Curvature, Torsion and Change of coordinates Is there a general way so we can find curvature and torsion of a curve $$\gamma : \mathbb{R} \rightarrow \mathbb{R}^n$$ where $$n=2,3$$ from a coordinate system to a new coordinate system?
 A: Case $n = 2$ and curvature $\kappa(t)$. 
The curve $\gamma(t) = \big(x(t),y(t)\big)^T$, has curvature
$$
\kappa(t) = \frac{x'y''-x''y'}{(x'^2 + y'^2)^{3/2}} = \frac{\det(\gamma',\gamma'')}{\|\gamma'\|^3}.
$$
Now, suppose $x(t) = x\big(\xi(t),\eta(t)\big)$, $y(t) = y\big(\xi(t),\eta(t)\big)$, then
\begin{align}
\frac{d x}{d t} &= x_\xi \xi' + x_\eta \eta'\\
\frac{d y}{d t} &= y_\xi \xi' + y_\eta \eta'
\end{align}
and let $\alpha(t) = \big(\xi(t),\eta(t)\big)^T$, we can write the above relation as
$$
\gamma' = \textbf{J} \, \alpha'
$$
where
$$
\textbf{J} = \begin{pmatrix}x_\xi & x_\eta \\ y_\xi & y_\eta\end{pmatrix}
$$
is the Jacobian matrix. Then
$$
\gamma'' = \frac{d \textbf{J}}{dt} \alpha' + \textbf{J} \alpha''
$$
and
$$
\kappa(t) = \frac{\det(\textbf{J}\,\alpha',\frac{d \textbf{J}}{dt} \alpha' + \textbf{J} \alpha'')}{\|\textbf{J}\,\alpha'\|^3}.
$$
Example.
Let $x(t) = r(t) \cos \theta(t)$ and $y(t) = r(t) \sin \theta(t)$, then
\begin{align}
\det(\textbf{J}\alpha',\textbf{J}'\alpha') &= 2 r'^2 \theta' + r^2 \theta'^3\\
\det(\textbf{J}\alpha',\textbf{J}\alpha'') &= -r r'' \theta' + r r' \theta''\\
\|\textbf{J}\alpha'\|^3 &= (r'^2 + r^2 \theta'^2)^{3/2}
\end{align}
and
$$
\kappa(t) = \frac{r^2 \theta'^3 + 2 r'^2 \theta' - r r'' \theta' + r r' \theta''}{(r'^2 + r^2 \theta'^2)^{3/2}}.
$$
In the special case $\theta = t$
$$
\kappa(t) = \frac{r^2 + 2r'^2 - r r''}{(r^2 + r'^2)^{3/2}}
$$
which, according to eq. (15) of Wolfram Mathworld, is correct.
Now you can do the rest.
