1. The problem gives three equations: $$u = p \sin\phi \cos\theta$$ $$y = p \sin\phi \sin\theta$$ $$z = p \cos\phi$$ 2. These equations represent the transformation from spherical coordinates $(p, \phi, \theta)$ to Cartesian-like coordinates $(u, y, z)$. 3. To derive a relationship or express $p$, $\phi$, and $\theta$ in terms of $u$, $y$, and $z$, we start by squaring and adding the first two equations: $$u^2 + y^2 = p^2 \sin^2\phi (\cos^2\theta + \sin^2\theta) = p^2 \sin^2\phi$$ since $\cos^2\theta + \sin^2\theta = 1$. 4. From the third equation: $$z = p \cos\phi$$ 5. Using the Pythagorean identity $\sin^2\phi + \cos^2\phi = 1$, we can write: $$p^2 = u^2 + y^2 + z^2$$ 6. To find $\phi$, use: $$\cos\phi = \frac{z}{p} = \frac{z}{\sqrt{u^2 + y^2 + z^2}}$$ 7. To find $\theta$, divide the first equation by the second: $$\frac{u}{y} = \frac{p \sin\phi \cos\theta}{p \sin\phi \sin\theta} = \frac{\cos\theta}{\sin\theta} = \cot\theta$$ thus, $$\theta = \arctan\left(\frac{y}{u}\right)$$ 8. Summary: $$p = \sqrt{u^2 + y^2 + z^2}$$ $$\phi = \arccos\left(\frac{z}{p}\right)$$ $$\theta = \arctan\left(\frac{y}{u}\right)$$ These formulas allow conversion from $(u,y,z)$ back to spherical coordinates $(p, \phi, \theta)$. This derivation shows the inverse transformation from the given equations.