Coordinate Systems | Sahithyan's Notes

A coordinate system assigns an ordered triple of numbers to each point in $\mathbb{R}^3$ . Cylindrical and spherical coordinates use distances and angles instead of 3 signed distances along the axes. They simplify regions and integrands that have an axis or a center of symmetry.

Rectangular Coordinates

Rectangular coordinates, also called Cartesian coordinates, locate a point $P$ by 3 signed distances from 3 mutually perpendicular coordinate planes.

P = (x, y, z)

Here:

$x$ : signed distance from the $yz$ -plane
$y$ : signed distance from the $zx$ -plane
$z$ : signed distance from the $xy$ -plane

Cylindrical Coordinates

Cylindrical coordinates extend polar coordinates in the $xy$ -plane with the height $z$ .

x = r\cos\theta, \quad y = r\sin\theta, \quad z = z, \quad r \ge 0

Here:

$r$ : distance from the $z$ -axis
$\theta$ : angle in the $xy$ -plane from the positive $x$ -axis
$z$ : signed height above the $xy$ -plane

Spherical Coordinates

Spherical coordinates are defined by

x = \rho\sin\phi\cos\theta, \quad y = \rho\sin\phi\sin\theta, \quad z = \rho\cos\phi

Here:

$\rho$ : distance from the origin to the point
$\theta$ : angle in the $xy$ -plane from the positive $x$ -axis
$\phi$ : angle from the positive $z$ -axis

Standard ranges are $\rho \ge 0$ , $0 \le \theta \le 2\pi$ , $0 \le \phi \le \pi$ .

Common Spherical Regions

Spherical shell between $\rho = \rho_1$ and $\rho = \rho_2$ , $\rho_1 < \rho_2$
$\rho_1 \le \rho \le \rho_2$ , $0 \le \phi \le \pi$ , $0 \le \theta \le 2\pi$
Sphere $\rho = \rho_0$ restricted to the first octant
$0 \le \rho \le \rho_0$ , $0 \le \phi \le \frac{\pi}{2}$ , $0 \le \theta \le \frac{\pi}{2}$
Sphere $\rho = \rho_0$ cut by the cone $\phi = \phi_0$
$0 \le \rho \le \rho_0$ , $0 \le \phi \le \phi_0$ , $0 \le \theta \le 2\pi$
Sphere $\rho = \rho_0$ cut by 2 cones $\phi = \phi_1$ and $\phi = \phi_2$ , $\phi_1 < \phi_2$
$0 \le \rho \le \rho_0$ , $\phi_1 \le \phi \le \phi_2$ , $0 \le \theta \le 2\pi$
Solid enclosed laterally by the cone $\phi = \phi_0$ , $0 < \phi_0 < \frac{\pi}{2}$ , and above by the plane $z = a$ , $a > 0$
$0 \le \rho \le a\sec\phi$ , $0 \le \phi \le \phi_0$ , $0 \le \theta \le 2\pi$

The region between the outer sphere and the inner sphere. A quarter is cut away to show the hollow.

The part of a ball of radius ρ₀ in the first octant. Its flat faces are the coordinate planes, and it is bounded above by the sphere.

A ball cut by the cone φ = φ₀. The sphere caps the cone, which has its tip at the origin.

A ball between 2 cones φ = φ₁ and φ = φ₂, capped by the sphere. The cones cut a hole through the top and bottom.

A solid enclosed laterally by the cone φ = φ ₀ and above by the plane z = a.