cp - 2n (see Figure 2).

The identification of the dual space T:*(M) of T:(M) with Yq(M) is unaffected by a change of charts, since «(L - 1)') - 1)' = L. 3) 1. In elementary vector calculus one encounters the gradient of a function f as an example of a vector. Here we shall recognize it as an element of T:(M). A function f E Coo(M) defines a mapping Tq(f): Yq(M) -+ Tf(qllR) = IR, which is therefore an element of T:(M). We may denote this mapping by dfiq and call it the differential off at the point q. On a chart we write the usual formula dfiq(v) = Vi ;lJq I, i \tv E Yq(M).