Let (Omega=left{mathbf{v}(mathbf{x}) in L_{mathcal{D}}^{2} cap C_{mathcal{D}}^{2} ight}) be the phase space for the compact flow domain (mathcal{D}

Let \(\Omega=\left\{\mathbf{v}(\mathbf{x}) \in L_{\mathcal{D}}^{2} \cap C_{\mathcal{D}}^{2}\right\}\) be the phase space for the compact flow domain \(\mathcal{D} \subset R^{3}\) with n.e. smooth boundary \(\partial \mathcal{D}, \Omega\) is then Hilbert space w.r.t.

\[ \|\mathbf{v}\|^{2}=(\mathbf{v} \cdot \mathbf{v}) \]

and scalar product

\[ (\mathbf{v}, \mathbf{w}) \equiv \int_{\mathcal{D}} d u v_{\alpha}(\mathbf{x}) w_{\alpha}(\mathbf{x}) \]

(Cartesian coordinates), let \(F[\mathbf{v}]: \Omega \rightarrow R^{1}\) be a linear functional defined by

\[ F[\mathbf{v}] \equiv \int_{\mathcal{D}} d u K_{\alpha}(\mathbf{x}) v_{\alpha}(\mathbf{x}) \]

for \(K_{\alpha}(\mathbf{x}) \in \Omega\).

(4.4.1): Compute the first Gateaux derivative of the functional \(F[\mathbf{v}]\) w.r.t. \(\mathbf{v}(\mathbf{x})\).
(4.4.2): Compute the first Gateaux derivative of (4.4.1) w.r.t. the ONS vector basis \(\mathcal{B} \equiv\left\{\mathbf{f}^{k}(\mathbf{x}), k=1, \ldots, \infty\right\}\).
(4.4.3): Compute the second Gateaux derivative w.r.t. v(x).