Variation of scalar field action I am reading Polchinski's  review on AdS/CFT .
I have a very simple question, and please help me out. Thanks in advanced.
The question is about formula (3.19).
The scalar effective bulk action is given by
$$ S_{0,\rm cl}=-\frac{1}{2L^{D-1}}\int dzd^Dx  \phi_{\rm cl}(\Box-m^2) \phi_{\rm cl}+\frac{\eta}{2}\epsilon^{1-D}\int d^Dx \phi_{\rm cl} \partial_\epsilon \phi_{\rm cl}, \tag{3.17}$$
where a typo $d\to D$ has been corrected. The boundary action is
$$S_{\rm b}=\frac{\eta \Delta_-}{2}\epsilon^{-D} \int d^D x \phi^2(\epsilon, x). \tag{3.18} $$
The variation of the whole action is given by
$$ \delta \big( S_{0,\rm cl}+S_{\rm b}\big)={\eta}\epsilon^{-D}\int d^Dx \delta \phi_{\rm cl} \big( \epsilon\partial_\epsilon-\Delta_-\big) \phi_{\rm cl} . \tag{3.19} $$
My question is why there is no such term $\int d^Dx  \phi_{\rm cl}  \epsilon\partial_\epsilon \delta\phi_{\rm cl} $ in the variation of the action?
 A: I think I understand the situation.
The term is there, and in principle you are right. 
In the integral you wrote you have something that goes like $\sim \partial_{\epsilon} \delta \phi_{cl}$ which is the derivative of the variation on the boundary or to put it in a better way the variation of the derivative on the boundary and, most of the times in physics, you can set it to zero for consistency reasons.
Polchinski mentioned it actually above eq.(3.18)
"The boundary term must be such that
the boundary terms in the variation of the action vanish for variations that respect the
boundary conditions; this is in order to have a good variational principle."
Just ask yourself, what is the point -physically- to vary the derivative of the field on the boundary. (this is taken from a professor of mine teaching QFT)
If that was not sufficient, try to do power-counting. 
The variation of the scalar effective bulk action plus the boundary term contains powers of variation and derivatives of the following form, i.e $\sim \delta \phi \partial \phi$, and the piece you wrote and said that it is not there goes like $\sim \partial_{\epsilon} \delta \phi_{cl}$. 
Starting from a quadratic action you want to get linear solutions to the equations of motion and study them analytically. This is linearization.
Cheers!!!
