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: Is a contravariant vector

  1. Apr 29, 2012 #1
    1. The problem statement, all variables and given/known data
    B is a third order tensor. Show that [itex]B^{ij}_{i}[/itex] is a contravariant vector.

    3. The attempt at a solution
    Well... I just thought about a simple solution but I don't think I'm right. But anyways.
    Considering [itex]B^{ij}_{i}[/itex]. If I raise the index i: [itex]g^{ij}B^{ij}_{i} = B^{ijj}[/itex]
    And so I can say that B is a first order tensor = vector. And this is a contravariant vector.
    Is this right?
  2. jcsd
  3. Apr 29, 2012 #2


    User Avatar
    Science Advisor

    Re: Tensors

    You can't have the same index three times! Also, using the "summation convention" you sum over one "upper" and one "lower" index so even if you wrote [itex]g^{kj}B^{ij}_k[/itex], you would not be summing. Perhaps what you mean is [itex]g^k_jB^{ij}_k= B^i[/itex]

    Well, what is the definition of a contravariant vector? Can you show that this satisfies that definition? Remember that it is NOT enough just to show that something can be written with a single index- you can write an array, in a given coordinate system, indexed with a single index- that does not make it a "vector". A vector must change coordinates correctly as the coordinate system changes. If I remember correctly (it's been a while!) one test for a contravariant vector, written as [itex]v^i[/itex], is that the combination [itex]g_{ij}v^iv^j[/itex] (essentially the vector length) is a scalar. Which, again, does not just mean "a number" in a given coordinate system. A scalar (0 order tensor) must not change when you change coordinate systems.
  4. Apr 30, 2012 #3
    Re: Tensors

    Equally well you can show that [itex] B^{ij}_i [/itex] transforms as a vector, ie. if I do a coordinate transform [itex] x^{i} \rightarrow x^{i'} [/itex], B should transform as
    [tex] B^{i' j'}_{i'} = \frac{\partial x^{j'}}{\partial x^{j}} B^{ij}_i [/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook