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

  • 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.

  • Triple Integrals

    • Looking at the domain, it seems convenient to adopt the passage in spherical coordinates; in fact, the intervals of the variables that delimit the new $T$ region are obviously:

  • Polar Coordinates

    • This is called the Cartesian coordinate system.
    • Such definitions are called polar coordinates.
    • Polar coordinates in $r$ and $\theta$ can be converted to Cartesian coordinates $x$ and $y$.
    • A set of polar coordinates.
    • The $x$ Cartesian coordinate is given by $r \cos \theta$ and the $y$ Cartesian coordinate is given by $r \sin \theta$.

  • Double Integrals in Polar Coordinates

    • In $R^2$, if the domain has a cylindrical symmetry and the function has several particular characteristics, you can apply the transformation to polar coordinates, which means that the generic points $P(x, y)$ in Cartesian coordinates switch to their respective points in polar coordinates.
    • The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates $x$ and $y$ by using the trigonometric functions sine and cosine:
    • The Cartesian coordinates $x$ and $y$ can be converted to polar coordinates $r$ and $\varphi$ with $r \geq 0$ and $\varphi$ in the interval $(−\pi, \pi]$:
    • In general, the best practice is to use the coordinates that match the built-in symmetry of the function.
    • This figure illustrates graphically a transformation from cartesian to polar coordinates

  • Triple Integrals in Cylindrical Coordinates

    • When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.
    • When the function to be integrated has a cylindrical symmetry, it is sensible to change the variables into cylindrical coordinates and then perform integration.
    • Also in switching to cylindrical coordinates, the $dx\, dy\, dz$ differentials in the integral become $\rho \, d\rho \,d\varphi \,dz$.
    • Finally, it is possible to apply the final formula to cylindrical coordinates:
    • Cylindrical coordinates are often used for integrations on domains with a circular base.