By Todhunter, I. (Isaac)

The beneficial reception which has been granted to my historical past of the Calculus of diversifications throughout the 19th Century has inspired me to adopt one other paintings of an identical style. the topic to which I now invite realization has excessive claims to attention because of the sophisticated difficulties which it includes, the precious contributions to research which it has produced, its very important useful functions, and the eminence of these who've cultivated it.

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**Extra resources for A history of the mathematical theory of probability : from the time of Pascal to that of Laplace**

**Example text**

9 (c), is a submanifold of N. , topology and smooth structure) on A which makes it an immersed submani- fold of M. 40 that there is at most one such smooth structure for a given topology on A. (c) is an example of the next proposition which says that an immersion is locally an embedding. 34 Proposition A smooth map f : M n → N m is an immersion if and only if one of the following equivalent assertions are satisfied : (1) Every point p in M has is an open neighbourhood U such that f |U is an embedding.

82. 47 48 SMOOTH MANIFOLDS AND FIBRE BUNDLES PROOF OF (2) : Assume first that X is a vector field on M, and let f ∈ F (V ) for an open set V in M. If (x,U) is a local chart on M with U ⊂ V and a : U → Rn is the smooth map defined in (1), we have that n XV ( f )| U = ∑ ai i=1 ∂f ∂ xi showing that XV ( f )| U ∈ F (U). Since this is true for every coordinate neighbourhood U contained in V , we have that XV ( f ) ∈ F (V ). Conversely, assume that X satisfies (2) and let (x,U) be a local chart on M.

If p1 and p2 lie in the same fibre π −1 (q) with q∈Uα , they can be separated by the neighbourhoods tα−1 (O1 ) and tα−1 (O2 ), where O1 and O2 are disjoint neighbourhoods of tα (p1 ) and tα (p2 ) in Uα × Rn . This shows that E is a smooth manifold with a unique smooth structure such that the bijections tα : π −1 (Uα ) → Uα × Rn are diffeomorphisms for every α ∈A. We see that the projection π is smooth since π Uα = pr1 ◦ tα is the composition of the smooth maps tα and pr1 . We finally show that there is a unique vector space structure on each fibre π −1 (p) for p ∈M such that (tα , π −1 (Uα )) is a local trivialization for each α ∈A.

### A history of the mathematical theory of probability : from the time of Pascal to that of Laplace by Todhunter, I. (Isaac)

by Anthony

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