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

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Homework Help Overview

The discussion revolves around solving for the variables theta, phi, and rho in the context of R^3 using the Inverse Function Theorem. The original poster is examining the conditions under which the determinant of the Jacobian matrix becomes zero, which affects the solvability of these variables in terms of x, y, and z.

Discussion Character

  • Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to calculate the determinant of the Jacobian matrix formed by the partial derivatives of the transformation equations. They express uncertainty about their results, particularly regarding the conditions leading to the determinant being zero. Other participants question the correctness of the original poster's determinant expression and suggest reviewing the calculations.

Discussion Status

The discussion is ongoing, with participants providing feedback on the original poster's calculations. Some guidance is offered in the form of suggestions to re-evaluate the determinant, but no consensus has been reached regarding the correctness of the expressions involved.

Contextual Notes

The original poster indicates they have checked their work multiple times, suggesting a potential constraint of time or pressure to find a solution. There is also a focus on the specific mathematical expressions used in the determinant calculation.

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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?
 
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That doesn't look right. What is the equation you got for the determinant?
 
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.
 
I get something different. All I can suggest is go back through it carefully.
 
thanks statusx for your time..i appreciate it
 

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