1. Not finding help here? Sign up for a free 30min 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!

3D Vectors

  1. Nov 23, 2008 #1
    1. Variables

    Given a generalized basis in three dimensions: [tex]e_{1},e_{2},e_{3}[/tex] and the standard Kronecker delta [tex]\delta_{ij}[/tex], and using Einstein summation.
    With the vector [tex]\textbf{x},\textbf{y},\textbf{z}[/tex] i'm trying to simplify this problem:

    2. Problem
    [tex]\delta_{il} . \delta_{jm} . x_{j}[/tex]

    3. My attempt
    [tex]\delta_{il}.\delta_{jm} . \textbf{x} . e_{j}
    = (e_{i}. e_{l}) . (e_{j} . e_{m}) . \textbf{x} . e_{j}
    = (e_{j}. e_{j}) . (e_{l} . e_{m} . e_{j}) . \textbf{x}
    = 1 . (e_{l} . e_{m} . e_{j}) . \textbf{x} [/tex]

    Surely this leads to [tex]\delta_{il} . \delta_{jm} . x_{j} = 0[/tex] as [tex]e_{l} , e_{m} , e_{j}[/tex] are all orthagonal ?

    Ultimately I'm trying to prove that
    [tex](\delta_{il} . \delta_{jm} - \delta_{jl} . \delta_{im}).x_{j}.y_{l}.z_{m}
    = y_{i}.x_{j}.z_{j} - z_{i}.x_{j}.y_{j}[/tex]
     
  2. jcsd
  3. Nov 24, 2008 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi timscully! Welcome to PF! :smile:

    (have a delta: δ :wink:)

    Forget about the basis vectors!

    All δij does is replace i by j (or vice versa) in anything else.

    So δijxj = xi, δijxi = xj.

    So just plug 'n' play! :biggrin:
     
  4. Nov 24, 2008 #3
    Thanks for the welcome.

    It looks like a great forum.

    So, is [tex]\delta_{ij} . x_{m}[/tex] zero, because m is neither i nor j ?
     
  5. Nov 25, 2008 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    not a sum

    Nooo:bugeye:

    δijxj is a sum over all values of j, so it only depends on i: δijxj = xi.

    But δijxm is not a sum; there is no "contraction"; it still depends on i j and m: δijxm = xm if i = j and = 0 if i≠j. :smile:
     
  6. Nov 25, 2008 #5
    Got it. Much appreciated.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: 3D Vectors
  1. Points on a 3D vector (Replies: 2)

  2. 3D vector (Replies: 9)

  3. 3d-Vector question (Replies: 6)

  4. Plane and 3d vector (Replies: 1)

  5. Vectors in 3D (Replies: 2)

Loading...