Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system that uses a combination of radial distance, polar angle, and azimuthal angle to describe points in space. This coordinate system is particularly useful for describing points and regions with spherical symmetry, such as spheres, spherical shells, and spherical caps.

In spherical coordinates, a point in space is represented by the tuple (r, θ, φ), where r is the radial distance from the origin, θ is the polar angle (measured from the positive z-axis), and φ is the azimuthal angle (measured from the positive x-axis in the xy-plane). Spherical coordinates can be converted to Cartesian coordinates using the following equations: x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), and z = r cos(θ).

When performing integration in spherical coordinates, it is essential to consider the Jacobian determinant, which in this case is r² sin(θ). This scaling factor accounts for the changing volume element as the radial distance r and polar angle θ change. Spherical coordinates are often used to solve problems in physics and engineering, especially when dealing with problems that exhibit spherical symmetry, such as the behavior of electric or magnetic fields around point charges or the distribution of mass in celestial bodies.