Set-Builder Notation for Spherical Coordinates of the Sphere $x^2 + y^2 + z^2 = a^2$

Question

I have a problem that requires me to convert the equation of the sphere $x^2 + y^2 + z^2 = a^2$ into spherical coordinates. When writing out my reasoning for this problem, I want to represent the spherical coordinates for this sphere in set-builder notation. I have never taken a class that focuses specifically on set theory, so my knowledge of the subject is only whatever rudimentary understanding I was taught when learning other fields such as linear algebra.

My understanding is that set-builder notation is $\{$ elements in set $\mid$ condition $\}$ where $\mid$ means such that. In implementing this understanding, I came up with $S = \{ \rho \ge 0, 2\pi \ge \theta \ge 0, \pi \ge \phi \ge 0 \mid \rho^2 \cos^2(\theta) \sin^2(\phi) + \rho^2 \sin^2(\theta) \sin^2(\phi) + \rho^2 \cos^2(\phi) \le a^2 \}$. Is this the correct formal representation of the sphere using set-builder notation? If not, then what are the errors and what is the correct representation?

I would greatly appreciate it if people could please take the time to clarify this.

Answer 1

This isn't quite right - in this case $\rho$ represents the radius of the sphere, so it would be simpler to write: $$ S = \{\rho, \theta, \phi : \rho = a, 0 \leq \theta \leq 2\pi, 0 \leq \phi \leq \pi\}$$.

Otherwise, your conceptual understanding of set-builder notation seems good. Keep trying to write other surfaces as sets.

Answer 2

When using set-builder notation, we want to define our set by describing what each element must satisfy. It's better to describe each element as a single object. Fox example, instead of writing $\rho, \theta, \phi$ and viewing them as three values, it's better to write them as $(\rho, \theta, \phi)$ and view them as a triple. Also, since your condition is quite long, I would define a function $f: [0,\infty)\times[0,2\pi]\times[0,\pi]\to\mathbb{R}$ by $$f(\rho,\theta,\phi)=\rho^2\cos^2(\theta)\sin^2(\theta)+\rho^2\sin^2(\theta)\sin^2(\phi)+\rho^2\cos^2(\phi),$$ and then write the set as $$ S = \{ (\rho, \theta, \phi) \in [0,\infty)\times[0,2\pi]\times[0,\pi] \mid f(\rho,\theta,\phi)\leq a^2\}. $$ Since $f$ only makes sense for triples in $[0,\infty)\times[0,2\pi]\times[0,\pi]$, some would simply write $$ S = \{ (\rho, \theta, \phi) \mid f(\rho,\theta,\phi)\leq a^2\}. $$