By J. Demailly

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3 Xo. x(t) such that x (O) Xo x(t) Xo + tv . V (0) = L � (xo) x i (0) ax i=l 11 = L i= l (� a "' ) (xo)v i ax The assignment v � w is, from this expression, independent of the curve and defines a linear transformation, the differential of at F Xo F .. )'O * m il mr x (t) chosen, ( Ll ) 8 MANIFOLDS AND VECTOR FIELDS (ay" /axi)(xo). whose matrix is simply the Jacobian matrix This interpretation of the Jacobian matrix, as a linear transformation sending tangents to curves into tangents to the image curves under F, can sometimes be used to replace the direct computation of matrices.

If n ::: r, we demand that F be of differentiability class C I, whereas if n - r = k > 0, we demand that F be of class C H I . The proof of Sard's theorem is delicate, especially if n > r; see, for example, [A , M, R ] . 3e. Change o f Coordinates The inverse function theorem is perhaps the most important theoretical result in all of differential calculus. M n -7 is a differentiable map between manifolds of the same dimension, and if at Xo E M the differential F* is an isomorphism, that is, it is 1 : I and onto, then F is a local diffeomorphism near Xo .

Xn ) , which can be abbreviated to y = F (x ) or y = y (x) . If, as we Mil M TANGENT VECTORS AND MAPPINGS 27 shall assume, the functions F a are differentiable functions of the x 's, we say that F is differentiable. As usual, such functions are, in particular, continuous. When n = r, we say that F is a diffeomorphism provided F is 1 : 1, onto, and if, in addition, F - I is also differentiable. 2a) with a differentiable inverse. (If F- I does exist and the Jacobian determinant does not vanish, a (y I , .

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