their derivatives, e.g. $v_{(1)} = \{v^\alpha_i\}, \ v^\alpha_i = D_i (v^\alpha).$ We use $\delta/\delta u^\alpha$ for the Euler-Lagrange operator $$\frac{\delta}{\delta u^\alpha} =
\frac{\partial}{\partial u^\alpha} + \sum_{s=1}^\infty (-1)^s
D_{i_1}\cdots D_{i_s}\,\frac{\partial}{\partial u^\alpha_{i_1\cdots
i_s}}\,,\quad \alpha=1, \ldots, m,$$ so that $$\frac{\delta (v^\beta F_\beta)}{\delta u^\alpha} =
 \frac{\partial (v^\beta F_\beta)}{\partial u^\alpha} -
 D_i \bigg(\frac{\partial (v^\beta F_\beta)}{\partial u^\alpha_i} \bigg)
 + D_i D_k \bigg(\frac{\partial (v^\beta F_\beta)}{\partial u^\alpha_{ik}} \bigg)
 - \cdots\,.$$ The total differentiation (\[sa:cl.diff1\]) is extended to the new dependent variables: $$\label{sa:cl.diff2}
  D_i = \frac{\partial}{\partial x^i} +
 u^\alpha_{i}\frac{\partial}{\partial u^\alpha} +
 v^\alpha_{i}\frac{\partial}{\partial v^\alpha} +
 u^\alpha_{ij}\frac{\partial}{\partial u^\alpha_j} +
 v^\alpha_{ij}\frac{\partial}{\partial v^\alpha_j} +
 \cdots\,.$$

The adjointness of *nonlinear* equations is not a symmetric relation. In other words, nonlinear equations, unlike the linear ones, do not obey the condition (\[adad\]) of mutual adjointness. Instead, the following equation holds: $$\label{nsa.linearized}
 \left(F^*\right)^* = \hat F$$ where $\hat F$ is the *linear approximation* to $F$ defined as follows. We use the temporary notation $F[u]$ for the left-hand side of Eq. (\[sa:eveq.int.1\]) and consider $F[u + w]$ by letting $w \ll
 1.$ Then neglecting the nonlinear terms in $w$ we define $\hat F$ by the equation $$\label{nsa.linearize1}
 F[u + w] \approx F[u] + \hat F[w]$$ For linear equations we have $\hat F = F,$ and hence Eq. 