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Homework Help: Sum of singular 1-cubes = boundary of a singular 2-cube?

  1. Aug 17, 2014 #1
    1. The problem statement, all variables and given/known data
    I'm doing question 23 in Chapter 4 of Spivak's Calculus on Manifolds. The question asks,
    For R > O, and n an integer, define the singular l-cube,
    [itex] c_{R,n} :[0,1] \rightarrow \mathbb {R}^2 - 0 [/itex] by [itex] c_{R,n} (t) = (Rcos2\pi nt, Rsin2\pi nt).[/itex] Show that there is a singular 2-cube [itex] c:[0,1]\rightarrow \mathbb {R}^2 - 0 [/itex] such that [itex] c_{R_1,n} - c_{R_2,n} = \partial c [/itex]

    2. Relevant equations
    [itex] \partial c = \sum_{i =1}^{n} \sum_{\alpha = 0,1} (-1)^{i+ \alpha} c_{i, \alpha} [/itex]
    [itex] c_{i, \alpha} = c(I^n_{i,\alpha})[/itex]


    3. The attempt at a solution
    What I'm confused about is probably the notation of
    [itex] c_{R_2,n} [/itex] and it's relation with [itex] c_{(i, \alpha)}[/itex], are they the same thing since
    [itex]c_{R,n} (t) = (Rcos2\pi nt, Rsin2\pi nt)[/itex]
    , which I find difficult to connect with the
    [itex](i,\alpha)[/itex]
    -face of c and I.
    To solve the question you use [itex] \partial [/itex] c, to get the (i, n) - face of c, from which c can be obtained. The solution is apparently
    [itex]c(t_1,t_2) = (t_1R_1 + (1-t_1)R_2)(cos2\pi nt_2, sin2\pi nt_1) [/itex]
     
    Last edited: Aug 17, 2014
  2. jcsd
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