Spherical co-ordinates with Implicit function thm

[trap101]

So I'm asked to determine near which points of R^3 can we solve for ρ, δ, θ in terms of x,y,z:

x = ρ sinδ cosθ
y= ρ sinδ sinθ
z= ρcosδ

so the spherical co-ordinates using IFT.

Attempt:

Ok so in order to determine solutions, I need to first find where the determinant of the freceht derivative does not equal zero. So I set it up as so:

\begin{bmatrix} sinδ cosθ & sinδ sinθ & cosδ\\ ρ cosδ cosθ & ρ cosδ sinθ & -ρ sinδ\\ -ρ sinδ sinθ & ρ sinδ sinθ & 0\end{bmatrix}

so I take the determinant of that and I suppose whichever points do not make the determinant 0 are the points where the system can be solved.

the issue is when I took the determinant, it didn't really simplify out how I hoped...is this what I'm suppose to do or is there some trick to this?

 

[trap101]

my mistake, I entered the matrix values wrong (darn latex). So it's actually the transpose of the above matrix is the one I get.

 

[HallsofIvy]

That doesn't really matter since the determinant of a matrix and its transpose are the same. However you have differentiated incorrectly. The term in the middle of the third row (as you have it) should be $\rho sin(\delta) cos(\theta)$, not $\rho sin(\delta)sin(\theta)$.