Chapter 30

Stokes and Green’s Theorems30.1 Green’s Theorem

Green’s theorem is an important theorem which relates line integrals to integrals over asurface in the plane. It can be used to establish the seemingly more general Stoke’s theorembut is interesting for it’s own sake. Historically, theorems like it were important in thedevelopment of complex analysis.

Here is a proof of Green’s theorem from the divergence theorem.

Theorem 30.1.1 (Green’s Theorem) Let U be an open set in the plane for whichthe divergence theorem holds. Let ∂U be piecewise smooth, and let

F (x,y) = (P(x,y) ,Q(x,y))

be a C1 vector field defined near U. Then∫∂U

F ·dR=∫

U

(∂Q∂x

(x,y)− ∂P∂y

(x,y))

dA.

Proof: Suppose the divergence theorem holds for U . Consider the following picture.

(x′,y′)(y′,−x′)

U

Counter clockwise motion around the curve is determined by imagining you stand up-right with your left hand over U and walk in the direction you are facing. Since it isassumed that motion around U is counter clockwise, the tangent vector (x′,y′) is as shown.The unit exterior normal is a multiple of(

x′,y′,0)× (0,0,1) =

(y′,−x′,0

).

Use your right hand and the geometric description of the cross product to verify this. Thiswould be the case at all the points where the unit exterior normal exists.

Now let G(x,y) = (Q(x,y) ,−P(x,y)). Also note the area (length) element on the

bounding curve ∂U is√

(x′)2 +(y′)2dt. Suppose the boundary of U consists of m smoothcurves, having a well defined outer normal, the ith of which is parameterized by (xi,yi) withthe parameter t ∈ [ai,bi]. Then by the divergence theorem,∫

U(Qx −Py)dA =

∫U

div(G)dA =∫

∂UG ·ndS

595