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Let $({M},{g})$ be a smooth $4d$ spacetime manifold with lorentzian metric $g$ and local coordinates $\xi^{\alpha}$ and let further $({N},{q})$ be a smooth $3d$ manifold with metric $q$ and local coordinates $\psi^{\alpha}$. Consider then the smooth embedding map $\Sigma\in{C}^{\infty}(N,M)$ which induces a hypersurface $\Sigma(N)$. The tangent basis vectors along the hypersurface are then given by the pushforward of the embedding map. Where $\Sigma^{\alpha}=\xi^{\alpha}\circ\Sigma$ and ${x}\in{N}$. \begin{align*} \Sigma_{\star}\bigg(\frac{\partial}{\partial\psi^{\alpha}}\bigg)_{\hspace{-0.2em}{x}}=\frac{\partial\Sigma^{\omega}}{\partial\psi^{\alpha}}\bigg\vert_{x}\bigg(\frac{\partial}{\partial\xi^{\omega}}\bigg)_{\hspace{-0.2em}\Sigma(x)} \end{align*} Let ${Z}\in\Gamma({T}^{\star}{M})$ be a covectorfield normal to the hypersurface $\Sigma(N)$ such that ${g}^{-1}({Z},{Z})={1}$ and $\Sigma^{\star}{Z}=0$. One can then study how functionals of the embedding map $\Sigma$ change under a deformation of the hypersurface $\Sigma(N)$ given through the vectorfield ${V}^{\omega}(\Sigma[\psi])={A}[\psi]{\,}{Z}^{\omega}(\Sigma[\psi])+{A}^{\alpha}[\psi]{\,}\partial\Sigma^{\omega}[\psi]/\partial\psi^{\alpha}$ where ${A}\in{C}^{\infty}(N)$ and ${A}^{\alpha}\partial/\partial\psi^{\alpha}\in\Gamma(TN)$ divide the total deformation into normal and tangent directions. The notation being ${A}[\psi]:={A}(\psi^{-1}(\psi^{1},\psi^{2},\psi^{3}))$. The normal and tangent deformation operators are given respectively by \begin{align*} \mathcal{H}(A)=\int_{N}\mathrm{d}^{3}\psi{\,}{A}[\psi]{\,}{Z}^{\omega}(\Sigma[\psi])\frac{\delta}{\delta\Sigma^{\omega}[\psi]} \end{align*} \begin{align*} \mathcal{D}({A}^{\alpha}\partial/\partial\psi^{\alpha})=\int_{N}\mathrm{d}^{3}\psi{\,}{A}^{\alpha}[\psi]\frac{\partial\Sigma^{\omega}[\psi]}{\partial\psi^{\alpha}}\frac{\delta}{\delta\Sigma^{\omega}[\psi]} \end{align*} To now calculate the hypersurface deformation algebra one needs to commute these two operators separately, which requires to know how to take the functional derivative of each deformation operator. The algebra is given by \begin{align*} [\mathcal{H}(A),\mathcal{H}(B)]&=-\mathcal{D}({q}^{\alpha\beta}({B}{\,}\partial{A}/\partial\psi^{\alpha}-{A}{\,}\partial{B}/\partial\psi^{\alpha})\partial/\partial\psi^{\beta})\\ [\mathcal{D}({A}^{\alpha}\partial/\partial\psi^{\alpha}),\mathcal{H}(B)]&=-\mathcal{H}({A}^{\alpha}\partial{B}/\partial\psi^{\alpha})\\ [\mathcal{D}({A}^{\alpha}\partial/\partial\psi^{\alpha}),\mathcal{D}({B}^{\beta}\partial/\partial\psi^{\beta})]&=-\mathcal{D}(({A}^{\alpha}\partial{B}^{\beta}/\partial\psi^{\alpha}-{B}^{\alpha}\partial{A}^{\beta}/\partial\psi^{\alpha})\partial/\partial\psi^{\beta}) \end{align*} My question is now what exactly is meant by taking the functional derivative of the operators and how to derive the algebra from it? I have already tried looking it up elsewhere but was not able to find any source that explains this in further detail.

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