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Linearization, some doubts

  1. Oct 7, 2008 #1
    Linearization, some doubts :)


    I was working on some Non-Linear PDE's but was quite doubtful about some calculus involved,

    I think these equations are right but i felt i could get a consult from someone good in
    calculus the confusion being the differentiation of a variable that belongs in a gradient of that variable.

    [tex]\delta w[/tex] is a test function (just another variable)

    [tex] w[/tex] is a scalar variable

    I want to linearize this, [tex]\intop_{\Omega}\nabla\delta w\cdot\left(\exp\left(\frac{-\varphi }{U_{t}}\right)\nabla w\right)}[/tex]

    Linearization w.r.t [tex]w[/tex],

    \frac{\partial\intop_{\Omega}\nabla\delta w\cdot\left(\exp\left(\frac{-\varphi}{U_{t}}\right)\nabla w\right)}{\partial w}\triangle w & = & \underset{_{\Omega}}{\int}\left[\frac{d}{dl}\left(\nabla\delta\left(w+l\triangle w\right)\cdot\left(\exp\left(\frac{-\varphi}{U_{t}}\right)\nabla\left(w+l\triangle w\right)\right)\right)\right]_{l=0}\\
    & = & \underset{_{\Omega}}{\int}\nabla\delta w\cdot\left(\exp\left(\frac{-\varphi}{U_{t}}\right)\nabla\triangle w\right)\end{eqnarray*}[/tex]

    Thanks a lot :)
    Last edited: Oct 7, 2008
  2. jcsd
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