a map $a: V \times V \times {\mathcal{P}}\to {\mathbb{R}}$, $\left(u, v, \mu\right) \mapsto a_\mu(u, v)$, we denote the Fréchet derivatives of $a$ w.r.t. the first, second and third argument of said map in the direction of $w\in V$, $\nu \in \mathbb{R}^P$ by $\partial_u a_\mu(u, v)[w] \in {\mathbb{R}}$, $\partial_v a_\mu(u, v)[w] \in {\mathbb{R}}$ and $\partial_\mu a_\mu(u, v)\cdot \nu \in {\mathbb{R}}$, respectively. Similarly, interpreting the linear functional $l$ as a map $l: V \times {\mathcal{P}}\to {\mathbb{R}}$, $(v, \mu) \mapsto l_\mu(v)$, we denote the Fréchet derivatives of $l$ w.r.t. the first and second argument of said map in the direction of $w \in V$, $\nu \in \mathbb{R}^P$ by $\partial_v l_\mu(v)[w] \in {\mathbb{R}}$ and $\partial_\mu l_\mu(v) \cdot \nu \in {\mathbb{R}}$, respectively. We omit the word Fréchet when referring to the derivatives of ${\mathcal{J}}$, $a$, $l$, $j$ and $k$, in order to simplify the notation, unless it is strictly necessary to specify it.

We apply this notation for Fréchet and partial derivatives for functionals and bilinear forms throughout this manuscript. Note that we denote the derivatives w.r.t. the symbol of the argument in the original definition of the functional or bilinear form, not w.r.t. the symbol of the actual argument, i.e. we use $\partial_u {\mathcal{J}}(u_\mu, 