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: A tensor problem

  1. Sep 12, 2008 #1
    1. The problem statement, all variables and given/known data
    A two dimensional space has a metric tensor field
    [tex]g=dq^1 \otimes dq^1 + 2 dq^1 \otimes dq^2 + 2dq^2 \otimes dq^1 + 3dq^2 \otimes dq^2[/tex].

    With the help of g, find the covariant components of the contravariant vector field
    [tex]\textbf{v}=3\partial _1 -4\partial _2[/tex]

    3. The attempt at a solution
    Not much here as I don't really understand how this works.

    g11=1, g12=2, g21=2, g22=3

    [tex]\begin{center}G=\left(\begin{array}{ll} 1 & 2 \\ 2 & 3 \end{array}\right) \Rightarrow
    G^{-1}=\left(\begin{array}{ll} {-3} & \ 2 \\ \ 2 & {-1} \end{array}\right)\end{center}[/tex]

    g11=-3, g12=2, g21=2, g22=-1

    Now the covariant components ai of v are given from the contravariant components ai like this:
    a1=a1g11+a1g12=3*(-3)+3*2=-3
    a2=a2g21+a2g22=(-4)*2+(-4)*(-1)=-4

    So [tex]\textbf{v}=-3dq^1 -4dq^2[/tex]

    Does this make any sense at all?
     
  2. jcsd
  3. Sep 12, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    [itex]a^i=a_k g^{ik}[/itex], yes? So don't you mean [itex]a^1=a_1 g^{11}+a_2 g^{12}[/itex]?
     
  4. Sep 13, 2008 #3
    Right, of course. I should never ask anything when tired.

    Thanks!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook