1.2.1 First-order Variation of the Free Energy

In the following we impose that the first-order variation $\delta G$ vanishes for any variation $\delta\textbf{{m}}$ of the vector field $\textbf{{m}}$, compatible with the constraint $\vert\textbf{{m}}+\delta\textbf{{m}}\vert=1$ (which in turn corresponds to $\vert\textbf{M}+\delta\textbf{M}\vert=M_s$). This will allow us to derive the equilibrium condition [4] and, therefore, the equilibrium configuration for magnetization within the body. We approach separately each term of the free energy (1.53).