spherical coordinate

(noun)

a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith

Related Terms

Examples of spherical coordinate in the following topics:

  • Triple Integrals in Spherical Coordinates

    • When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
    • When the function to be integrated has a spherical symmetry, it is sensible to change the variables into spherical coordinates and then perform integration.
    • It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:
    • Points on $z$-axis do not have a precise characterization in spherical coordinates, so $\theta$ can vary from $0$ to $2 \pi$.
    • Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.

  • Cylindrical and Spherical Coordinates

    • Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
    • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
    • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.
    • The spherical coordinates (radius $r$, inclination $\theta$, azimuth $\varphi$) of a point can be obtained from its Cartesian coordinates ($x$, $y$, $z$) by the formulae:
    • Spherical coordinates ($r$, $\theta$, $\varphi$) as often used in mathematics: radial distance $r$, azimuthal angle $\theta$, and polar angle $\varphi$.

  • Three-Dimensional Coordinate Systems

    • Often, you will need to be able to convert from spherical to Cartesian, or the other way around.
    • This is a three dimensional space represented by a Cartesian coordinate system.
    • The spherical system is used commonly in mathematics and physics and has variables of $r$, $\theta$, and $\varphi$.
    • The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.
    • Identify the number of parameters necessary to express a point in the three-dimensional coordinate system

  • Introduction to Spherical and Cylindrical Harmonics

    • In this section we will apply separation of variables to Laplace's equation in spherical and cylindrical coordinates.
    • Spherical coordinates are important when treating problems with spherical or nearly-spherical symmetry.
    • On the other hand, if we tried to use Cartesian coordinates to solve a boundary value problem on a spherical domain, we couldn't represent this as a fixed value of any of the coordinates.
    • Obviously this would be much simpler if we used spherical coordinates, since then we could specify boundary conditions on, for example, the surface $x = r \cos \phi \sin \theta$ constant.
    • To derive an expression for the Laplacian in spherical coordinates we have to change variables according to: $x = r \cos \phi \sin \theta$, $y = r \sin \phi \sin \theta$, $z = r \cos \theta$.

  • Hydrostatics

  • Surfaces in Space

    • To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined.
    • For example, the surface of the Earth is (ideally) a two-dimensional surface, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian).
    • In spherical coordinates, the surface can be expressed simply by $r=R$.

  • Change of Variables

    • To do so, the function must be adapted to the new coordinates.
    • Changing to cylindrical coordinates may be useful depending on the setup of problem.

  • Parametric Surfaces and Surface Integrals

    • Using the spherical coordinates, the unit sphere can be parameterized by $\vec r(\theta,\phi) = (\cos\theta \sin\phi, \sin\theta \sin \phi, \cos\phi), 0 \leq \theta < 2\pi, 0 \leq \phi \leq \pi$.
    • For example, the coordinate $z$-plane can be parametrized as $\vec r(u,v)=(au+bv,cu+dv, 0)$ for any constants $a$, $b$, $c$, $d$ such that $ad - bc \neq 0$, i.e. the matrix $\begin{bmatrix}a & b\\ c & d\end{bmatrix}$ is invertible.

  • Mapping the Earth

    • Geographic locations can be described in terms of coordinates of latitude and longitude.
    • In cartography, any place or object can also be referenced by its absolute location: its coordinates in latitude and longitude.
    • Across the spherical Earth, longitude lines, also called meridians, stretch vertically from the North Pole to the South Pole.
    • In most coordinate systems, the prime meridian passes through Greenwich, England.
    • Map projections are what enable the reshaping of the Earth through mathematically transformations of spherical coordinates (x, y, and z) into 2-dimensional (x and y) space.

  • Spherical and Plane Waves

    • Spherical waves come from point source in a spherical pattern; plane waves are infinite parallel planes normal to the phase velocity vector.
    • In 1678, he proposed that every point that a luminous disturbance touches becomes itself a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time.
    • Since the waves all come from one point source, the waves happen in a spherical pattern.
    • All the waves come from a single point source and are spherical .
    • When waves are produced from a point source, they are spherical waves.