By Hermann Weyl
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36 4 Vector Fields, Differential Forms, and Derivatives The tangent space is isomorphic with an n-dimensional real vector space ∂ i through the canonical application θ : Tx X → Rm , θx (ξ i (x) ∂x i ) = (ξ ). The collection of all tangent spaces corresponding to all points of X is called tangent bundle and it is denoted as T X = ∪x∈X Tx X. By the property of overlapping and differentiability of charts in the atlas, all the tangent spaces on a manifold can be smoothly connected. Definition 6. A differentiable function v : X → Tx X is called a vector field on the smooth manifold X.
A local comR2 pact linear topological space has finite dimension. , but we do not need these concepts in our book. Basically, they occur whenever we relax one of the three properties defining compactness [11–13] (see Fig. 4). 3 Weierstrass–Stone Theorem How is it possible for the Taylor series to exist? That is, how is it possible to know all the values of a continuous function from just knowing a countable sequence of number, the coefficients of the Taylor series. The answer is related to the separation axioms and it is the Weierstrass–Stone theorem.
V n , } be a finite set of n vector fields defined on a smooth manifold X. We call integral submanifold of S a submanifold Y ⊂ X whose tangent space Tp Y is spanned by the system S at every point p ∈ N . The system at every point S is integrable if through every point p ∈ X there passes an integral submanifold. Definition 15. A finite system of vector fields S = {v 1 , v 2 , . . , if ∀p(x) ∈ X, ∀i, j = 1, . . , n n ckij (x)v k , [v i , v j ] = k=1 where ckij (x) are differentiable real functions on X, and [, ] is the Lie bracket defined by the action (see Definition 9) of two smooth vector fields on functions f : M → R [v, w]f = v(w(f )) − w(v(f )).