Calculus: Analytic Geometry and Calculus, with Vectors

13.3 Double integrals 675

of this problem. One example is sufficientto expose an interesting idea that

enables us to establish (21) for less simplesets S in the xy plane. To establish

(21) for the set S of Figure 13.391, we split Sinto two subsets S, and S2

as in

6 S CI

LLA

'S2

Figure 13.391 Figure 13.392
Figure 13.392. Writing the formulas obtainedby applying (21) to the simpler
sets S2 and S2 and adding the results shows that (21) is validfor S because the
curve integrals over the common boundary of SI and S2come uith opposite
signs and cancel out of the sum. After having proved that (21) is valid forsets
of particular types that lie in the xy plane, thenext step is to recognize that, when
C is a suitable plane curve, the right side of (21) hasan intrinsic meaning which
is independent of coordinate systems. As can be suspected, this factcan be
used to show that the left member of (21) and thecurl itself also have intrinsic
meanings. To be appropriately narrow-minded about this matter, let S be a
plane triangular set or plane circular disk in E3 which is boundedby an oriented
triangle or oriented circle C. Then (21) is valid because the simpler Green
formula shows that it is valid when the coordinatesystem is chosen such that
S lies in the xy plane. The method that was applied to the planesets of Figures
13.391 and 13.392 can now be employed toprove that (21) holds when S is a
triangulated oriented surface in E3 consisting ofa finite set of plane triangular
faces bounded by oriented triangles provided the topologicalstructure and
orientations are such that if a side of a triangle
is a part of the boundaries of more than one
triangular face, then the side is a part of
boundaries of exactly two such faces and, as
in Figure 13.393, the side has opposite orienta-
tions in the two triangles that contain it.
While a full treatment of the matter lies
beyond the scope of this book, teachers of
courses in electricity and magnetism and aero-
dynamics (among others) require knowledge of
consequences of the idea that the Stokes for-

Figure 13.393

mula (21) is valid when S is a patch of surface in E3 that can be satisfactorily
approximated by a triangulated oriented surface of the type described above.
One of the important applications of the Green and Stokes formulas involves
conservative force fields. Let the vector V(x,y,z) in (11) be the force on a par-
ticle when the particle is at the point (x,y,z). The force field determined by the
vector function V is said to be conservative if the work done in moving the particle
around a closed curve C is 0 when C belongs to a class of curves which is not
always carefully defined but which certainly includes circles. If V is conserva-
tive over a region in E3, then the right member of (21) must therefore be 0 when
C is a circle in the region. Then the left member of (21) must be 0 when S is a
plane circular disk in the region and therefore (as can be proved) the hypothesis