To convert from cylindrical coordinates $(r, \theta, z)$ to spherical coordinates $(\rho, \phi, \theta)$, we use the following formulas:

1. $\rho = \sqrt{r^2 + z^2}$
2. $\phi = \tan^{-1}\left(\frac{r}{z}\right)$
3. $\theta = \theta$ (unchanged)

Given cylindrical coordinates:
$r = -4, \quad \theta = \frac{\pi}{3}, \quad z = 4$

Step-by-Step Conversion:

1. Calculate $\rho$: $\rho = \sqrt{r^2 + z^2} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

2. Calculate $\phi$: $\phi = \tan^{-1}\left(\frac{r}{z}\right) = \tan^{-1}\left(\frac{-4}{4}\right) = \tan^{-1}(-1)$ This gives $\phi = -\frac{\pi}{4}$, as $\tan^{-1}(-1)$ is $-\frac{\pi}{4}$ in the standard range.

3. The angle $\theta$ remains unchanged: $\theta = \frac{\pi}{3}$

Thus, the spherical coordinates are: $(\rho, \phi, \theta) = (4\sqrt{2}, -\frac{\pi}{4}, \frac{\pi}{3})$

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Would you like further details or clarifications on this conversion?

Here are 5 related questions to explore:

1. What is the relationship between spherical and cylindrical coordinates?
2. How do you handle the case when the cylindrical coordinate $r$ is negative?
3. How can you convert from spherical coordinates back to cylindrical coordinates?
4. What is the geometric interpretation of spherical coordinates?
5. What happens to spherical coordinates if $z = 0$ in cylindrical coordinates?

Tip: Always ensure that you correctly interpret the angle $\phi$ when converting from cylindrical to spherical coordinates, as it depends on the signs of $r$ and $z$.