$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

The linearization of $\delta W_{\text{ext}}$ in (3.3-3) depends on whether natural boundary conditions are prescribed as area densities or total net values over an area. Thus, in the case when $\mathbf{t}\,da$ (net force), $w_{n}da$ (net volumetric flow rate), or $j_{n}da$ (net molar flow rate) are prescribed over the elemental area $da$ , there is no variation in $\delta W_{\text{ext}}$ and it follows that $D\delta W_{\text{ext}}=0$ . Alternatively, in the case when $\mathbf{t}$ , $w_{n}$ or $j_{n}$ are prescribed, the linearization may be performed by evaluating the integral in the parametric space of the boundary surface $\partial b$ , with parametric coordinates $\left(\eta^{1},\eta^{2}\right)$ . Accordingly, for a point $\mathbf{x}\left(\eta^{1},\eta^{2}\right)$ on $\partial b$ , surface tangents (covariant basis vectors) are given by $$\begin{equation} \mathbf{g}_{\alpha}=\frac{\partial\mathbf{x}}{\partial\eta^{\alpha}},\quad\left(\alpha=1,2\right)\label{eq260} \end{equation}$$ and the outward unit normal is $$\begin{equation} \mathbf{n}=\frac{\mathbf{g}_{1}\times\mathbf{g}_{2}}{\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|}\,.\label{eq261} \end{equation}$$ The elemental area on $\partial b$ is $da=\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|d\eta^{1}d\eta^{2}$ . Consequently, the external virtual work integral may be rewritten as $$\begin{equation} \delta W_{\text{ext}}=\int_{\partial b}\left(\delta\mathbf{v}\cdot\mathbf{t}+\delta\tilde{p}\,w_{n}+\delta\tilde{c}\,j_{n}\right)\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|d\eta^{1}d\eta^{2}\,.\label{eq262} \end{equation}$$ The directional derivative of $\delta W_{\text{ext}}$ may then be applied directly to its integrand, since the parametric space is invariant [23].