What is happening to the sin(phi) factor in the spherical curl?

[FrankJ777]

Homework Statement

Take the curl in the shperical coordinate system.

Relevant Equations

Same as below.

This is from my E&M textbook.
I'm doing a problem where I need to take the Curl in spherical coordinates but I'm getting the wrong answer.
I tried applying the matrix, but it doesn't seem like it make sense with the expansion that they show in the textbook (screenshot below).
If I apply the matrix formula I end up with an extra sinθ that don't get if I use the formula in the expansion.
It looks to me like in the aφ component there should be a 1/sinθ factor outside the brackets?
I don't see what is happening in the aφ compentent that is making the 1/sinθ factor next to the matrix disapear when we get to the aφ component.
Am I missing something.

Thanks

 

[fresh_42]

This looks like a typo. But I have some trouble with a factor $r$ in the second term, too, missing $r^{-1}$. I suggest to deal with the determinant and apply the factor outside last.

 

[FrankJ777]

This looks like a typo. But I have some trouble with a factor $r$ in the second term, too, missing $r^{-1}$. I suggest to deal with the determinant and apply the factor outside last.

I get the wrong answere when I use the determinant, even though I divide through with the r^2 sin(theta) last.
I do get the right answer if I use the expanded notation as a guide. So not sure if I'm applying the matrix wrong, or if there's a typo somewhere?