Variational Derivative Relationships

Question

I am interested in a free-energy functional given by: $$ E[\phi] = \int f(\phi) - \frac{\epsilon^2}{2}|\nabla\phi|^2 \ d\vec{x} = \int F(\phi)$$ My understanding of variational derivatives is quite weak, and so, I am wondering what to make of: $$ \frac{\delta E}{\delta (1-\phi)}=\ ???? $$ From what I understand, the variational derivative with respect to $\phi$ can be calculated as: $$ \frac{\delta E}{\delta \phi} = \frac{\partial F}{\partial \phi}-\nabla\cdot\frac{\partial F}{\partial \nabla\phi} = f'(\phi)-\epsilon^2\nabla^2\phi $$ My question then: is it true that: \begin{equation} \pmb{\frac{\delta E}{\delta (1-\phi)}=-\frac{\delta E}{\delta \phi}}\ ???? \end{equation} If it matters, in this context $\phi=\phi(\vec{x},t)$ represents the composition of component A in a binary mixture. (Hence, $1-\phi$ is the composition of component B).

Answer 1

Yes, that's right.

As a simple explanation, changing $1-\phi$ by $\epsilon \psi$ amounts to changing $\phi$ by $-\epsilon \psi$, so $E$ responds as if you had done that change to $\phi$.