Could someone explain how the attached picture is calculated... I'm not sure... there is a typo in the last row (phi component) on the very bottom, but that's minor... thanks

 

Could someone explain how the attached picture is calculated... I'm not sure... there is a typo in the last row (phi component) on the very bottom, but that's minor... thanks

Looks like very unusual notation to me, and the typo doesn't help in trying to interpret it. You basically want the correct formula for curl, and you have it without the matrix notation.

Sometimes people get too tied up in making shorthand notation, and just make things more confusing. A literal interpretation of what is written would be to take the determinant of each matrix separately and then multiply the first constant, times the resulting scalar from the lefthand determinant, times the resulting vector from the righthand determinant. But, that doesn't seem to give the right answer.

I've attached a pdf of the correct formulas in general. You'll note that this reference drops the attempt to use a matrix determinant to represent the curl in spherical and general curvilinear coordinates. They use that notation for rectangular and cylindrical coordinates because the form is simple enough.