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Vector and Tensor Exercise

  1. Mar 21, 2015 #1
    1. The problem statement, all variables and given/known data

    I'm currently trying to work through some issues I'm having with tensor and vector analysis. I have an equation of the form
    $$\textbf{a} \bullet \textbf{b} = \textbf{c} \bullet \textbf{d}$$
    where all quantities here are vectors. I want to solve for ##\textbf{b}## by finding an equation of the form
    $$\textbf{b} = \overline{\textbf{T}} \bullet \textbf{d}$$
    where ##\overline{\textbf{T}}## is a tensor. However I'm not sure the proper mathematical procedure to go about this. Any suggestions?

    2. Relevant equations

    That's what I'm here for.

    3. The attempt at a solution

    No idea.
     
  2. jcsd
  3. Mar 21, 2015 #2
    Hi. You can't possibly do that without imposing more constraints on your vector:
    In tensor language, the vectors are contracted on both sides so you can't "solve" for b.
    This may be confusing because a⋅b looks like a vector expression but it's really a scalar; if you want to solve for a vector you need a vector expression.
    Look at the simplest example in 2 dimensions:
    a⋅b = cd ⇔a1b1 + a2b2 = c1d1 + c2d2
    You see that you would need two equations to solve for the two variables b1 and b2 , and for every additional dimension you need an additional constraint...
     
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