 
Summary: Vector fields and divergence.
Let U be an open subset of Rn
.
Definition. We let
F(U)
be the algebra of smooth real valued functions on U and we let
X(U)
be the F(U) module of smooth Rn
valued functions on U. We call the members of X(U) (smooth) vector
fields on U. We let
Fc(U) and Xc(U)
be the members of F(U) and X(U), respectively, whose support is a compact subset of U.
Theorem. Suppose X X(U) and is its flow. Then
d
dt
det t(x)
t=0
= trace X(x) for x U.
Proof. Exercise for the reader. Use the Chain Rule and the fact that
det (iRn ) = trace
