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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