Summary: Homotopies and the Poincar´e Lemma.
Let I = [0, 1] and let T be the vector field on R × Rn
which assigns the vector (1, 0) to each point
of R × Rn
. Let p and q be the projections of R × Rn
on R and Rn
, respectively. For each t R let
it : Rn
R × Rn
assign (t, x) to x Rn
.
Proposition. Suppose is a smooth m-form defined on some open subset of R × Rn
. Then
d(T ) + T (d) = T .
Proof. We have
(1) d = p T +
n
j=1
ej
q (0,ej ).

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

Collections: Mathematics