Defining the Metric for a Standard Parametrization of a Cylinder This is very simple. Consider a cylinder in $\mathbb{R}^{3}$. Let the axis of the cylinder coincide with the $z$-axis. Allow the cylinder to be paramterized as follows:
\begin{align*}
x(\varphi,h) &= a \cos \varphi \\
y(\varphi,h) &= a \sin \varphi \\
z(\varphi,h)&= h \\
\end{align*}
The metric $g$ for this surface is 
\begin{equation*}
g = a^{2} \text{d}\varphi^{2} + \text{d}h^{2}
\end{equation*}
How do we know this is the metric? Like, derive it for me please from the given parameterization. Thanks.

If you choose the usual Euclidean (Pythagorean) metric, then the metric tensor (and its pull-back to cylindrical coordinates) is $g=dx^2+dy^2+dz^2=\\(\cos\phi da-a\sin\phi d\phi)^2+(\sin\phi da+a\cos\phi d\phi)^2+(dh)^2=\\da^2+a^2d\phi^2+dh^2.$
