vector field~F = h0, xi .6 Green’s theorem allows t o express the coordinates of the centroid = center of mass(Z ZGx dA/A,Z ZGy dA/A)using line integrals. With the vector field~F = h0, x2i we haveZ ZGx dA =ZC~F~dr .7 An important application of Green is to compute area. With the vector fields~F (x, y) =hP, Qi = h−y, 0i or~F (x, y) = h0, xi have vorticity curl(~F )(x, y) = 1. For~F (x, y) = h0, xi ,the right hand side in Green’s theorem is the area of G:Area(G) =ZCx(t) ˙y(t) dt .8 Let G be the region under the graph of a function f(x) on [a, b]. The line integral aroundthe boundary of G is 0 from (a, 0) to (b, 0) because~F (x, y) = h0, 0i there. The line integralis also zero from (b, 0) to (b, f (b)) and (a, f (a)) to (a, 0) because N = 0. The line integralalong the curve (t, f(t)) is −Rbah−y(t), 0i·h1, f′(t)i dt =Rbaf(t) dt. Green’s theorem confirmsthat this is the area of the region below the gra ph.It had b een a consequence of the fundamental theorem of line integrals thatIf~F is a gradient field then curl(F ) = 0 everywhere.Is t he conver se true? Here is the answer:A region R is called simply connected if every closed loop in R can be pulledtogether to a point in R.If curl(~F ) = 0 in a simply connected region G, then~F is a gradient field.Pro of. Given a closed curve C in G enclosing a region R. Green’s theorem assures thatRC~F~dr = 0.So~F has the closed loop property in G and is therefore a gradient field there.In the homework, you look at an example of a not simply connected region where the curl(~F ) = 0does no t imply that~F is a gradient field.An engineering application of Greens theorem is the planimeter, a mechanical device for mea-suring areas. We will demonstrate it in class. Historically it had been used in medicine to measurethe size of the cross-sections of tumors, in biology to measure the area of leaves or wing sizes ofinsects, in agriculture to measure the area of forests, in engineering to measure the size of profiles.There is a vector field~F associated t o a planimeter which is obtained by placing a unit vectorperpendicular to the arm).One can prove that~F has vorticity 1. The planimeter calculates the line int egr al of~F along agiven curve. Green’s theorem assures it is the area.Homework1 Calculate the line integralRC~F~dr with~F = h2y + x sin(y) , x2cos(y) − 3y200 sin(y)i along atriangle C with edges (0, 0), (π/2, 0) and (π/2, π/2).2 Evaluate the line integral of the vector field~F (x, y) = hxy2, x2i along the rectangle withvertices (0, 0), (2, 0), (2, 3), ( 0, 3).3 Find the a rea of the regio n bounded by the hypocycloid~r(t) = hcos3(t), sin3(t)iusing Green’s theorem. The curve is parameterized by t ∈ [0, 2π].4 Let G be the r egion x6+ y6≤ 1. Compute the line integral of the vector field~F (x, y) =hx6, y6i along the boundary.5 Let~F (x, y) = h−y/(x2+ y2), x/(x2+ y2)i. Let C : ~r(t) = hcos(t), sin(t)i, t ∈ [0, 2π].a) ComputeRC~F ·~dr.b) Show that curl(~F ) = 0 everywhere for (x, y) 6= (0, 0).c) Let f (x, y) = arctan(y/ x). Verify tha t ∇f =~F .d) Why do a) and b) not contradict the fact that a gradient field has the closed loop property?Why does a) and b) not contradict Green’s
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