Volume form for a warped product I would like to know if there exists some useful formulas that let us compute the volume form for a warped product, that is a $n$-dimensional Riemannian manifold of the form $dr^2 + \phi(r)^2 g_{N^{n-1}}$, where $N^{n-1}$ is a $n-1$-dimensional riemannian manifold and $g_{N^{n-1}}$ is the metric induced on it.
Also the case where $N^{n-1}$ is the hypershere in $\mathbb{R}^n$ would be useful: in this case computations could be carried out by direct computation, but I find it awkward. 
I thank you in advance for any suggestion,
Mattia.

Let me call your warped product manifold as $(M,g)$. 
Choose a local coordinate system $\{x^i\}_{i=1}^{n-1}$ on $N$, and denote $\partial_i:=\frac{\partial}{\partial x^i}$. Then $\partial_1,\ldots,\partial_{n-1},\partial_r$ form a basis for the tangent spaces of $M$. Denote $[g]$ to be the matrix whose entries are the components of the warped product metric $g$ with respect to this basis. Then we have
\begin{align}
[g]=\begin{pmatrix}
1 & \\ & \phi(r)^2[g_N]
\end{pmatrix}
\end{align}
where $[g_N]$ has the similar meaning. Then I guess it is not difficult to compute $\det[g]$. Of course it will be in terms of $\det[g_N]$ and the warping function $\phi$.
