Unique volume element in a vector space

  1. Given two orthonormal bases [itex]v_1,v_2,\cdots,v_n[/itex] and [itex]u_1,u_2,\cdots,u_n[/itex] for a vector space [itex]V[/itex], we know the following formula holds for an alternating tensor [itex]f[/itex]:
    [tex]f(u_1,u_2,\cdots,u_n)=\det(A)f(v_1,v_2,\cdots,v_n)[/tex]
    where A is the orthogonal matrix that changes one orthonormal basis to another orthonormal basis.

    The volume element is the unique [itex]f[/itex] such that [itex]f(v_1,v_2,\cdots,v_n)=1[/itex] for any orthonormal basis [itex]v_1,v_2,\cdots,v_n[/itex] given an orientation for [itex]V[/itex].

    Here is my question:
    If [itex]\det(A)=-1[/itex], then [itex]f(u_1,u_2,\cdots,u_n)=-1[/itex], which makes [itex]f[/itex] not unique
     
  2. jcsd
  3. micromass

    micromass 19,661
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    Hi yifli! :smile:
    I don't quite see why your reasoning implies that f is not unique. What's the other tensor that makes [itex]f(v_1,...,v_n)=1[/itex]?
     
  4. Sorry for the confusion. Actually I mean [itex]f(u_1,u_2,\cdots,u_n)[/itex] is not necessarily equal to 1, it may be equal to -1 also, even though [itex]f(v_1,v_2,\cdots,v_n)[/itex] is made to be 1
     
  5. micromass

    micromass 19,661
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    Yes, but the point is that f of every basis with the same orientation will give you 1. If you take the opposite orientation, then you will get -1 of course.
     
  6. So given a fixed orientation, it's impossible for [itex]\det(A)[/itex] to be -1?

    What does it mean for [itex]\det(A)[/itex] to be -1?
     
  7. micromass

    micromass 19,661
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    The transition matrix between two orthonormal bases has det(A)=1 if and only if the bases have the same orientation and has det(A)=-1 if they have opposite orientation.
     
  8. I like Serena

    I like Serena 6,194
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    A orthonormal transformation matrix with det(A)=1 is a rotation.
    A orthonormal transformation matrix with det(A)=-1 is a rotation combined with a reflection.

    So the volume sign would be invariant.
     
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