By J. Demailly
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This monograph is an annotated translation of what's thought of to be the world’s first calculus textbook, initially released in French in 1696. That anonymously released textbook on differential calculus was once in keeping with lectures given to the Marquis de l’Hôpital in 1691-2 by means of the good Swiss mathematician, Johann Bernoulli.
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Extra info for Complex Analytic and Differential Geometry
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 , .
Complex Analytic and Differential Geometry by J. Demailly