6.
V. Guillemin and S. Sternber g
§2. The language of jets . Let M, and M
?
be differentiabie
manifolds and let f be a differentiable m a p of a neighborhood, U , of a
point x 6 M, into a neighborhood of a point y JVU , with f(x) = y . If we
a r e intereste d in the purel y local propertie s of f about x , it is natura l to
r e g a r d two map s f and ff as equivalent if they agre e in som e neighborhood
of x . The equivalence class containing f is known as the g e r m of f at x.
Frequentl y v/e study propertie s which requir e us to know f only "up to
t e r m s of o r d e r k " at x . This can be m a d e p r e c i s e by introducing a
c o a r s e r equivalence relatio n as follows:
DEFINITION 2. 1. Two differentiable map s f and V as above
a r e said to agre e to o r d e r k at x if they have the s a m e Taylor expansion
through o r d e r k in t e r m s of s o m e (and hence all) choice of coordinate
chart s about x and y . Her e 0 k oo . The equivalence class of all
map s agreein g with f to orde r k at x is called the k-je t of f at x and
denoted by j , (f)(x) . The point x is called the sourc e of the jet j , (f)(x)
and the point y is called its target .
If g is a differentiable ma p defined about y then it is eas y to se e
that j , (g o f)(x) depends only on j , (g)(y) and j , (f)(x) . We can thus writ e
(2-1) j
k
(g f)(x) = j
k
(g)(y) . j
k
(f)(x)
wher e the composition on the right is defined by the equation. It is obvious
that the usual rules for composition a r e obeyed by the composition of jets .
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