Spherical coordinate system can be considered as the three dimensional counterpart of polar coordinate system of two dimensional plane. While the polar coordinate system defines point with one angle and one distance, in spherical coordinate system, points are defined using two angles and one distance. In spherical coordinate system, as the name hints, all points in space are viewed to be placed on the surfaces of concentric spheres.

Any point P is referred using the coordinates ($\rho$, $\phi$, $\theta$) 
where, $\rho$ is the distance of the point from the origin O
$\phi$ is angle between line OP and positive z - axis
$\theta$ is the angle from the positive x - axis to the the projection of OP on XY plane.

The spherical coordinates are specially useful in solving problems, where we find symmetry about a point. The origin is shifted to this point and the equations are transformed to spherical system.

Spherical coordinates are also useful in dealing with some multiple integrals which cannot be evaluated using rectangular or cylindrical coordinates.

Spherical coordinates represent a point P in three dimensional space by ordered triples ($\rho$, $\phi$, $\theta$), in which
  1. $\rho$ is the distance of point P from the origin or $\rho$ = |$\overrightarrow{OP}$|
  2. $\phi$ is the angle $\overrightarrow{OP}$ makes with the positive z - axis.
  3. $\theta$ is the angle measured from the positive x - axis to $\overrightarrow{OR}$, the projection of $\overrightarrow{OP}$ on XY - plane.

Spherical Coordinates Definition


It can be observed from the above description that $\rho$ $\geq$ 0 and 0 $\leq$ $\theta$ $\leq$ $\pi$.

The relationships between the cartesian coordinates (x, y, z) to the corresponding spherical coordinates are as follows:
x = $\rho$ sin($\phi$) cos($\theta$)     
y = $\rho$ sin($\phi$) sin($\theta$)         
z = $\rho$ cos($\phi$)

Using distance formula, we also get
$\rho$2 = x2 + y2 + z2

Here, both $\phi$ and $\theta$ are measured in radians.

Let us see with the help of an example how to convert spherical coordinates to rectangular coordinates.

Solved Example

Question: Change the spherical coordinates $(2\sqrt{2}, $$\frac{3 \pi}{2}$$, $$\frac{\pi}{2}$$)$ into rectangular coordinates.
Solution:
For the point given, $\rho$ = $2\sqrt{2}$, $\phi$ = $\frac{3 \pi}{2}$ and $\theta$ = $\frac{\pi}{2}$

To write this in the form (x, y, z), we use following the conversions formulas:
x = $\rho$ sin($\phi$) cos($\theta$) = $2\sqrt{2} \sin $$(\frac{3\pi}{2})$$ \cos $$(\frac{\pi}{2})$ = $2\sqrt{2} \times (-1) \times (0) = 0$

y = $\rho$ sin($\phi$) sin($\theta$) = $2\sqrt{2} \sin $$(\frac{3\pi}{2})$$ \sin $$(\frac{\pi}{2})$ = $2\sqrt{2} \times (-1) (1) = -2\sqrt{2}$

z = $\rho$ cos($\phi$) = $2\sqrt{2} \cos $$(\frac{3\pi}{2})$ = $2\sqrt{2} \times 0 = 0$

Hence, the coordinates in rectangular system are $(0, -2\sqrt{2}, 0)$
This is a point on the negative y - axis at a distance $2\sqrt{2}$ from the origin.

Let us write the rectangular form (1, $\sqrt{3}$, 2$\sqrt{3}$) as spherical coordinates.
We have x = 1, y = $\sqrt{3}$ and z = $2\sqrt{3}$

Spherical coordinates are as follows:
x = $\rho$ sin($\phi$) cos($\theta$)     
y = $\rho$ sin($\phi$) sin($\theta$)         
z = $\rho$ cos($\phi$)

Now,
$\rho$sin($\phi$) cos($\theta$)  = 1   ............(1)
$\rho$ sin($\phi$) sin ($\theta$)  = $\sqrt{3}$ ............(2)
$\rho$ cos($\phi$) = $2\sqrt{3}$         ............(3)

Dividing equation (2) by (1), we get

tan ($\theta$) = $\sqrt{3}$
 
$\theta$ = $\frac{\pi}{3}$   (As both x and y are positive)

$\rho ^2$ = $x^2 + y^2 + z^2$ = $1 + (\sqrt{3})^2 + (2\sqrt{3})^2$ = 1 + 3 + 12 =  16.
Hence, $\rho$ = 4
$\rho$ cos($\phi$) = $2\sqrt{3}$     
4 cos($\phi$) = $2\sqrt{3}$ 
cos($\phi$) = $\frac{\sqrt{3}}{2}$      

$\phi$ = $\frac{\pi}{6}$

Thus, the required spherical form of the coordinates are $(4, $$\frac{\pi}{6}$$, $$\frac{\pi}{3}$$)$
In rectangular coordinate system, unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$ are corresponding to the directions of three coordinate axes. In the same manner, we can think of unit vectors $\hat{\rho }$, $\hat{\phi }$ and $\hat{\theta }$ in spherical coordinates. The position vector of any point P<$\rho$, $\phi$, $\theta$> can be written in terms of these three mutually perpendicular vectors.

Geometrically, $\hat{\rho }$, $\hat{\phi }$ and $\hat{\theta }$ are correspondingly normal to the sphere of radius $\rho$, cone of angle $\phi$ and the vertical plane inclined at an angle $\theta$ to the x - axis.

The unit vectors in spherical and rectangular system are related as follows:
$\hat{\rho }$ = $\hat{i}$ sin($\phi$) cos($\theta$) + $\hat{j}$ sin($\phi$) sin($\theta$) + $\hat{k}$ cos($\phi$)
$\hat{\phi }$ = $\hat{i}$ cos($\phi$) cos($\theta$) +  $\hat{j}$ cos($\phi$) sin($\theta$) - $\hat{k}$ sin($\phi$)
$\hat{\theta }$ = - $\hat{i}$ sin($\theta$) + $\hat{j}$ cos($\theta$) 
Gradient vector of a function f of three variables is represented by $\bigtriangledown f$ and is defined as follows:

$\bigtriangledown f$ = $<f_x, f_y, f_z>$ = $\frac{\partial f}{\partial x}$$i + $$\frac{\partial f}{\partial y}$$j + $$\frac{\partial f}{\partial z}$$k$

The gradient in spherical coordinates is defined in terms of the components in the directions of $\hat{\rho }$, $\hat{\phi }$, $\hat{\theta }$ as follows:

$\bigtriangledown f$ = $\frac{\partial f}{\partial \rho }$$\hat{\rho }$ + $\frac{1}{\rho \sin\theta }\frac{\partial f}{\partial \phi }$$\hat{\phi }$ + $\frac{1}{\rho }\frac{\partial f}{\partial \theta }$$\hat{\theta }$.
Given below are some of the examples on spherical coordinates.

Solved Examples

Question 1: Write the equation x2 + y2 = 2y in cylindrical coordinates.
Solution:
Using the transformation equations x = $\rho$ sin($\phi$) cos($\theta$) and y = $\rho$ sin($\phi$) sin($\theta$), the equation can be rewritten as follows:

$\rho$2 sin2($\phi$) cos2($\theta$) + $\rho$2 sin2($\phi$) sin2($\theta$) = 2($\rho$ sin($\phi$) sin($\theta$))
$\rho$2 sin2($\phi$)[cos2($\theta$) + sin2($\theta$)] = 2$\rho$ sin($\phi$) sin($\theta$)
$\rho$2 sin2($\phi$) = 2$\rho$ sin($\phi$) sin($\theta$)  or
$\rho$ sin($\phi$) = 2 sin($\theta$)

Question 2: What do the equations $\rho$ = k, $\phi$ = k and $\theta$ = k represent in spherical coordinate system?
Solution:
The equation $\rho$ = k represents the surface of sphere with center at origin and radius = $\rho$.
The equation $\phi$ = k represents the top half cone with vertex at origin if 0 < k < $\frac{\pi}{2}$ and the bottom half cone if $\frac{\pi}{2}$ < k < $\pi$.

The equation $\theta$ = k represents the vertical half plane with the edge along the z- axis inclined at an angle k to the x axis.