# covariant derivative on TikZ

I would like to draw the picture as in the Thorpe's book, "elementary topics in differential geometry." One way is to first do the computation in R^3 and then draw the solid or dotted lines. Is there a better and quicker way then this.

\documentclass[border=5mm]{standalone}
\usepackage{tikz}

\tikzset{
3D/.style={
x={(-3.85mm, -3.85mm)},
y={(1cm, 0cm)},
z={(0cm, 1cm)},
},
}

\begin{document}
\begin{tikzpicture}[3D, scale=3]
\tikzset{>=stealth, shorten >=1pt}
\coordinate (sw) at (+1, -1, 0);
\coordinate (nw) at (-1, -1, 0);
\coordinate (ne) at (-1, +1, 0);
\coordinate (se) at (+1, +1, 0);
\coordinate (c)  at (0, 0, 0);

\draw (sw) -- (nw) -- (ne) node[right] {$S_{\alpha(t)}$} -- (se) -- cycle;
\draw[->] (c) -- ++(0, 0, .7) node[above left] {$N(\alpha(t))$};
\draw[->] (c) -- ++(-.3, .3, 0) node[below right] {$X'(t)$};
\draw[->] (c) -- ++(-.3, .3, .7) node[above right] {$\dot X(t)$};
\draw[dashed] (c) ++(-.3, .3, 0) -- ++(0, 0, .7);
\end{tikzpicture}
\end{document}