ABSTRACT
Now let us prove the theorem. The functions being represented by the images of points on which they "operate" and by remembering the properties of linearity and derivation of the operator
X po : O(U)~R:gHXpo(g),
we write the following in accordance with the usage that identifies the image of any point with the function:
= 0 + L(x~ -xDXpJg/x1,...,xn )] +Lgix~,...,x;).xpo (xi -x6). (1) j~ j~
Otherwise _ 1 og og g/O,... ,O) =J-j(0,... ,0) du = ~J (0, ... ,0)
o 0' vy leads to
- (I n) ag (I n)gj xo,· ..,xo =-j xo,· .. ,xo .ax Therefore, (1) is written
XpJg(x1, ... ,xn )] = i ~(x~,... ,x;)Xpo(xj). j=1 ax
Using again the habitual writing of functions, we obtain n og ,
Xp.<g) = L -,(xo)Xpo (x ). 1=1 ax
We have definitely found again the expression (2-4) of tangent vector at po. Consequently we can express a second definition equivalent to the first:
D <T A tangent vector to M, at Po, is a (linear) mapping ofderivation: X po :O(U)~R:gHXpo(g)·
Let po be a point ofa differentiable manifold M
2.1 DEFINITION OF A TANGENT SPACE
D <T The tangent (vector) space of M, at Po, is the set of equivalence classes of tangent curves at Po.
