Every independent variable has its own domain, but to show the complete domain of a multivariable function is to show the domain of each variable. Because each variable is its own "dimension", to show them altogether you would use the "times" notation.

Thus the domain of \(\displaystyle f\left( x, y, z \right) \) for example would be \(\displaystyle \textrm{Dom}\left( x \right) \times \textrm{Dom} \left( y \right) \times \textrm{Dom} \left( z \right)\).

If the domains happen to be the same, then we can use a power notation, the same way we would with multiplying the same amount by itself. So if the domain of x, y, z were all R, then we could write \(\displaystyle \mathbf{R} \times \mathbf{R} \times \mathbf{R} = \mathbf{R}^3\).