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Tangent Vectors We continue to let U C Rn be a fixed but arbitrary open set. We fix p E U and describe the tangent space Tp(U)of U at p. In calculus, it is customary to translate a tangent vector a' at p to the origin 0 E R n , thereby identifying a' canonically with an element of Rn. That is, we set Tp(U) = Rn. This will not do for our purposes since we are trying to set up a local calculus that will make sense on manifolds where, generally, there will be no preferred coordinate system. In standard calculus, the vector defines a directional derivative Dz at p by the formula Da(f ) = lim f(p+h;)-ff(p) h h+O = af C" a'-@), axi i=l where f is an arbitrary smooth function defined on an open neighborhood of p.

Dn,,) is a basis of the vector space T,(U). Suppose that n For the coordinate functions xj, 1 5 j 5 n, the Kronecker delta. Thus, for 1 5 j 5 n. This proves that { D l , , , . . , D,,,) is a linearly independent subset of T,. We must prove that it is also a spanning set. Let D E T,. Set a' = D[xi],, 1 5 i n. 2. 20. Then, Since [f], E 6 , is arbitrary, it follows that D = x:=,aiDi,,. The above proof would not work for deriwtives of the algebra 6 of germs of C ' functions, k < oo. The problem is that gi E c'-', 1 5 i 5 n, so DlpiIp is not even defined.

If Lp(x) = C + x7=lbizi is a derivative o f f at p, then 1 5 i 5 n. In particular, i f f is diflerentiable at p, these partial derivatives exist and the derivative L p is unique. Having seen that derivatives are given by partial derivatives, we center our attention on these more familiar operators. 3. If r 2 1, the class Cr (U) of functions f : U -* R that are smooth of order r is specified inductively by requiring that af /axa exist and belong to Cr-I (u), 1 5 i 5 n. The functions that are smooth of order r are also called Crsmooth.

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