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).