  Using a multivariable Faa di Bruno formula we give conditions on
transformations $\tau:[0,1]^m\to\mathcal{X}$ where $\mathcal{X}$ is a closed
and bounded subset of $\mathbb{R}^d$ such that $f\circ\tau$ is of bounded
variation in the sense of Hardy and Krause for all $f\in C^d(\mathcal{x})$. We
give similar conditions for $f\circ\tau$ to be smooth enough for scrambled net
sampling to attain $O(n^{-3/2+\epsilon})$ accuracy. Some popular symmetric
transformations to the simplex and sphere are shown to satisfy neither
condition. Some other transformations due to Fang and Wang (1993) satisfy the
first but not the second condition. We provide transformations for the simplex
that makes $f\circ\tau$ smooth enough to fully benefit from scrambled net
sampling for all $f$ in a class of generalized polynomials. We also find
sufficient conditions for the Rosenblatt-Hlawka-M\"uck transformation in
$\mathbb{R}^2$ and for importance sampling to be of bounded variation in the
sense of Hardy and Krause.
