Greens theorem in a plane

LINE, SURFACE AND VOLUME INTEGRALS
y
V
d
U
R
S
C
c
T
a
b
x
Figure 11.3 A simply connected region R bounded by the curve C.
These ideas can be extended to regions that are not planar, such as general
three-dimensional surfaces and volumes. The same criteria concerning the shrinking of closed curves to a point also apply when deciding the connectivity of such
regions. In these cases, however, the curves must lie in the surface or volume
in question. For example, the interior of a torus is not simply connected, since
there exist closed curves in the interior that cannot be shrunk to a point without
leaving the torus. The region between two concentric spheres of different radii is
simply connected.
11.3 Green’s theorem in a plane
In subsection 11.1.1 we considered (amongst other things) the evaluation of line
integrals for which the path C is closed and lies entirely in the xy-plane. Since
the path is closed it will enclose a region R of the plane. We now discuss how to
express the line integral around the loop as a double integral over the enclosed
region R.
Suppose the functions P (x, y), Q(x, y) and their partial derivatives are singlevalued, finite and continuous inside and on the boundary C of some simply
connected region R in the xy-plane. Green’s theorem in a plane (sometimes called
the divergence theorem in two dimensions) then states
0
∂Q ∂P
(P dx + Q dy) =
−
dx dy,
(11.4)
∂x
∂y
C
R
and so relates the line integral around C to a double integral over the enclosed
region R. This theorem may be proved straightforwardly in the following way.
Consider the simply connected region R in figure 11.3, and let y = y1 (x) and
384
11.3 GREEN’S THEOREM IN A PLANE
y = y2 (x) be the equations of the curves ST U and SV U respectively. We then
write
b y2 (x)
b y=y2 (x)
∂P
∂P
dx
dy
dx P (x, y)
dx dy =
=
y=y1 (x)
∂y
a
y1 (x)
a
R ∂y
b
P (x, y2 (x)) − P (x, y1 (x)) dx
=
a
=−
b
a
P (x, y1 (x)) dx −
a
0
P (x, y2 (x)) dx = −
b
P dx.
C
If we now let x = x1 (y) and x = x2 (y) be the equations of the curves T SV and
T UV respectively, we can similarly show that
R
∂Q
dx dy =
∂x
d
x2 (y)
dy
c
d
=
dx
x1 (y)
c
Q(x1 , y) dy +
=
d
x=x2 (y)
dy Q(x, y)
x=x1 (y)
c
Q(x2 (y), y) − Q(x1 (y), y) dy
c
∂Q
=
∂x
d
0
d
Q(x2 , y) dy =
Q dy.
C
c
Subtracting these two results gives Green’s theorem in a plane.
Show
that the area
/ of a region
/ R enclosed by a simple closed curve C is given by A =
/
1
(x dy −y dx) = C x dy = − C y dx. Hence calculate the area of the ellipse x = a cos φ,
2 C
y = b sin φ.
In Green’s theorem (11.4) put P = −y and Q = x; then
0
(x dy − y dx) =
(1 + 1) dx dy = 2
dx dy = 2A.
C
R
R
/
1
Therefore the area of the region
/ is A = 2 C (x dy − y dx). Alternatively, we could put/ P = 0
and Q = x and obtain A = C x dy, or put P = −y and Q = 0, which gives A = − C y dx.
The area of the ellipse x = a cos φ, y = b sin φ is given by
0
1 2π
1
(x dy − y dx) =
ab(cos2 φ + sin2 φ) dφ
A=
2 C
2 0
ab 2π
dφ = πab. =
2 0
It may further be shown that Green’s theorem in a plane is also valid for
multiply connected regions. In this case, the line integral must be taken over
all the distinct boundaries of the region. Furthermore, each boundary must be
traversed in the positive direction, so that a person travelling along it in this
direction always has the region R on their left. In order to apply Green’s theorem
385
LINE, SURFACE AND VOLUME INTEGRALS
y
R
C2
C1
x
Figure 11.4 A doubly connected region R bounded by the curves C1 and C2 .
to the region R shown in figure 11.4, the line integrals must be taken over
both boundaries, C1 and C2 , in the directions indicated, and the results added
together.
We may also use Green’s theorem in a plane to investigate the path independence (or not) of line integrals when the paths lie in the xy-plane. Let us consider
the line integral
B
(P dx + Q dy).
I=
A
For the line integral from A to B to be independent of the path taken, it must
have the same value along any two arbitrary paths C1 and C2 joining the points.
Moreover, if we consider as the path the closed loop C formed by C1 − C2 then
the line integral around this loop must be zero. From Green’s theorem in a plane,
(11.4), we see that a sufficient condition for I = 0 is that
∂Q
∂P
=
,
∂y
∂x
(11.5)
throughout some simply connected region R containing the loop, where we assume
that these partial derivatives are continuous in R.
It may be shown that (11.5) is also a necessary condition for I = 0 and is
equivalent to requiring P dx + Q dy to be an exact differential of some function
B
φ(x, y) such
/ that P dx + Q dy = dφ. It follows that A (P dx + Q dy) = φ(B) − φ(A)
and that C (P dx + Q dy) around any closed loop C in the region R is identically
zero. These results are special cases of the general results for paths in three
dimensions, which are discussed in the next section.
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