What is the geometric meaning of the variables $r$, $v$ and $\phi$? I have this mapping.
$f: \left[0,\infty \right)\times \left[0,2\pi\right] \times \left[-\frac{\pi}{2}, \frac{\pi}{2}  \right] \mapsto \mathbb R ^{3} $
How do I explain what is the geometric meaning of variables r,v and phi? I dont understand really what means "geometric meaning"?
$\begin{pmatrix} r \\ \phi \\ v \end{pmatrix} \mapsto \begin{pmatrix} r \cdot \cos v \cdot \cos\phi \\ r\cdot \cos v\cdot \sin\phi \\ r\cdot \sin v \end{pmatrix} $
 A: Realize that 
$$(r\cos v\cos\phi)^2+(r\cos v\sin\phi)^2+(r\sin v)^2=r^2$$
So, your map represents a sphere in $\mathbb{R}$. Here, $r$ is the radius of the sphere. $v$ is the angle that vector makes with the $xy$-plane and $\phi$ is the angle in the in the circle that is parallel to the $xy$-plane.
Look at the picture

A: By geometric meaning I think of: a way to graphically represent the points.
I will try to make an analogy...
Your mapping takes three values, $(r, \phi, v)$, from the sets $\left[0,\infty \right),  \left[0,2\pi\right],  \left[-\frac{\pi}{2}, \frac{\pi}{2}  \right]$.
By mapping, it is understood that, for each value of $(r, \phi, v)$, there is a unique $(x, y, z)$ correspondence in $ \mathbb R ^{3}$. (see the definition of map: http://mathworld.wolfram.com/Map.html)
Suppose that $$r \in \left[0,\infty \right), \phi \in \left[0,2\pi\right],  v \in \left[-\frac{\pi}{2}, \frac{\pi}{2}  \right]$$
and the map
$$\begin{pmatrix} r \\ \phi \\ v \end{pmatrix} \mapsto \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} r \cdot cosv \cdot cos\phi \\ r\cdot cos\cdot sin\phi \\ r\cdot sinv \end{pmatrix}$$
To make a graphic representation of that mapping, create three axis, each being one of the set.
The first axis is the values of $r \in \left[0,\infty \right)$. It will be a line that begins at a point $0$ and never ends, i.e., goes to $\infty$.

The second axis is defined by $\phi \in \left[0,2\pi\right]$. It is the angle of a circle:

Your third axis is the angle of a semicircle. $v \in \left[-\frac{\pi}{2}, \frac{\pi}{2}  \right]$

Now, how do we put them together to represent $(x,y,z)$?
Try to make them perpendicular to each other.
Begin with $r$ and choose a value. That is, from the line begining at $0$, "walk" up to $r$. Then, from the point $r$, walk in a circle up to the angle $\phi$ (your distance to the point $0$ will remain equal to $r$).

Then walk in a semicircle up to the angle $v$.
You have ended in a point $(r, \phi, v)$.
Because of the mapping that takes these three variables, you are "teleported" to a corresponding $(x,y,z)$ location. That teleport is proportional to the mapping and will make you end in the location 
$$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} r \cdot cosv \cdot cos\phi \\ r\cdot cos\cdot sin\phi \\ r\cdot sinv \end{pmatrix}$$
Edit
This teleport I'm talking about is a translation.
