Michael Corral: Vector Calculus

150 CHAPTER 4. LINE AND SURFACE INTEGRALS
4.3 Green’s Theorem
We will now see a way of evaluating the line integral of a smooth vector field around a
simpleclosedcurve. Avectorfieldf(x,y)=P(x,y)i+Q(x,y)jissmoothifitscomponent
functionsP(x,y)andQ(x,y)aresmooth. WewilluseGreen’sTheorem(sometimescalled
Green’s Theorem in the plane) to relate the line integral around a closed curve with a
double integral over the region inside the curve:
Theorem 4.7. (Green’s Theorem) Let R be a region in 2 whose boundary is a
simple closed curve C which is piecewise smooth. Let f(x,y)=P(x,y)i+Q(x,y)j be a
smooth vector field defined on both R and C. Then
∮ ( ) ∂Q
f·dr=
∂x −∂P dA, (4.21)
∂y
C
where C is traversed so that R is always on the left side of C.
R
Proof: We will prove the theorem in the case for a simple region R, that is, where the
boundary curve C can be written as C= C 1 ∪C 2 in two distinct ways:
C 1 = the curve y=y 1 (x) from the point X 1 to the point X 2 (4.22)
C 2 = the curve y=y 2 (x) from the point X 2 to the point X 1 , (4.23)
where X 1 and X 2 are the points on C farthest to the left and right, respectively; and
C 1 = the curve x= x 1 (y) from the point Y 2 to the point Y 1 (4.24)
C 2 = the curve x= x 2 (y) from the point Y 1 to the point Y 2 , (4.25)
where Y 1 and Y 2 are the lowest and highest points, respectively, on C. See Figure
4.3.1.
y
d
◭
y=y 2 (x)
Y 2
x= x 1 (y)
c
X 1
Y 1
C
X 2 x= x 2 (y)
R
◮
y=y 1 (x)
x
a b
Figure 4.3.1
Integrate P(x,y) around C using the representation C= C 1 ∪C 2 given by (4.23) and