It is very sloppy when you come from say, analysis, where things are precisely defined, especially in more applied statistics.
In my experience, any more applied than probability theory itself and you get omissions of very important details such as the probability space under consideration and the RVs involved in expectations, in addition to frequent inconsistencies and outright abuses of notation. These tendencies make it a harder subject for someone who is used to precise exposition to pick up.
It is hardly sloppy, the author just wrote it in the language of vector calculus. Another treatment, the one in my textbooks, addresses it with the determinant of the Jacobian matrix from the perspective of change-of-coordinates in multidimensional integrals. As always there are as many ways to skin a cat as there are closed curves on the 2-sphere.