1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Wedge product and change of variables

  1. Apr 21, 2013 #1
    1. The problem statement, all variables and given/known data

    The question is:

    Let [itex]\phi: \mathbb{R}^n\rightarrow\mathbb{R}^n[/itex] be a [itex]C^1[/itex] map and let [itex]y=\phi(x)[/itex] be the change of variables. Show that

    d[itex]y_1\wedge...\wedge [/itex]d[itex]y_n[/itex]=(detD[itex]\phi(x)[/itex])[itex]\cdot[/itex]d[itex]x_1\wedge...\wedge[/itex]d[itex]x_n[/itex].

    2. Relevant equations


    3. The attempt at a solution
    Take a look at here and the answer given by Michael Albanese:

    My question is can we prove it without using the fact "[itex]\det A = \sum_{\sigma\in S_n}\operatorname{sign}(\sigma)\prod_{i=1}^na_{i \sigma(j)}[/itex]"?
  2. jcsd
  3. Apr 21, 2013 #2


    User Avatar
    Science Advisor

    Do you know the definition of the pullback of a differential form ? This is a generalization to multilinear

    maps of the "induced map" L* , from W* to V*, given a linear map L:V-->W , both V,W vector spaces.

    I'm trying to avoid heavy machinery, but I think you need to understand this, unless you just want

    a quick-and-dirty answer ( I assume you don't since you would have accepted the answer from the link

    if you did.). You are basically doing a change of bases for multilinear maps, an extension of the idea of

    basis change for a linear map.
    Last edited: Apr 21, 2013
  4. Apr 21, 2013 #3


    User Avatar
    Science Advisor

    Or maybe you can tell us the approach you want to follow .
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted