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Problem 4-25 in Spivak's book

  1. Feb 5, 2010 #1
    Let c be a singular k-cube and [tex]p:[0,1]^k \to [0,1]^k [/tex] a 1-1 function such that [tex]p([0,1]^k ) = [0,1]^k

    [/tex] and [tex]\det p^' (x) \ge 0[/tex] for [tex]x \in [0,1]^k [/tex].If [tex]\omega [/tex] is a k-form,show that [tex]

    \int\limits_c \omega = \int\limits_{c \circ p} \omega[/tex]

    Note that
    [tex]\int\limits_c \omega = \int\limits_{[0,1]^k } {c*\omega } = \int\limits_{[0,1]^k } {(f \circ c)(\det c')dx^1

    \wedge ... \wedge dx^k } [/tex]
    [tex]\int\limits_{c \circ p} \omega = \int\limits_{[0,1]^k } {(c \circ p)*\omega } = \int\limits_{[0,1]^k } {(f \circ c

    \circ p)(\det (c \circ p)')dx^1 \wedge ... \wedge dx^k } = \int\limits_{[0,1]^k } {(f \circ c \circ p)((\det c') \cdot

    (\det p'))dx^1 \wedge ... \wedge dx^k } [/tex]

    did I deduce it right?If it's right,how to prove
  2. jcsd
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