# variation of the scalar field theory

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The question abou formula (3.19)
The scalarbulk effective  action is given by
$$S_{0,\rm cl}=-\frac{1}{2L^{D-1}}\int dzd^Dx \phi_{\rm cl}(\Box-m^2) \phi_{\rm cl}+\frac{\eta}{2}\epsilon^{1-D}\int d^Dx \phi_{\rm cl} \partial_\epsilon \phi_{\rm cl}$$
The bdy action is
$$S_{\rm b}=\frac{\eta \Delta_-}{2}\epsilon^{-D} \int d^D x \phi^2(\epsilon, x)$$
The variation of the whole action is given by
$$\delta \big( S_{0,\rm cl}+S_{\rm b}\big)={\eta}\epsilon^{-D}\int d^Dx \delta \phi_{\rm cl} \big( \epsilon\partial_\epsilon-\Delta_-\big) \phi_{\rm cl}$$

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?

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