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

Infinitesimal length

From the previous section we have that

\[\begin{align} \Delta \u{X}_{\epsilon} &=\mt{\u{\gamma}}'(0)\epsilon+o(\epsilon)\\ &= \u{N}\epsilon+o(\epsilon)\\ &=d\u{X}+o(\epsilon) \end{align}\] \[\begin{align} \Delta \u{x}_{\epsilon} &=\st{\u{\gamma}}'(0)\epsilon+o(\epsilon)\\ &=\u{n}\epsilon+o(\epsilon)\\ &=d\u{x}+o(\epsilon) \end{align}.\]

From the chain rule of differentiation we have that \(\begin{align} \st{\u{\gamma}}'(\xi)&=\u{\varphi}'(\mt{\u{\gamma}}(\xi))\circ\mt{\u{\gamma}}'(\xi)\\ \st{\u{\gamma}}'(\xi)&=\u{\varphi}'(\mt{\u{\gamma}}(\xi))\mt{\u{\gamma}}'(\xi), \end{align}\) then \(\begin{align} \st{\u{\gamma}}'(0)\epsilon=\u{\varphi}'(\mt{\u{\gamma}}(0))\mt{\u{\gamma}}'(0)\epsilon. \end{align}\)

Hence we can get \(\begin{align} \Delta \u{x}_{\epsilon}&=\u{F}(\u{X})\mt{\u{\gamma}}'(0)\epsilon+o(\epsilon)\\ \Delta \u{x}_{\epsilon}&=\u{F}(\u{X})(\Delta \u{X}_{\epsilon}-o(\epsilon))+o(\epsilon)\\ \Delta \u{x}_{\epsilon}&=\u{F}(\u{X})\Delta \u X_{\epsilon}+o(\epsilon), \end{align}\) and \(\begin{align} d\u{x}&=\u{F}(\u{X})d \u{X} \\ \epsilon\u{n} &=\epsilon\u{F}(\u{X})\u{N} \\ \u{n} &=\u{F}(\u{X})\u{N} \end{align}\)