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By E. H. Askwith

Initially released in 1917. This quantity from the Cornell collage Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 structure by way of Kirtas applied sciences. All titles scanned hide to hide and pages may well contain marks notations and different marginalia found in the unique quantity.

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Transition functions and chart representation of higher order tangent maps. For a good understanding of the structure of T k M we will need to know the structure of the transition functions T k ϕij of T k M , and more generally, the structure of higher order tangent maps T k f . For k = 1 , recall that, for a smooth map f : V ⊃ U → W (which represents a smooth map M → N with respect to charts), we have T f (x + εv) = f (x) + εdf (x)v. 6 (iv)), we may calculate d(T f ) and hence T T f . One gets T 2 f (x + ε1 v1 + ε2 v2 + ε1 ε2 v12 ) = f (x) + ε1 df (x)v1 + ε2 df (x)v2 + ε1 ε2 df (x)v12 + d2 f (x)(v1 , v2 ) .

3 by checking the definition of differentiability over K[ε] directly, leads to fairly long and involved calculations. ) 42 DIFFERENTIAL GEMETRY OVER GENERAL BASE FIELDS AND RINGS 7. Scalar extensions. 1. Higher order tangent bundles. For any manifold M , the second order tangent bundle is T 2 M := T (T M ) , and the k -th order tangent bundle is inductively defined by T k M := T (T k−1 M ) . 2. The k -th order tangent bundle T k M is, in a canonical way, a manifold over the ring of iterated dual numbers T k K = K[ε1 ] .

2. If M is a manifold of class C k+1 over K, modelled on V , then T M is, in a canonical way, a manifold of class C k over K[ε], modelled on the scalar extended K[ε] -module V ⊕ εV . If f : M → N is of class C k+1 over K, then T f : T M → T N is of class C k over K[ε] . Thus the tangent functor can be characterized as the “functor of scalar extension” from manifolds over K into manifolds over K[ε] agreeing on open submanifolds of K-modules with the algebraic scalar extension functor. Proof. Change of charts on T M is described by the transition maps T ϕij (x, v) = (ϕij (x), dϕij (x)v) .

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