Solving for Theta, Phi, and Rho in R^3 using the Inverse Function Theorem

[Haftred]

I'm trying to see near which points of R^3 I can solve for theta, phi, and rho in terms of x,y, and z. I know i need to find the determinant and see when it equals zero; however, I get the determinant to equal zero when sin(phi) = 0, and when tan(theta) = -cot(phi). The first is right, but I've checked my work many times and keep getting the last solution. I just calculated the determinant of the partial derivatives (dx/dtheta, dx / dphi, dx / drho...dy/dtheta, dy/dphi, dy/drho...dz/dtheta, dz/dphi, dz/drho). I've checked my work many times. Am I correct, or am I doing something wrong?

 

[StatusX]

That doesn't look right. What is the equation you got for the determinant?

 

[Haftred]

p = rho
a = phi
b = theta

p^2[cos(a)sin(a)(cosb)^3 + (sina)^2(sinb)^3 + cos(b)cos(a)sin(a)(sinb)^2 + sin(b)(sina)^2(sinb)^2].

I differentiated with respect to rho in the first column, phi in the second column, and theta in the third.

Thanks.

 

[StatusX]

I get something different. All I can suggest is go back through it carefully.

 

[Haftred]

thanks statusx for your time..i appreciate it