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Stokes' Theorem. Let n be a positive integer, let V be an open subset of Rn
 

Summary: Stokes' Theorem.
Let n be a positive integer, let V be an open subset of Rn
and let m be an integer such that 1 m n.
Stokes' Theorem will follow rather directly from the definition of the integral of a differential form over
a submanifold and the following Proposition.
Proposition. Suppose Am-1
0 (Um
). Then
(1)
Um
d(t)(e1, . . . , em) dt = 0
and
(2)
Um,m,+
d(t)(e1, . . . , em) dt = (-1)m
Um-1
im-1,m
#
(s)(e1, . . . , em-1) ds.
Proof. For each j = 1, . . . , m set fj = ej

  

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

 

Collections: Mathematics