By Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu
The aim of this ebook is to supply center fabric in nonlinear research for mathematicians, physicists, engineers, and mathematical biologists. the most objective is to supply a operating wisdom of manifolds, dynamical structures, tensors, and differential varieties. a few functions to Hamiltonian mechanics, fluid mechanics, electromagnetism, plasma dynamics and keep watch over 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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Example text
Proof. It suffices to prove the complex case. If α, β ∈ C, then 2 0 ≤ αe1 + βe2 , αe1 + βe2 = |α| 2 e1 , e1 + 2 Re(αβ e1 , e2 ) + |β| e2 , e2 . If we set α = e2 , e2 , and β = − e1 , e2 , then this becomes 0 ≤ e2 , e2 2 2 2 e1 , e1 − 2 e2 , e2 | e1 , e2 | + | e1 , e2 | e2 , e2 , and so 2 e2 , e2 | e1 , e2 | ≤ e2 , e2 2 e1 , e1 . If e2 = 0, equality results in the statement of the proposition and there is nothing to prove. If e2 = 0, the term e2 , e2 in the preceding inequality can be cancelled since e2 , e2 > 0 by CI4.
14 Theorem (Alexandroff’s Theorem). ble and hence is second countable. An arcwise connected locally compact metric space is separa- Proof (Pfluger [1957]). Since the metric space M is locally compact, each m ∈ M has compact neighborhoods that are disks. Let m ∈ M and denote by r(m) the least upper bound of the radii of such disks. If r(m) = ∞, since every metric space is first countable, M can be written as a countable union of compact disks. 5-2), these disks and also their union will be separable, and so the proposition is proved in this case.
Proof. To prove that (i) implies (ii), assume there is a nonempty proper set A that is both open and closed. Then S = A ∪ (S\A) with A, S\A open and nonempty. Conversely, if S = A ∪ B with A, B open and nonempty, then A is also closed, and thus A is a proper nonempty set of S that is both open and closed. The equivalences of the remaining assertions are similarly checked. Behavior under Mappings. Connectedness is preserved by continuous maps, as is shown next. 4 Proposition. , arcwise connected ) then so is f (S).