676 Iterated and multiple integrals
that V X V is continuous over the region implies that V X V = 0 over the
region. On the other hand, if V X V = 0, then the left member of (21) is clearly
0, so the right member must be 0 and V must be conservative This proves the
very useful nontrivial fact that if V and its scalar components P, Q, R are con-
tinuous and have continuous partial derivatives over a region in Ea, then V is
conservative over the region if and only if the formulas
aQ_aP OR aP aR_aQ
ax = ay' ax = az' ay - az
hold over the region.
11 There is much to be learned about the process of Problem 10 by which the
Stokes formula (21) is proved first for simpler surfaces composed of oriented
plane triangles suitably joined at their edges and then for curved surfaces that
can be satisfactorily approximated by the simpler surfaces Relatively few
people undertake to master all of the details, but everybody can see that some
quite delicate topological considerations are involved. Classical examples
involve ordinary bands and Mobius bands. When the ends of a strip of paper
a foot long and an inch wide are joined in the simplest way, the result is an
ordinary curved band that has two edges (a top and a bottom) and two sides
(an inside and an outside). It is easy to put a dozen diagonal creases in the
paper to obtain a band composed of a dozen plane triangular patches. Let S1
be the surface composed of the points on the outside of the latter band. It is
easy to orient the triangles as in the discussion of Figure 13.393 and to obtain
the Stokes formula for S1. To make a Mobius band, we start with another strip
of paper a foot long and an inch wide, but this time we put a twist (a half-turn)
in one of the ends before the two ends are joined. This strip can be creased to
obtain a band composed of plane triangles joined at their edges. It turns out
that the Mobius band has just one edge and just one side, there being no "side"
that is "an outside" that is different from "the inside." Inner secrets are
revealed to those who try to color only "the outside" of this band. Persons
interested in this matter may construct Mobius bands and study their properties.
It is quite easy to obtain the correct idea that topological considerations form an
essential part of rigorous (free from blunders) statements and proofs of theorems
setting forth conditions under which the
Figure 13.41 Stokes formula is valid.Y13.4 Rectangular coordinate ap-
plications of double and iterated
integrals This section illustrates
ideas that are often used when prob-
lems are being solved with the aid
of double and iterated integrals in-
volving rectangular coordinates. The
principal illustration involves a lamina(or flat plate) which, as in Figure
13.41, lies in the xy plane and is
bounded by the graphs of the equa-