If $\textbf{x}\in \Omega \subset\mathbb{R}^n,$ where $\Omega$ is a bounded open set, $u:\Omega\rightarrow\mathbb{R}, \;\eta:\Omega\rightarrow\mathbb{R},\;u'=\nabla u = \displaystyle\left(\frac{\partial u}{\partial x_1}\cdots\frac{\partial u}{\partial x_n}\right), \;\;\eta'=\nabla \eta = \displaystyle\left(\frac{\partial \eta}{\partial x_1}\cdots\frac{\partial \eta}{\partial x_n}\right)$ and I have the functions:
$$J(u)=\int_\Omega F(\textbf{x}, u, u')\;d\textbf{x}=\int_\Omega F(\textbf{x},u,u')\;dx_1\cdots dx_n$$ $$\Phi(\varepsilon)=J(u+\varepsilon\eta)=J(z)=\int_\Omega F(\textbf{x}, z,z')\;d\textbf{x},$$
where $\varepsilon$ is some small number and $F$ is a function of $\textbf{x}, z$ and $z'$. My problem is to calculate the derivative of $\Phi$ at the point $\varepsilon = 0$ and equating it to $0$. I get the following:
$$0=\Phi'(\varepsilon)=\int_\Omega F_z\eta+\sum_{i=1}^n\left(\frac{\partial F}{\partial p_i}\frac{\partial \eta}{\partial x_i}\right)\;d\textbf{x},\;\;\;(1)$$
where $p_i=\displaystyle\frac{\partial u}{\partial x_i}+\varepsilon\frac{\partial \eta}{\partial x_i}$. Now by integrating by parts I should get $(1)$ to be equal to: (this is given to me in reference material)
$$\int_\Omega F_z\eta+\sum_{i=1}^n\left(\frac{\partial F}{\partial p_i}\frac{\partial \eta}{\partial x_i}\right)\;d\textbf{x}=F_z-\sum_{i=1}^n\frac{d}{dx_i}\left(\frac{\partial F}{\partial p_i}\right)\;\;\;(2)$$
Can someone show me the steps how does one get from $(1)$ to $(2)$. My teacher tells me that simple integration by parts is applied, but I can't get it. Any advices? thank you =)
Maybe it's better to put it in an answer:
$$\int_\Omega F_z\eta+\sum_{i=1}^n\left(\frac{\partial F}{\partial p_i}\frac{\partial \eta}{\partial x_i}\right)\;d\textbf{x}=\int_{\Omega}\eta \left( F_z-\sum_{i=1}^n\frac{d}{dx_i}\left( \frac{\partial F}{\partial p_i} \right) \right)\text{d}\textbf{x} \rightarrow \\ F_z-\sum_{i=1}^n\frac{d}{dx_i}\left( \frac{\partial F}{\partial p_i} \right)=0 $$
As mentioned in the comment, we used integration by parts for every term in the sum to get the first line.
Rememeber that our goal was to find the minimum of our functional derivative $ \Phi'(0)$ therefore we introduced the auxillary function $\eta$. It should be clear that our final equation should be independent of this quantity. To assure this the term in the second line has to be zero.
For further informations please have a look at this excellent book Mathematics for Physics - M. Stone & P. Goldbart