Let Tm be the set of tangent vectors at m (that is, the tangent space), and define
by assigning to a typical tangent vector its n local coordinates. Define an inverse
by the formula
|G(v1, v2, . . . , vn)||=|
First of all, F and G are both linear functions, by construction (proof of that fact is left as n exercise!). More important, we can verify that F and G are inverses as follows:
|F(G(v))||=||F(G(v1, v2, . . . , vn))||=||
where F converts each of the summands to local coordinates. But we have seen that the local coordinates of /xi are given by the Kronecker delta:
|j th coordinate of F(G(v))||=||F(G(v))j||=||viij||=||vj||=||j th coordinate of v.|
In other words, F(G(v)) = v.
where wi are the local coordinates of the vector w. Is this the same vector as w? Well, let us look at its ambient coordinates; since if two vectors have the same ambient coordinates, they are certainly the same vector! But we already know that
|j th Ambient coordinate of|
Thus, the j th ambient coordinate of the vector G(F(w)) above is
where the last step is the conversion formula for ambient coordinates. In other words, G(F(w)) = w, and we are done.
That is why we use local coordinates; there is no need to specify a path every time we want a tangent vector! All we need to do is specify its local coordinates.
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