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.

Hint: Let $(a,\phi,h) = (\theta^1,\theta^2,\theta^3)$. Write
$$dx = \frac{\partial x}{\partial \theta^i}d\theta^i\ \ dy = \frac{\partial x}{\partial \theta^i}d\theta^i\ \ dz = \frac{\partial z}{\partial \theta^i}d\theta^i \tag{1}$$
we are using Einstein Notation in (1).
Compute $dx^2$, $dy^2$ and $dz^2$. 
Finally the metric is given by $$g_{\mathbb{R}^3} = dx^2+dy^2+dz^2$$ 
