So, we're going over the Born interpretation of the function, and it gives rise to the normalization condition:

$\displaystyle \int |\Psi (\bold{r}, t)|^2 ~d^3\bold{r} = 1$

My professor wasn't very descriptive of it, but the question I have is what does the $\displaystyle d^3\bold{r}$ mean? I know it represents a volume, and in the integral it is over all space... So, what does it mean mathematically?