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

Volume change

Consider the material region \(\u{V}_{\epsilon}=\set{\u{X}+\alpha_i\lsc[i]{\hat{\u{N}}}}{\le \alpha_i \le \epsilon}\). The volume of this materials region is \(\epsilon^3(\lsc[1]{\hat{\u{N}}}\times \lsc[2]{\hat{\u{N}}})\cdot \lsc[3]{\hat{\u{N}}}\). The volume of the region \(\u{\kappa}_{\tau}(\u{V}_{\epsilon})=(\lsc[1]{\Delta {\u{x}}}\times \lsc[2]{\Delta {\u{x}}})\cdot \lsc[3]{\Delta{\u{x}}}\), where \(\begin{align} \lsc[i]{\Delta \u{x}}&=\epsilon \u{F}(\u{X})\lsc[i]{\hat{\u{N}}}+o(\epsilon). \end{align}\)

Hence the volume of the region \(\u{\kappa}_{\tau}(\u{V}_{\epsilon})\) is, \(\begin{align} \u{v}_{\epsilon}&=(\lsc[1]{\Delta {\u{x}}}\times \lsc[2]{\Delta {\u{x}}})\cdot \lsc[3]{\Delta{\u{x}}}\\ &=\epsilon^3\pr{\pr{\u{F}(\u{X})\lsc[1]{\hat{\u{N}}}\times \u{F}(\u{X})\lsc[2]{\hat{\u{N}}}}\cdot\pr{\u{F}(\u{X})\lsc[3]{\hat{\u{N}}}}}+o\pr{\epsilon^3}\\ &={\rm Det}\pr{\u{F}(\u{X})}\u{V}_{\epsilon}+o\pr{\epsilon^3}. \end{align}\)

Then \(\begin{align} \lim_{\epsilon \to 0}\frac{\u{v}_{\epsilon}}{\u{V}_{\epsilon}}={\rm Det}\pr{\u{F}(\u{X})}=J(\u{X}) \end{align}\)