\[\renewcommand{\u}[1]{\boldsymbol{#1}} \renewcommand{\b}[1]{\mathbb} \newcommand{\set}[2]{\left\{#1\,\big|\,#2\right\}} \newcommand{\pr}[1]{\left(#1\right)} \newcommand{\st}[2][]{#2_{\mathscr{s}#1}} \newcommand{\mt}[2][]{#2_{\mathscr{m}#1}} \newcommand{\lsc}[2][\mathscr{l}]{ { }^{#1}\!#2 }\]

Area Changes

Consider the material curve \(\mt[\epsilon]{\u{\gamma}}(\xi)=\u{N}\xi\). \(\begin{align} \Delta \u X_{\epsilon} &=\u{N}\epsilon+o(\epsilon)\\ &=d\u{X}+o(\epsilon) \label{eq:matInfinitesimal} \end{align}\)

\[\begin{align} \Delta \u{A}_{\epsilon} &=\lsc[1]{\Delta \u X}_{\epsilon}\times \lsc[2]{\Delta \u X}_{\epsilon}\\ &=\epsilon^2\,\, \lsc[1]{\hat{\u N}}_{\epsilon}\times \lsc[2]{\hat{\u N}}_{\epsilon}+o(\epsilon)\\ &=d\u{A}+o(\epsilon) \end{align}\]

From the calculation of volume, it can shown that

\[\begin{align} \text{vol}(V_{\epsilon})&= \lsc[3]{\Delta \u{X}}\cdot \Delta \u{A}_{\epsilon}\\ &=\epsilon^3 \pr{\lsc[3]{\u{N}}\cdot (\lsc[1]{\u{N}}\times \lsc[2]{\u{N}})}+o(\epsilon^3)\\ \end{align}\]

To calculate the deformed area, \(\begin{align} \Delta \u x_{\epsilon} &=\u{F}(\u{X})\u{N}\epsilon+o(\epsilon)\\ &=\u{F}(\u{X})d\u{X}+o(\epsilon)\\ &=\u{n}\epsilon+o(\epsilon) \end{align}\)

\[\begin{align} \Delta \u{a}_{\epsilon} &=\lsc[1]{\Delta\u{x}_{\epsilon}}\times \lsc[2]{\Delta \u{x}_{\epsilon}}\\ &=\epsilon^2 \pr{\lsc[1]{\u n}\times \lsc[2]{\u n}}+o(\epsilon^2)\\ &= d\u{a}+o(\epsilon) \end{align}.\] \[\begin{align} \text{vol}(v_{\epsilon}) &=\lsc[3]{\Delta\u{x}_{\epsilon}}\cdot \Delta \u a_{\epsilon}\\ &=\pr{\epsilon\,\, \lsc[3]{\u{n}}+o(\epsilon)}\cdot \Delta \u{a}_{\epsilon}\\ &=\epsilon^3 \,\lsc[3]{\u{n}}\cdot \pr{\lsc[1]{\u{n}}\times \lsc[2]{\u{n}}}+o(\epsilon^3) \end{align}\]

From the relation between the undeformed volume and deformed volume, we can get,

\[\begin{align} \epsilon^3 \,\lsc[3]{\u{n}}\cdot \pr{\lsc[1]{\u{n}}\times \lsc[2]{\u{n}}}&=\epsilon^3 J(\u{X})\pr{\lsc[3]{\u{N}}\cdot (\lsc[1]{\u{N}}\times \lsc[2]{\u{N}})}+o(\epsilon^3)\\ \u{F}(\u{X})\,\lsc[3]{\u{N}}\cdot \pr{\lsc[1]{\u{n}}\times \lsc[2]{\u{n}}}&= J(\u{X})\pr{\lsc[3]{\u{N}}\cdot (\lsc[1]{\u{N}}\times \lsc[2]{\u{N}})}\\ \lsc[3]{\u{N}}\cdot \u{F}(\u{X})^{\rm T}\pr{\lsc[1]{\u{n}}\times \lsc[2]{\u{n}}}&= J(\u{X})\pr{\lsc[3]{\u{N}}\cdot (\lsc[1]{\u{N}}\times \lsc[2]{\u{N}})}\\ \u{F}(\u{X})^{\rm T}\pr{\lsc[1]{\u{n}}\times \lsc[2]{\u{n}}}&= J(\u{X})\pr{ (\lsc[1]{\u{N}}\times \lsc[2]{\u{N}})}\\ \pr{\lsc[1]{\u{n}}\times \lsc[2]{\u{n}}}&= J(\u{X})\u{F}(\u{X})^{\rm -T} (\lsc[1]{\u{N}}\times \lsc[2]{\u{N}}) \end{align}\]

Multiplying by $\epsilon^2$ on both sides of the above equation we get \(\begin{align} d\u{a}=J(\u{X})\u{F}(\u{X})^{\rm -T}d\u{A} \end{align}\)