Divergent theorem in the plane.

�� � 29. Conservative vector fields revisitedLet U ⊂ R2 be an open subset. Given a smooth function f : U −→ R we get a smooth vector field by taking F�= grad f. Given a smooth vector field F�: U −→ R2 we get a function by taking f = curl F�. Suppose that M ⊂ U is a smooth 2-manifold with boundary. If we start with F�, then we have curl F�dA = F�d�s.· M ∂M Suppose we start with f, and let C be a smooth oriented curve. Pick a parametrisation, �x:[a,b] −→ U, such that �x(a)= P and �x(b)= Q. Then we have �� b grad f d�s = grad f(�x(t)) �x�(t)dt·· Ca� bd = f(�x(t))dt dta = f(�x(b)) − f(�x(a)) = f(Q) − f(P ). Definition 29.1. We say that X ⊂ Rn is star-shaped with respect to P ∈ X, if given any point Q ∈ X then the point P + t−→PQ ∈ X, for every t ∈ [0, 1]. In other words, the line segment connecting P to Q belongs to X. Theorem 29.2. Let U ⊂ R2 be an open and star-shaped let F�: U −→ R2 be a smooth vector field. The following are equivalent: (1) curl F�=0. (2) F�= grad f. Proof. (2) implies (1) is easy. We check (1) implies (2). Suppose that U is star-shaped with respect to P =(x0,y0). Parametrise the line L from P to Q =(x,y) as follows P + t−→PQ =(x0 + t(x − x0),y0 + t(y − y0)) = Pt, for 0 ≤ t ≤ 1. 1 � �� � Define f(x,y)= F�d�s· L� 1 = xF1(Pt)+ yF2(Pt)dt. 0 Then � 1∂f ∂ =(xF1(x0 + t(x − x0),y0 + t(y − y0)) + yF2(x0 + t(x − x0),y0 + t(y − y0))) dt ∂x 0 ∂x � 1 ∂F1 ∂F2 = F1(Pt)+ tx (Pt)+ ty (Pt)dt. ∂x ∂x 0 On the other hand, ∂ ∂F1 ∂F2 ∂t (tF1(x0 + t(x − x0),y0 + t(y − y0))) = F1(Pt)+tx∂t F1(Pt)+ty ∂y (Pt). Since curl F�= 0, we have ∂F1 ∂F2 = ,∂y ∂x and so � 1∂f ∂x = 0 ∂F1 ∂t (Pt) dt = F1(x, y). Similarly ∂f = F2(x,y). ∂y It follows that F�= grad f. � Definition 29.3. Let F�: U −→ R2 be a vector field. Define another vector field by the rule ∗F : U −→ R2 where ∗ F =(−F2,F1). Theorem 29.4 (Divergence theorem in the plane). Suppose that M ⊂R2 is a smooth 2-manifold with boundary ∂M. If F�: U −→ R2 is a smooth vector field, then div F�dA = F�nˆds,· M ∂M where nˆis the unit normal vector of the smooth oriented curve C = ∂M which points out of M. 2 Proof. Note that curl(∗F�) = div �F, and ∗F · d�s =( F�· nˆ)ds, and so the result follows from Green’s theorem applied to ∗F�. � 3 MIT OpenCourseWarehttp://ocw.mit.edu 18.022 Calculus of Several Variables Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.