2.5 : Spherical Coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and gravitational fields. One of the other common applications of spherical coordinates is in the Earth's latitude and longitude system, which is used for navigational purposes.

Spherical coordinates belong to the family of curvilinear coordinates. These are the extension of polar coordinates and are used to describe a vector's position in three-dimensional space. A vector in a spherical coordinate system is defined using the radial, polar, and azimuthal scalar components. The radial component, which ranges from zero to infinity, specifies the vector's distance from its origin. The polar angle ranges from zero to π and measures the angle between the positive z-axis and the vector. The azimuthal angle, which ranges from zero to 2π, measures the angle between the x-axis and the orthogonal projection of the vector onto the xy-plane. A surface with a constant radius traces a sphere in a three-dimensional spherical coordinate system. On the other hand, a surface with a constant polar angle forms a half-cone, and a surface with a constant azimuthal angle forms a half-plane.

The transformation equations are used to convert a vector in spherical coordinates to Cartesian coordinates and cylindrical coordinates.