Kenneth Kuttler

442 CHAPTER 23. STOKES AND GREEN’S THEOREMS

Suppose ∣∣∣∣∣ xu xv

yu yv

∣∣∣∣∣> 0

so the unit normal is then just k. Then Stoke’s theorem applied to this special case yields∫∂V

F ·dR=∫

U(Qx (x(u,v) ,y(u,v))−Px (x(u,v) ,y(u,v)))k ·k

∣∣∣∣∣ xu xv

yu yv

∣∣∣∣∣dA

Now by the change of variables formula, this equals

=∫

V(Qx (x,y)−Px (x,y))dA

This is just Green’s theorem for V . Thus if U is a region for which Green’s theorem holdsand if V is another region, V =R(U) , where |Ru×Rv| ̸= 0, R is one to one, and twicecontinuously differentiable with Ru×Rv in the direction of k, then Green’s theorem holdsfor V also.

This verifies the following theorem.

Theorem 23.4.1 (Green’s Theorem) Let V be an open set in the plane and let ∂V be piece-wise smooth and let F (x,y) = (P(x,y) ,Q(x,y)) be a C1 vector field defined near V. Thenif V is oriented counter clockwise, it is often2 the case that∫

∂VF ·dR=

∫V

(∂Q∂x

(x,y)− ∂P∂y

(x,y))

dA. (23.4)

In particular, if there exists U such as the simple convex in both directions case consideredearlier for which Green’s theorem holds, and V =R(U) where R : U → V is C2

(U ,R2

)such that

∣∣Rx×Ry∣∣ ̸= 0 and Rx×Ry is in the direction of k, then 23.4 is valid where the

orientation around ∂V is consistent with the orientation around U.

This is a very general version of Green’s theorem which will include most of what willbe of interest.

23.4.1 Conservative Vector FieldsDefinition 23.4.2 A vector field F defined in a three dimensional region is said to be con-servative3 if for every piecewise smooth closed curve C, it follows

∫C F ·dR= 0.

Definition 23.4.3 Let (x,p1, · · · ,pn,y) be an ordered list of points in Rp. Let

p(x,p1, · · · ,pn,y)

denote the piecewise smooth curve consisting of a straight line segment from x to p1 andthen the straight line segment from p1 to p2 · · · and finally the straight line segment frompn to y. This is called a polygonal curve. An open set in Rp, U, is said to be a region ifit has the property that for any two points x,y ∈U, there exists a polygonal curve joiningthe two points.

2For a general version see the advanced calculus book by Apostol. This is presented in the next section also.The general versions involve the concept of a rectifiable Jordan curve. You need to be able to take the area integraland to take the line integral around the boundary.

3There is no such thing as a liberal vector field.

442 CHAPTER 23. STOKES AND GREEN’S THEOREMSSupposetu | 50Yu Vvso the unit normal is then just k. Then Stoke’s theorem applied to this special case yields[ F-dR= [ (Q, (x (u,v) ,y(u,v)) — Py (x (u,v) ,y(u,v)))k-k aa 7ov JU Yu YvNow by the change of variables formula, this equals= I (Qx (x,y) — Py (x, y)) dAThis is just Green’s theorem for V. Thus if U is a region for which Green’s theorem holdsand if V is another region, V = R(U), where |R, x R,| 40, R is one to one, and twicecontinuously differentiable with R,, x R, in the direction of k, then Green’s theorem holdsfor V also.This verifies the following theorem.Theorem 23.4.1 (Green’s Theorem) Let V be an open set in the plane and let OV be piece-wise smooth and let F (x,y) = (P(x,y),Q(x,y)) be aC! vector field defined near V. Thenif V is oriented counter clockwise, it is often* the case that_ [ (92). 9Pj eaR= | (Bo) yy (x9) dA. (23.4)In particular, if there exists U such as the simple convex in both directions case consideredearlier for which Green’s theorem holds, and V = R(U) where R: U > V is C2 (U, R’)such that | Rx x R,| # 0 and R, x R, is in the direction of k, then 23.4 is valid where theorientation around OV is consistent with the orientation around U.This is a very general version of Green’s theorem which will include most of what willbe of interest.23.4.1 Conservative Vector FieldsDefinition 23.4.2 A vector field F defined in a three dimensional region is said to be con-servative if for every piecewise smooth closed curve C, it follows Jc F-dR=0.Definition 23.4.3 Let (x,p,,--- ,P,,y) be an ordered list of points in R?. Letp(x,p1,° - Pn)denote the piecewise smooth curve consisting of a straight line segment from x to p, andthen the straight line segment from p, to py--- and finally the straight line segment fromp, to y. This is called a polygonal curve. An open set in R?, U, is said to be a region ifit has the property that for any two points x,y € U, there exists a polygonal curve joiningthe two points.2For a general version see the advanced calculus book by Apostol. This is presented in the next section also.The general versions involve the concept of a rectifiable Jordan curve. You need to be able to take the area integraland to take the line integral around the boundary.3There is no such thing as a liberal vector field.