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12. Theorem.If{ = (x', , x") is a coordinate system in M at p, then
its coordinate vectors d, lp, …… 0,l, forma basis for the tangent space T,(M);
and
D= E(x) 。
i=1
for all ve T(M).
Proof. By the preceding remarks we can work solely on the coordinate
neighborhood of G. Since u(c) = Othere is no loss of generality in assuming
ど(p) = 0eR". Shrinking W if necessary gives E(W) = {qe R":|q| < } for
some 8.
Ifg is a smooth function on E(W) then for each 1 <isndefine
og
(tq) dt
du
g(9) =
for all qe {(W).
It follows using the fundamental theorem of calculus that
g= g(0) + E&,u' on (W).
Thus if fe &(M), setting g = f。' yields
f= f(P) + Ex on
U.
Applying d/ax' gives f(p) = (f /0x)(P). Thus applying the tangent vector
e to the formula gives
(f) = 0+ E(x'(p) + E Ap)u(x) = E(Px).
ず
ax
Since this holds for all f e &(M), the tangent vectors v and Z Ux') d,l, are
equal.
It remains to show that the coordinate vectors are linearly independent.
But if ) a, o.l, = 0, then application to x' yields
dxi
0=24
(P) = 2q d」= 4.
In particular the (vector space) dimension of T,(M) is the same as the
dimension of M.
