Page 39 - Curvature and Homology

carriers of the g,., these neighborhoods for all P E S form a covering of S.
Since S is compact, it has a finite sub-covering, and so there is at most
a finite number of gj different from zero. Since $gja is defined, we put

That the integral of a over M so defined is independent of the choice
of the neighborhood containing the carrier of gj as well as the covering
{(Ii} and its corresponding partition of unity is not difficult to show.
Moreover, it is convergent and satisfies the properties (i) and (ii).
The uniqueness is obvious.
Suppose now that M is a compact orientable manifold and let /3 be
an (n - 1)-form defined over M. Then,

To prove this, we take a partition of unity (g,) and replace /3 by &$.
This result is also immediate from the theorem of Stokes which we
now proceed to establish.
Stokes' theorem expresses a relation between an integral over a
domain and one over its boundary. Its applications in mathematical
physics are many but by no means outstrip its usefulness in the theory
of harmonic integrals.
Let M be a differentiable manifold of dimension n. A domain D
with regular boundary is a point set of M whose points may be classified
as either interior or boundary points. A point P of D is an interior point
if it has a neighborhood in D. P is a boundary point if there is a coordinate
neighborhood U of P such that U n D consists of those points Q E U
satisfying un(Q) 2 un(P), that is, D lies on only one side of its boundary.
That these point sets are mutually exclusive is clear. (Consider, as an
example, the upper hemisphere including the rim. On the other hand,
a closed triangle has singularities). The boundary aD of D is the set of
all its boundary points. The following theorem is stated without proof:
The boundary of a domain with regular boundary is a closed sub-
manifold of M. Moreover, if M is orientable, so is aD whose orientation
is canonically induced by that of D.
Now, let D be a compact domain with regular boundary and let h
be a real-valued function on M with the property that h(P) = 1* if
P E D and is otherwise zero. Then, the integral of an n-form a may be
defined over D by the formula