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How do we get this equality about bilinear form

  1. May 6, 2012 #1
    1. The problem statement, all variables and given/known data

    [itex]B(u,u)=\int_{0}^{L}a\frac{du}{dx}\frac{du}{dx}dx[/itex]
    B(.,.) is bilinear and symmetric, δ is variational operator.

    In the following expression, where does [itex]\frac{1}{2}[/itex] come from? As i know variational operator is commutative why do not we just pull δ to the left?

    [itex]B(\delta u,u)=\int_{0}^{L}a\frac{d\delta u}{dx}\frac{du}{dx}dx=\delta\int_{0}^{L}\frac{a}{2}\left(\frac{du}{dx}\right)^{2}dx=\frac{1}{2}δ\int_{0}^{L}a\frac{du}{dx}\frac{du}{dx}dx=\frac{1}{2}δ\left[B(u,u)\right]
    [/itex]
     
  2. jcsd
  3. May 6, 2012 #2

    HallsofIvy

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    First, the "binlinear form" you have written makes no sense since it does not operate on two things in order to be bilinear. Is it possible that the form is
    [tex]B(u, v)= \int_0^L \frac{\partial u}{\partial x}\frac{\partial v}{\partial x}dx[/tex]
    ?
     
  4. May 6, 2012 #3

    Ray Vickson

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    It comes from the same place as it does in [itex] x\, dx = \frac{1}{2} d(x^2),[/itex] and does so for exactly the same reason.

    RGV
     
  5. May 7, 2012 #4
    Yes, it is.

    @Ray vickson

    Could you explain in more details?
     
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