# Linearization, some doubts

1. Oct 7, 2008

### tim85ruhruniv

Linearization, some doubts :)

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 :)

Last edited: Oct 7, 2008