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Integration of a differential form over an oriented submanifold. Let m and n be positive integers, let V be an open subset of V and let M Mm(V ) and let o be an
 

Summary: Integration of a differential form over an oriented submanifold.
Let m and n be positive integers, let V be an open subset of V and let M Mm(V ) and let o be an
orientation for M. We want to define a linear mapping
M
: Am
0 (V ) R
which "does the right thing"; what the "right thing" is will be spelled out below.
Lemma. Suppose Am
0 (V ),
(Ui, i) P(M, V ), i = 1, 2
and
spt U1 U2.
Then
o(U1, 1)
-1
1 [M]
1
#
(t1)(e1, . . . , em) dt1 = o(U2, 2)
-1

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics