By Jerrold E. Marsden and Tudor Ratiu, with the collaboration of Ralph Abraham
The aim of this ebook is to supply center fabric in nonlinear research for mathematicians, physicists, engineers, and mathematical biologists. the most aim is to supply a operating wisdom of manifolds, dynamical structures, tensors, and differential kinds. a few functions to Hamiltonian mechanics, fluid mechanics, electromagnetism, plasma dynamics and regulate idea are given utilizing either invariant and index notation. the necessities required are stable undergraduate classes in linear algebra and complex calculus.
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Additional info for Manifolds, Tensor Analysis, and Applications (2007 version)
Example text
1-5. A sequence {en } ⊂ E, where E is an inner product space, is said to be weakly convergent to e ∈ E iff all the numerical sequences v, en converge to v, e for all v ∈ E. Let ∞ 2 (C) = 2 an ∈ C and {an } |an | < ∞ n=1 and put ∞ {an } , {bn } = an bn . n=1 Show that: (i) in any inner product space, convergence implies weak convergence; (ii) 2 (C) is an inner product space; (iii) the sequence (1, 0, 0, . . ), (0, 1, 0, . . ), (0, 0, 1, . . ), . . is not convergent but is weakly convergent to 0 in 2 (C).
Let fn : [a, b] → R be uniformly bounded continuous functions. Set x Fn (x) = fn (t) dt, a ≤ x ≤ b. a Prove that Fn has a uniformly convergent subsequence. 5-10. Let X and Y be topological spaces. A map f : X → Y is called proper if the inverse image of any compact set in Y is compact in X. (i) Show that the composition of two proper maps is again proper. (ii) Let X be a compact space, Y be a Hausdorff space and suppose that f : X → Y is a continuous map. Show that f is proper. (iii) Let f : X → Y be continuous and proper.
Hint: Use the Baire category theorem. (iii) Show that if f ∈ E, then f has no derivative at any point. 7-4. Prove that in a complete metric space (M, d) with no isolated points, no countable dense set is a Gδ -set. Hint: Suppose E = {x1 , x2 , . . } is dense in M and is also a Gδ set, that is, E = n>0 Vn with Vn open, n = 1, 2, . . Conclude that Vn is dense in M . Let Wn = Vn \{x1 , . . , xn }. Show that Wn is dense in M and that n>0 Wn = ∅. This contradicts the Baire property. Page 37 2 Banach Spaces and Differential Calculus This version: January 8, 2007 Manifolds have enough structure to allow differentiation of maps between them.