By Ralph Abraham

The aim of this publication is to supply middle fabric in nonlinear research for mathematicians, physicists, engineers, and mathematical biologists. the most target is to supply a operating wisdom of manifolds, dynamical structures, tensors, and differential kinds. a few purposes to Hamiltonian mechanics, fluid mechanics, electromagnetism, plasma dynamics and keep watch over thought are given utilizing either invariant and index notation. the must haves required are good undergraduate classes in linear algebra and complex calculus.

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**Extra resources for Jerrold E Marsden Tudor Ratiu Ralph Abraham Manifolds Tensor Analysis and Applications**

**Example text**

Xn ) ∈ Rn such that |||(x1 , . . , xn )||| = 1. Thus, for all (x1 , . . , xn ) ∈ Rn , we have n xi ei ≤ M2 |||(x1 , . . , xn )|||, M1 |||(x1 , . . , xn )||| ≤ i=1 that is, M1 |||e||| ≤ e ≤ M2 |||e|||, where e = shows that ||| · ||| and · are equivalent norms. n i=1 xi ei . 9 (iii) It is enough to observe that n (x1 , . . , an isometry) between (Rn , ||| · |||) and (E, ||| · |||). 1. 1. The unit spheres for various norms The foregoing proof shows that compactness of the unit sphere in a ﬁnite-dimensional space is crucial.

These statements are readily checked. Finite Dimensional Spaces. In the ﬁnite dimensional case equivalence and completeness are automatic, according to the following result. 10 Proposition. Let E be a ﬁnite-dimensional real or complex vector space. Then (i) there is a norm on E; (ii) all norms on E are equivalent; (iii) all norms on E are complete. Proof. Let e1 , . . , en denote a basis of E, where n is the dimension of E. (i) A norm on E is given, for example, by n n |||e||| = |ai |, i=1 (ii) Let · ai ei .

Ii) If Bα is a family of connected subsets of S and Bα ∩ B = ∅, then B∪ Bα α is connected. Proof. If A is not connected, A is the disjoint union of U1 ∩ A and U2 ∩ A where U1 and U2 are open in S. 9(i), U1 ∩ B = ∅ and U2 ∩ B = ∅, so B is not connected. We leave (ii) as an exercise. 10 Corollary. The components of a topological space are closed. Also, S is the disjoint union of its components. If S is locally connected, the components are open as well as closed. 11 Proposition. Let S be a ﬁrst countable compact Hausdorﬀ space and {An } a sequence of closed, connected subsets of S with An ⊂ An−1 .