6 CHAPTER I. Manifolds and Vector Fields
only on the germ of / at a and we have Xa(f-g) = Xa(f) g(a) + f (a) Xa(g)
(the derivation property).
If conversely D :
C°°(Rn)
R is linear and satisfies
D(f-g) = D(f).g(a) + f(a)-D(g)
(a derivation at a), then D is given by the action of a tangent vector with
foot point a. This can be seen as follows. For / G
C°°(Rn)
we have
f(x) = f(a)+ f ftf(a + t(x-a))dt
Jo
n
= f(a) +
'£fhi(x)(xi-ai).
i=i
On the constant function 1 the derivation gives D(l) = D(l 1) = 21?(1),
so I?(constant) = 0. Therefore,
n
D(f) = D(f(a) +
J2hi(xi-ai))
i = l
n n
= 0 + J2 DihiW
~ai)
+ Yl hiiaXDix*) - 0)
i = l t = l
= EU(«)^').
z = l
where
xl
is the i-th coordinate function on
Rn.
So we have
D(f) = 5"(*«)&|tt(/), D = f(z)&|0.
i=l
i = l
Thus D is induced by the tangent vector (a, X^ILi
D(xl)ei),
where (e^) is the
standard basis of
Rn.
1.8. The tangent space of a manifold. Let M be a manifold and let
x G M and dimM = n. Let TXM be the vector space of all derivations at x
of C£°(M,R), the algebra of germs of smooth functions on M at x. Using
(1.5), it may easily be seen that a derivation of C°°(M) at x factors to a
derivation of C™(M, R).
So TXM consists of all linear mappings Xx : C°°(M) R with the property
Xx(f' g) Xx(f) - g{x) + f(x) - Xx(g). The space TXM is called the tangent
space of M at x.
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