I'm trying to understand change of variables for for multivarate integrals, particularly for applications in multidimensional mechanism design. An Econometrica paper has a key passage that does not make sense. There is a vector $\alpha$ of dimension $m$ and scalar $r$. The paper is performing an integral change of variables, and it says:
Making the change of variables $\hat{\alpha}=r\alpha$ and using the fact that $d\hat{\alpha}=r^md\alpha$ gives ...
Unfortunately I could not follow this. How does $d\hat{\alpha} = r^m d\alpha$? This makes no sense to me, and I don't understand what operation is being performed by "$d\hat{\alpha}.$''
But then again, I probably don't understand what it means to have "d[vector]." Maybe it's something like: Taking the derivative of each element in the vector with respect itself?
This appears in the paper below on page 62 near the top.
Armstrong, M. (1996). Multiproduct nonlinear pricing. Econometrica: Journal of the Econometric Society, 51-75.