By A. I. Fetísov

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In fact, it can be shown that such an operator may be represented as the derivative along a smooth curve as described above. We formalise these changes of emphasis in a new definition of a tangent vector: a tangent vector at a point in an affine space A is an operator on smooth functions which maps functions to numbers and is linear and satisfies Leibniz's rule as set out above. We shall denote by T,, A the set of tangent vectors at xo E A. As we have remarked above, association of a tangent vector with an element of V gives a natural with V.

Show that the open subset of A obtained by deleting the half-plane on which z2 = 0, x1 < 0 is a suitable domain for (r,>9,'p), and that no larger open subset of A will do. Verify that these functions do define a coordinate chart; identify the corresponding coordinate patch (in terms of the affine coordinates). Compute the components of the affine coordinate differentials and vectors in terms of the curvilinear coordinates, and vice-versa. 0 The great majority of coordinate formulae carry over to the case of curvilinear coordinates without change of appearance, but it must be remembered that in general they hold only locally, that is on the coordinate patch.

A linear map is determined completely by its action on a basis. If {ea} is a basis for V and {/a} a basis for 1V, where a - 1, 2,... , m = dim 1V, we may write A(e,) = Ac' /Q. The AQ are the entries of the matrix representing A with respect to the given bases. The action of A on an arbitrary vector in V is given by A(v) = AQ va f0, where (v°) is the n-tuple of components of v with respect to the basis of V. This amounts to the left multiplication of the column vector of components of v by the n x m matrix (An).