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!

Linear Algebra Proof

  1. May 14, 2008 #1
    1. The problem statement, all variables and given/known data

    A = A(t), B = constant

    [tex]\frac{d}{dt} \left[A \cdot (\dot{A} \times B) \right] = \frac{d}{dt} \left[A \cdot (\ddot{A} \times B) \right] [/tex]

    3. The attempt at a solution

    In Einstein notation I get

    [tex]LHS = \frac{d}{dt} \left [ A_i (\dot{A} \times B)_i \right] = \frac{d}{dt} \left [ \epsilon_{ilm} A_i \dot{A}_l B_m \right] = \epsilon_{ilm} B_m \frac{d}{dt} \left [A_i \dot{A}_l \right] [/tex]

    Is this right so far? Product rule on A and A-dot from here?
    Last edited: May 14, 2008
  2. jcsd
  3. May 14, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    I think you have an extra d/dt on the right side of what you are trying to prove. But yes, now product rule. Then what?
  4. May 14, 2008 #3
    I think I got it:

    After using product rule I get

    [tex]A \cdot ( \ddot{A} \times B) + \dot{A} \cdot ( \dot{A} \times B)[/tex]

    Where the second term would be 0

    So the second term must expand as (A . A) x ( A . B) ?
  5. May 14, 2008 #4
    Yeah the RHS time derivative shouldn't be there.
  6. May 14, 2008 #5


    User Avatar
    Science Advisor
    Homework Helper

    Yes, but you don't expand it like that A.A and A.B are scalars. How can you cross them? In terms of vectors axb is perpendicular to a and b. In terms of your tensor expansion the product of the two A dots is symmetric, the corresponding indices of the epsilon tensor are antisymmetric. So?
  7. May 14, 2008 #6
    Ah. This way just seemed quicker but I see where it makes no sense.

    Thanks for your help.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Linear Algebra Proof
  1. Linear Algebra Proof (Replies: 8)

  2. Linear Algebra proof (Replies: 21)