Definition:Spherical Coordinate System

Definition

A distinct point $O$ is identified, and referred to as the origin.

Having identified the polar axis of a spherical coordinate system, one then selects a distinct horizontal straight line, also passing through the origin $O$, perpendicular to the polar axis

Hence, let $P$ be an arbitrary point $P$ in space.

Let $\mathbf r$ be the radius vector of $P$ with respect to $O$.

The position of $P$ is specified in spherical coordinates by:

$(1): \quad$ the length of $\mathbf r$, that is, the distance of $P$ from the origin $O$, denoted by $r$.

$(2): \quad$ the angle between $\mathbf r$ and the polar axis, known as the colatitude of $P$, denoted by $\theta$

$(3): \quad$ the angle between the horizontal axis and the projection of $\mathbf r$ onto the horizontal plane, known as the longitude of $P$, denoted by $\phi$.

A system of spherical coordinates is also known as a system of spherical polar coordinates.