Proof

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

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:

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:

Substituting gives

In other words, F(G(v)) = v.

Conversely,

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

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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