Linearization, some doubts :)

[tim85ruhruniv]

Hi,

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.


\delta w is a test function (just another variable)

w is a scalar variable



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


Linearization w.r.t w,

\begin{eqnarray*}
\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*}

Thanks a lot :)