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but repeating what we did in Example 4.9.1.2 would now get very messy and so we
shall not attempt to do so.
4.9.1.4. Theorem.
(Stokes' Theorem) Let
A
Õ
R
n
. If w is a (k - 1)-form on
A
and if
c is a k-chain on
A
, then
Ú
Ú
d
ww
=
.
c
∂
Proof.
See [Spiv65]. The proof is really not very hard and boils down mainly to using
the definitions of the various quantities that are involved and showing that both sides
of the equation are the same, starting first with the special case c = I
k
.
If one writes out what Theorem 4.9.1.4 says in the case of the 1-chain I
1
in
R
, one
will see that what one has is just the classical Fundamental Theorem of Calculus
(Theorem D.1.3). (Actually, the proof of Theorem 4.9.1.4 uses that special case and
assumes that it has been proved separately.) One should therefore not be surprised
when Theorem 4.9.1.4 is often called the
Generalized Fundamental Theorem of Calcu-
lus
. To quote from [Spiv65]:
(1) It is trivial.
(2) It is trivial because the terms in it have been properly defined.
(3) It has significant consequences.
One sees the truth of point (2) over and over in mathematics. Making the right defi-
nitions can isolate the essential aspects needed to arrive at a solution to a problem.
The validity of point (3) in our current context of integration will become more appar-
ent in Section 8.12. For now we have finished with our outline of differential forms
and their relation to integration.
4.10
E
XERCISES
Section 4.2
Prove that the sets
R
n
and f are
both
open and closed.
4.2.1.
4.2.2.
Prove that a single point is always a closed set.
4.2.3.
Is {1/n | n = 1, 2,...} a closed set? Prove or disprove your answer.
Prove that if
X
is a closed set in
R
n
, then cl(
X
) =
X
.
4.2.4.
4.2.5.
Prove that the interior of a set is an open set.
4.2.6.
Give examples of sets that show the following statements are false:
(a)
If
X
Õ
Y
, then bdry(
X
) Õ bdry(
Y
).
(b)
bdry(
X
) = bdry(cl(
X
)).
(c)
bdry(
X
) = bdry(int(
X
)).
(d)
int(
X
) = int(cl(
X
)).
