By Philip M. Morse, Herman Feshbach

Brown conceal

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I> = 1. Only one chart is necessary in this case. This is also true for the somewhat more general case of an open subset of IR". 1 ________ ~ ____ ~~m Compatibility of two charts. 2. M = {(Xl' X2) E 1R2: xf + X~ = I} is called the one-dimensional sphere Sl or the one-dimensional torus Tl. M is compact and therefore not homeomorphic to an open subset of IR. 2). Tl x Tl . 1 Manifolds is the n-dimensional torus, and the n-dimensional sphere is defined as Sft = {(x;) E IRft + I : xi + x~ + ... + x; + I = 1}.

1). Then quently, d x I di" f ,=0 = of oqj 7). q = f 0 *~q = f(q(t)). Conse- of 7). X;(q) = Lx f· q, 5. 12), let U, be the domain of I/! (Ud = II x VI' II C IR, VI C IR m- 1 . 4) guarantees the existence ofa local solution u(t; Xl, ... *X 0 u = Ii, using this chart. , Xm): 12 X V2 ...... IRm, 12 C 11, V2 C VI, has the derivative Df(O) = 1: IRm...... IR m, because the components J; are found to satisfy "oj; ut I= 0 X;(O) = (1,0,0, ... ,0), -oj; OX2 I= 0 bj2 , etc. 5]. Becausef(O, X2,"" xm) = (O'X2' .. *

2. Let Matn{R) denote the manifold of real n x n matrices. Addition and multiplication of two n x n matrices are Coo mappings Matn{R) x Matll{~) -+ Matn{R). 3. If fl and f2 E CP, then their composition fl 0 f2: M 34M 2 :4 M 1, is also a CP-mapping (Problem 6). 4. If M is the product manifold MIx M 2 and f = fl X f2' then when the fi are cP-mappings, so is f. 5. Ml = I c ~. Thinking of I as an interval oftime and M2 as space, we will refer to the function f or the curve f(I) as a trajectory. 6. M 2 = R The p-times differentiable functioll£ in this case are denoted by CP(M 1)' They form an algebra with the usual product in IR because of the elementary rules for differentiation of sums and products.