Understanding the Relationship Between Rectangular and Spherical Coordinate Systems

The relation between rectangular (Cartesian) and spherical coordinate systems allows mathematical representation of points in space by expressing coordinates in both systems and establishing precise algebraic transformations between them.

Coordinate Expressions in Rectangular and Spherical Systems

A point $P$ in rectangular coordinates has position $(x, y, z)$, while in the spherical coordinate system it is represented as $(r, \theta, \phi)$, where $r$ is the distance from the origin to $P$, $\theta$ the azimuthal angle in the $xy$-plane from the positive $x$-axis, and $\phi$ the polar angle from the positive $z$-axis.

Transformation from Spherical to Rectangular Coordinates

The relations connecting spherical and rectangular coordinates are given by the equations:

$x = r \sin\phi \cos\theta$

$y = r \sin\phi \sin\theta$

$z = r \cos\phi$

Transformation from Rectangular to Spherical Coordinates

The reverse mapping from rectangular to spherical coordinates is:

$r = \sqrt{x^2 + y^2 + z^2}$

$\theta = \tan^{-1}\left(\dfrac{y}{x}\right)$ (with quadrant correction)

$\phi = \cos^{-1}\left(\dfrac{z}{r}\right)$

Care must be taken in evaluating the angle $\theta$, as it is undefined when $x$ and $y$ are both zero, and requires principal value selection to match the correct quadrant.

Variable Ranges and Notation in Spherical Representation

In the spherical coordinate system, the standard ranges are $r \geq 0$, $\theta \in [0, 2\pi)$, and $\phi \in [0, \pi]$. The notation $r$ can also be found as $\rho$ in alternative conventions, and $\theta$, $\phi$ assignments may vary in some texts, so context-specific definitions must be checked. For coordinate axes and analytic geometry, the above assignment is standard.

Further foundational aspects can be reviewed in Analytical Geometry Overview.

Interpretation and Application in Coordinate Geometry

The transformation formulas demonstrate that every Cartesian point uniquely maps to a spherical triple, except for the degenerate case at the origin. These conversions are fundamental in problems involving radial symmetry, three-dimensional integration, and evaluating regions bounded by spheres and cones.

Connections to other coordinate systems are examined in Coordinate Geometry Principles.

• Meaning of $r$, $\theta$, $\phi$ in space
• Standard transformation equations
• Typical variable ranges
• Conversion strategy for problem solving
• Quadrant considerations in $\tan^{-1}(y/x)$

Calculation Example: Rectangular to Spherical Conversion

Given the rectangular coordinates $(x, y, z) = (2,\,2,\,2)$, the spherical coordinates are computed as follows.

$r = \sqrt{2^2 + 2^2 + 2^2} = \sqrt{12} = 2\sqrt{3}$

$\theta = \tan^{-1}\left(\dfrac{2}{2}\right) = \tan^{-1}(1) = \dfrac{\pi}{4}$

$\phi = \cos^{-1}\left(\dfrac{2}{2\sqrt{3}}\right) = \cos^{-1}\left(\dfrac{1}{\sqrt{3}}\right)$

Solution: The point $P$ corresponds to $\left(2\sqrt{3},\, \dfrac{\pi}{4},\, \cos^{-1}\left(\dfrac{1}{\sqrt{3}}\right)\right)$ in spherical coordinates.

For additional transformation examples, consult Geometry of Complex Numbers.

Calculation Example: Spherical to Rectangular Conversion

For the spherical coordinates $(r,\; \theta,\; \phi) = (5,\, \dfrac{\pi}{3},\, \dfrac{\pi}{4})$, evaluate the corresponding rectangular coordinates.

$x = 5 \sin\left(\dfrac{\pi}{4}\right) \cos\left(\dfrac{\pi}{3}\right) = 5 \cdot \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{5\sqrt{2}}{4}$

$y = 5 \sin\left(\dfrac{\pi}{4}\right) \sin\left(\dfrac{\pi}{3}\right) = 5 \cdot \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} = \dfrac{5\sqrt{6}}{4}$

$z = 5 \cos\left(\dfrac{\pi}{4}\right) = 5 \cdot \dfrac{\sqrt{2}}{2} = \dfrac{5\sqrt{2}}{2}$

Example: The rectangular coordinates are $\left(\dfrac{5\sqrt{2}}{4},\,\dfrac{5\sqrt{6}}{4},\,\dfrac{5\sqrt{2}}{2}\right)$.

For advanced study, refer to Introduction to Dimensions and Graphs of Sine and Cosine Functions.

Common Misinterpretations in Coordinate Transformation

A frequent error arises from confusing the angular definitions: $\theta$ always refers to the $xy$-plane angle, while $\phi$ refers to the angle with respect to the $z$-axis. Assignments can differ in physics literature, highlighting the need to confirm conventions used in the question context.

Common Error: Substituting $\phi$ and $\theta$ values interchangeably leads to incorrect point positioning in space and was often observed in previous JEE Main papers.

Coordinate system linkages with vector operations are also discussed in Vector Algebra Concepts.