Spherical coordinate system

File:3D Spherical.svg

The physics convention. Spherical coordinates (Template:Mvar, Template:Mvar, Template:Mvar) as commonly used: (ISO 80000-2:2019): radial distance Template:Mvar (slant distance to origin), polar angle Template:Mvar (theta) (angle with respect to positive polar axis), and azimuthal angle Template:Mvar (phi) (angle of rotation from the initial meridian plane). This is the convention followed in this article.

In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are

• the radial distance Template:Mvar along the line connecting the point to a fixed point called the origin;
• the polar angle Template:Mvar between this radial line and a given polar axis;Template:Efn and
• the azimuthal angle Template:Mvar, which is the angle of rotation of the radial line around the polar axis.Template:Efn

(See graphic regarding the "physics convention".)

Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).

Terminology

{{#invoke:Hatnote|hatnote}}

The radial distance from the fixed point of origin is also called the radius, or radial line, or radial coordinate. The polar angle may be called inclination angle, zenith angle, normal angle, or the colatitude. The user may choose to replace the inclination angle by its complement, the elevation angle (or altitude angle), measured upward between the reference plane and the radial lineTemplate:Mdashi.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The depression angle is the negative of the elevation angle. (See graphic re the "physics convention"Template:Mdashnot "mathematics convention".)

Both the use of symbols and the naming order of tuple coordinates differ among the several sources and disciplines. This article will use the ISO convention<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> frequently encountered in physics, where the naming tuple gives the order as: radial distance, polar angle, azimuthal angle, or <math>(r,\theta,\varphi)</math>. (See graphic re the "physics convention".) In contrast, the conventions in many mathematics books and texts give the naming order differently as: radial distance, "azimuthal angle", "polar angle", and <math>(\rho,\theta,\varphi)</math> or <math>(r,\theta,\varphi)</math>Template:Mdashwhich switches the uses and meanings of symbols θ and φ. Other conventions may also be used, such as r for a radius from the z-axis that is not from the point of origin. Particular care must be taken to check the meaning of the symbols.

File:3D Spherical 2.svg

The mathematics convention. Spherical coordinates Template:Math as typically used: radial distance Template:Mvar, azimuthal angle Template:Mvar, and polar angle Template:Mvar. + The meanings of Template:Mvar and Template:Mvar have been swappedTemplate:Mdashcompared to the physics convention. The 'south'-direction x-axis is depicted but the 'north'-direction x-axis is not. (As in physics, Template:Mvar (rho) is often used instead of Template:Mvar to avoid confusion with the value Template:Mvar in cylindrical and 2D polar coordinates.)

According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). There are a number of celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees; (note 90 degrees equals Template:Fraction radians). And these systems of the mathematics convention may measure the azimuthal angle counterclockwise (i.e., from the south direction Template:Mvar-axis, or 180°, towards the east direction Template:Mvar-axis, or +90°)Template:Mdashrather than measure clockwise (i.e., from the north direction x-axis, or 0°, towards the east direction y-axis, or +90°), as done in the horizontal coordinate system.<ref>Duffett-Smith, P and Zwart, J, p. 34.</ref> (See graphic re "mathematics convention".)

The spherical coordinate system of the physics convention can be seen as a generalization of the polar coordinate system in three-dimensional space. It can be further extended to higher-dimensional spaces, and is then referred to as a hyperspherical coordinate system.

Definition

To define a spherical coordinate system, one must designate an origin point in space, Template:Mvar, and two orthogonal directions: the zenith reference direction and the azimuth reference direction. These choices determine a reference plane that is typically defined as containing the point of origin and the xTemplate:Ndash and yTemplate:Ndashaxes, either of which may be designated as the azimuth reference direction. The reference plane is perpendicular (orthogonal) to the zenith direction, and typically is designated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a point Template:Mvar then are defined as follows:

• The radius or radial distance is the Euclidean distance from the origin Template:Mvar to Template:Mvar.
• The inclination (or polar angle) is the signed angle from the zenith reference direction to the line segment Template:Mvar. (Elevation may be used as the polar angle instead of inclination; see below.)
• The azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the radial line segment Template:Mvar on the reference plane.

The sign of the azimuth is determined by designating the rotation that is the positive sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system definition. (If the inclination is either zero or 180 degrees (= Template:Pi radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.)

The elevation is the signed angle from the x-y reference plane to the radial line segment Template:Mvar, where positive angles are designated as upward, towards the zenith reference. Elevation is 90 degrees (= Template:Sfrac radians) minus inclination. Thus, if the inclination is 60 degrees (= Template:Sfrac radians), then the elevation is 30 degrees (= Template:Sfrac radians).

In linear algebra, the vector from the origin Template:Mvar to the point Template:Mvar is often called the position vector of P.

Conventions

Several different conventions exist for representing spherical coordinates and prescribing the naming order of their symbols. The 3-tuple number set <math>(r,\theta,\varphi)</math> denotes radial distance, the polar angleTemplate:Mdash"inclination", or as the alternative, "elevation"Template:Mdashand the azimuthal angle. It is the common practice within the physics convention, as specified by ISO standard 80000-2:2019, and earlier in ISO 31-11 (1992).

As stated above, this article describes the ISO "physics convention"Template:Mdashunless otherwise noted.

However, some authors (including mathematicians) use the symbol ρ (rho) for radius, or radial distance, φ for inclination (or elevation) and θ for azimuthTemplate:Mdashwhile others keep the use of r for the radius; all which "provides a logical extension of the usual polar coordinates notation".<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> As to order, some authors list the azimuth before the inclination (or the elevation) angle. Some combinations of these choices result in a left-handed coordinate system. The standard "physics convention" 3-tuple set <math>(r,\theta,\varphi)</math> conflicts with the usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates, where Template:Mvar is often used for the azimuth.<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html" />

Angles are typically measured in degrees (°) or in radians (rad), where 360° = 2Template:Pi rad. The use of degrees is most common in geography, astronomy, and engineering, where radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context, as occurs in applications of the 'unit sphere', see applications.

When the system is used to designate physical three-space, it is customary to assign positive to azimuth angles measured in the counterclockwise sense from the reference direction on the reference planeTemplate:Mdashas seen from the "zenith" side of the plane. This convention is used in particular for geographical coordinates, where the "zenith" direction is north and the positive azimuth (longitude) angles are measured eastwards from some prime meridian.

Major conventions

coordinates set order	corresponding local geographical directions Template:Math	right/left-handed
Template:Math	Template:Math	right
Template:Math	Template:Math	right
Template:Math	Template:Math	left

Note: [[Easting and northing|Easting (Template:Mvar), Northing (Template:Mvar)]], Upwardness (Template:Mvar). In the case of Template:Math the local azimuth angle would be measured counterclockwise from Template:Mvar to Template:Mvar.

Unique coordinates

Any spherical coordinate triplet (or tuple) <math>(r,\theta,\varphi)</math> specifies a single point of three-dimensional space. On the reverse view, any single point has infinitely many equivalent spherical coordinates. That is, the user can add or subtract any number of full turns to the angular measures without changing the angles themselves, and therefore without changing the point. It is convenient in many contexts to use negative radial distances, the convention being <math>(-r,\theta,\varphi)</math>, which is equivalent to <math>(r,\theta{+}180^\circ,\varphi)</math> or <math>(r,90^\circ{-}\theta,\varphi{+}180^\circ)</math> for any Template:Mvar, Template:Mvar, and Template:Mvar. Moreover, <math>(r,-\theta,\varphi)</math> is equivalent to <math>(r,\theta,\varphi{+}180^\circ)</math>.

When necessary to define a unique set of spherical coordinates for each point, the user must restrict the range, aka interval, of each coordinate. A common choice is:

• radial distance: Template:Math
• polar angle: Template:Math, or Template:Math,
• azimuth : Template:Math, or Template:Math.

But instead of the interval Template:Closed-open, the azimuth Template:Mvar is typically restricted to the half-open interval Template:Open-closed, or Template:Open-closed radians, which is the standard convention for geographic longitude.

For the polar angle Template:Mvar, the range (interval) for inclination is Template:Closed-closed, which is equivalent to elevation range (interval) Template:Closed-closed. In geography, the latitude is the elevation.

Even with these restrictions, if the polar angle (inclination) is 0° or 180°Template:Mdashelevation is −90° or +90°Template:Mdashthen the azimuth angle is arbitrary; and if Template:Mvar is zero, both azimuth and polar angles are arbitrary. To define the coordinates as unique, the user can assert the convention that (in these cases) the arbitrary coordinates are set to zero.

Plotting

To plot any dot from its spherical coordinates Template:Math, where Template:Mvar is inclination, the user would: move Template:Mvar units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle (Template:Mvar) about the origin from the designated azimuth reference direction, (i.e., either the x- or y-axis, see Definition, above); and then rotate from the z-axis by the amount of the Template:Mvar angle.

Applications

[[File:Spherical coordinate system.svg|thumb|upright=1.2|right|In the mathematics convention: A globe showing a unit sphere, with tuple coordinates of point Template:Mvar (red): its radial distance Template:Mvar (red, not labeled); its azimuthal angle Template:Mvar (not labeled); and its polar angle of inclination Template:Mvar (not labeled). The radial distance upward along the [[zenith|zenithTemplate:Ndashaxis]] from the point of origin to the surface of the sphere is assigned the value unity, or 1. + In this image, Template:Mvar appears to equal 4/6, or .67, (of unity); i.e., four of the six 'nested shells' to the surface. The azimuth angle Template:Mvar appears to equal positive 90°, as rotated counterclockwise from the azimuth-reference xTemplate:Ndashaxis; and the inclination Template:Mvar appears to equal 30°, as rotated from the zenithTemplate:Ndashaxis. (Note the 'full' rotation, or inclination, from the zenithTemplate:Ndashaxis to the yTemplate:Ndashaxis is 90°).]]

Just as the two-dimensional Cartesian coordinate system is usefulTemplate:Mdashhas a wide set of applicationsTemplate:Mdashon a planar surface, a two-dimensional spherical coordinate system is useful on the surface of a sphere. For example, one sphere that is described in Cartesian coordinates with the equation Template:Math can be described in spherical coordinates by the simple equation Template:Math. (In this systemTemplate:Mdashshown here in the mathematics conventionTemplate:Mdashthe sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored, see graphic.)

This (unit sphere) simplification is also useful when dealing with objects such as rotational matrices. Spherical coordinates are also useful in analyzing systems that have some degree of symmetry about a point, including: volume integrals inside a sphere; the potential energy field surrounding a concentrated mass or charge; or global weather simulation in a planet's atmosphere.

File:Bosch 36W column loudspeaker polar pattern.png

The output pattern of the industrial loudspeaker shown here uses spherical polar plots taken at six frequencies

Three dimensional modeling of loudspeaker output patterns can be used to predict their performance. A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. Polar plots help to show that many loudspeakers tend toward omnidirectionality at lower frequencies.

An important application of spherical coordinates provides for the separation of variables in two partial differential equationsTemplate:Mdashthe Laplace and the Helmholtz equationsTemplate:Mdashthat arise in many physical problems. The angular portions of the solutions to such equations take the form of spherical harmonics. Another application is ergonomic design, where Template:Mvar is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player's position<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In geography

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also

Instead of inclination, the geographic coordinate system uses elevation angle (or latitude), in the range (aka domain) Template:Math and rotated north from the equator plane. Latitude (i.e., the angle of latitude) may be either geocentric latitude, measured (rotated) from the Earth's centerTemplate:Mdashand designated variously by Template:MathTemplate:Mdashor geodetic latitude, measured (rotated) from the observer's local vertical, and typically designated Template:Mvar. The polar angle (inclination), which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography.

The azimuth angle (or longitude) of a given position on Earth, commonly denoted by Template:Mvar, is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian); thus its domain (or range) is Template:Math and a given reading is typically designated "East" or "West". For positions on the Earth or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation.

Instead of the radial distance Template:Mvar geographers commonly use altitude above or below some local reference surface (vertical datum), which, for example, may be the mean sea level. When needed, the radial distance can be computed from the altitude by adding the radius of Earth, which is approximately Template:Convert.

However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be inaccurate by several kilometers. The precise standard meanings of latitude, longitude and altitude are currently defined by the World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about Template:Convert) and many other details.

Planetary coordinate systems use formulations analogous to the geographic coordinate system.

In astronomy

A series of astronomical coordinate systems are used to measure the elevation angle from several fundamental planes. These reference planes include: the observer's horizon, the galactic equator (defined by the rotation of the Milky Way), the celestial equator (defined by Earth's rotation), the plane of the ecliptic (defined by Earth's orbit around the Sun), and the plane of the earth terminator (normal to the instantaneous direction to the Sun).

Coordinate system conversions

Template:Also As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.

Cartesian coordinates

The spherical coordinates of a point in the ISO convention (i.e. for physics: radius Template:Mvar, inclination Template:Mvar, azimuth Template:Mvar) can be obtained from its Cartesian coordinates Template:Math by the formulae

<math display="block">\begin{align} r &= \sqrt{x^2 + y^2 + z^2} \\ \theta &= \arccos\frac{z}{\sqrt{x^2 + y^2 + z^2}} = \arccos\frac{z}{r}= \begin{cases}

\arctan\frac{\sqrt{x^2+y^2}}{z} &\text{if } z > 0 \\
\pi +\arctan\frac{\sqrt{x^2+y^2}}{z} &\text{if } z < 0 \\
+\frac{\pi}{2} &\text{if } z = 0 \text{ and } \sqrt{x^2+y^2} \neq 0 \\
\text{undefined} &\text{if } x=y=z = 0 \\

\end{cases} \\ \varphi &= \sgn(y)\arccos\frac{x}{\sqrt{x^2+y^2}} = \begin{cases}

\arctan(\frac{y}{x}) &\text{if } x > 0, \\
\arctan(\frac{y}{x}) + \pi &\text{if } x < 0 \text{ and } y \geq 0, \\
\arctan(\frac{y}{x}) - \pi &\text{if } x < 0 \text{ and } y < 0, \\
+\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\
-\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0, \\
\text{undefined} &\text{if } x = 0 \text{ and } y = 0.

\end{cases} \end{align}</math>

The inverse tangent denoted in Template:Math must be suitably defined, taking into account the correct quadrant of Template:Math, as done in the equations above. See the article on atan2.

Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian Template:Mvar plane from Template:Math to Template:Math, where Template:Mvar is the projection of Template:Mvar onto the Template:Mvar-plane, and the second in the Cartesian Template:Mvar-plane from Template:Math to Template:Math. The correct quadrants for Template:Mvar and Template:Mvar are implied by the correctness of the planar rectangular to polar conversions.

These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian Template:Mvar plane, that Template:Mvar is inclination from the Template:Mvar direction, and that the azimuth angles are measured from the Cartesian Template:Mvar axis (so that the Template:Mvar axis has Template:Math). If θ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the Template:Math and Template:Math below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius Template:Mvar, inclination Template:Mvar, azimuth Template:Mvar), where Template:Math, Template:Math, Template:Math, by <math display="block">\begin{align}

x &= r \sin\theta \, \cos\varphi, \\
y &= r \sin\theta \, \sin\varphi, \\
z &= r \cos\theta.

\end{align}</math>

Cylindrical coordinates

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Cylindrical coordinates (axial radius ρ, azimuth φ, elevation z) may be converted into spherical coordinates (central radius r, inclination θ, azimuth φ), by the formulas

<math display="block">\begin{align}

r &= \sqrt{\rho^2 + z^2}, \\
\theta &= \arctan\frac{\rho}{z} = \arccos\frac{z}{\sqrt{\rho^2 + z^2}}, \\
\varphi &= \varphi.

\end{align}</math>

Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae

<math display="block">\begin{align}

\rho &= r \sin \theta, \\
\varphi &= \varphi, \\
z &= r \cos \theta.

\end{align}</math>

These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle Template:Mvar in the same senses from the same axis, and that the spherical angle Template:Mvar is inclination from the cylindrical Template:Mvar axis.

Generalization

Template:See also It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates.

Let P be an ellipsoid specified by the level set

<math display="block">ax^2 + by^2 + cz^2 = d.</math>

The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: radius Template:Mvar, inclination Template:Mvar, azimuth Template:Mvar) can be obtained from its Cartesian coordinates Template:Math by the formulae

<math display="block">\begin{align}

x &= \frac{1}{\sqrt{a}} r \sin\theta \, \cos\varphi, \\
y &= \frac{1}{\sqrt{b}} r \sin\theta \, \sin\varphi, \\
z &= \frac{1}{\sqrt{c}} r \cos\theta, \\
r^{2} &= ax^2 + by^2 + cz^2.

\end{align}</math>

An infinitesimal volume element is given by

<math display="block"> \mathrm{d}V = \left|\frac{\partial(x, y, z)}{\partial(r, \theta, \varphi)}\right| \, dr\,d\theta\,d\varphi =

\frac{1}{\sqrt{abc}} r^2 \sin \theta \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi =
\frac{1}{\sqrt{abc}} r^2 \,\mathrm{d}r \,\mathrm{d}\Omega.

</math>

The square-root factor comes from the property of the determinant that allows a constant to be pulled out from a column:

<math display="block"> \begin{vmatrix}

ka & b & c \\
kd & e & f \\
kg & h & i

\end{vmatrix} = k \begin{vmatrix}

a & b & c \\
d & e & f \\
g & h & i

\end{vmatrix}. </math>

Integration and differentiation in spherical coordinates

File:Kugelkoord-lokb-e.svg

Unit vectors in spherical coordinates

The following equations (Iyanaga 1977) assume that the colatitude Template:Mvar is the inclination from the positive Template:Mvar axis, as in the physics convention discussed.

The line element for an infinitesimal displacement from Template:Math to Template:Math is <math display="block"> \mathrm{d}\mathbf{r} = \mathrm{d}r\,\hat{\mathbf r} + r\,\mathrm{d}\theta \,\hat{\boldsymbol\theta } + r \sin{\theta} \, \mathrm{d}\varphi\,\mathbf{\hat{\boldsymbol\varphi}},</math> where <math display="block">\begin{align}

\hat{\mathbf r} &= \sin \theta \cos \varphi \,\hat{\mathbf x} +
 \sin \theta \sin \varphi \,\hat{\mathbf y} + \cos \theta \,\hat{\mathbf z}, \\
\hat{\boldsymbol\theta} &= \cos \theta \cos \varphi \,\hat{\mathbf x} +
 \cos \theta \sin \varphi \,\hat{\mathbf y} - \sin \theta \,\hat{\mathbf z}, \\
\hat{\boldsymbol\varphi} &= - \sin \varphi \,\hat{\mathbf x} +
 \cos \varphi \,\hat{\mathbf y}

\end{align}</math> are the local orthogonal unit vectors in the directions of increasing Template:Mvar, Template:Mvar, and Template:Mvar, respectively, and Template:Math, Template:Math, and Template:Math are the unit vectors in Cartesian coordinates. The linear transformation to this right-handed coordinate triplet is a rotation matrix, <math display="block">R = \begin{pmatrix}

    \sin\theta\cos\varphi&\sin\theta\sin\varphi&\hphantom{-}\cos\theta\\
    \cos\theta\cos\varphi&\cos\theta\sin\varphi&-\sin\theta\\
   -\sin\varphi&\cos\varphi   &\hphantom{-}0
  \end{pmatrix}.

</math>

This gives the transformation from the Cartesian to the spherical, the other way around is given by its inverse. Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

The Cartesian unit vectors are thus related to the spherical unit vectors by: <math display="block">\begin{bmatrix}\mathbf{\hat x} \\ \mathbf{\hat y} \\ \mathbf{\hat z} \end{bmatrix}

 = \begin{bmatrix} \sin\theta\cos\varphi & \cos\theta\cos\varphi & -\sin\varphi \\
                   \sin\theta\sin\varphi & \cos\theta\sin\varphi & \hphantom{-}\cos\varphi \\
                   \cos\theta         & -\sin\theta        & \hphantom{-}0 \end{bmatrix}
   \begin{bmatrix} \boldsymbol{\hat{r}} \\ \boldsymbol{\hat\theta} \\ \boldsymbol{\hat\varphi} \end{bmatrix}</math>

The general form of the formula to prove the differential line element, is<ref name="q74503">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\mathrm{d}\mathbf{r} =

\sum_i \frac{\partial \mathbf{r}}{\partial x_i} \,\mathrm{d}x_i =
\sum_i \left|\frac{\partial \mathbf{r}}{\partial x_i}\right|
 \frac{\frac{\partial \mathbf{r}}{\partial x_i}}{\left|\frac{\partial \mathbf{r}}{\partial x_i}\right|} \, \mathrm{d}x_i =
\sum_i \left|\frac{\partial \mathbf{r}}{\partial x_i}\right| \,\mathrm{d}x_i \, \hat{\boldsymbol{x}}_i,

</math> that is, the change in <math>\mathbf r</math> is decomposed into individual changes corresponding to changes in the individual coordinates.

To apply this to the present case, one needs to calculate how <math>\mathbf r</math> changes with each of the coordinates. In the conventions used, <math display="block">\mathbf{r} = \begin{bmatrix}

r \sin\theta \, \cos\varphi \\
r \sin\theta \, \sin\varphi \\
r \cos\theta

\end{bmatrix}, x_1=r, x_2=\theta, x_3=\varphi.</math>

Thus, <math display="block"> \frac{\partial\mathbf r}{\partial r} = \begin{bmatrix}

\sin\theta \, \cos\varphi \\
\sin\theta \, \sin\varphi \\
\cos\theta

\end{bmatrix}=\mathbf{\hat r}, \quad \frac{\partial\mathbf r}{\partial \theta} = \begin{bmatrix}

r \cos\theta \, \cos\varphi \\
r \cos\theta \, \sin\varphi \\
-r \sin\theta

\end{bmatrix}=r\,\hat{\boldsymbol\theta }, \quad \frac{\partial\mathbf r}{\partial \varphi} = \begin{bmatrix}

-r \sin\theta \, \sin\varphi \\
\hphantom{-}r \sin\theta \, \cos\varphi \\
0

\end{bmatrix} = r \sin\theta\,\mathbf{\hat{\boldsymbol\varphi}} . </math>

The desired coefficients are the magnitudes of these vectors:<ref name="q74503" /> <math display="block"> \left|\frac{\partial\mathbf r}{\partial r}\right| = 1, \quad \left|\frac{\partial\mathbf r}{\partial \theta}\right| = r, \quad \left|\frac{\partial\mathbf r}{\partial \varphi}\right| = r \sin\theta. </math>

The surface element spanning from Template:Mvar to Template:Math and Template:Mvar to Template:Math on a spherical surface at (constant) radius Template:Mvar is then <math display="block"> \mathrm{d}S_r =

\left\|\frac{\partial {\mathbf r}}{\partial \theta} \times \frac{\partial {\mathbf r}}{\partial \varphi}\right\| \mathrm{d}\theta \,\mathrm{d}\varphi =

\left|r {\hat \boldsymbol\theta} \times r \sin \theta {\boldsymbol\hat \varphi} \right|\mathrm{d}\theta \,\mathrm{d}\varphi=

r^2 \sin\theta \,\mathrm{d}\theta \,\mathrm{d}\varphi  ~.

</math>

Thus the differential solid angle is <math display="block">\mathrm{d}\Omega = \frac{\mathrm{d}S_r}{r^2} = \sin\theta \,\mathrm{d}\theta \,\mathrm{d}\varphi.</math>

The surface element in a surface of polar angle Template:Mvar constant (a cone with vertex at the origin) is <math display="block">\mathrm{d}S_\theta = r \sin\theta \,\mathrm{d}\varphi \,\mathrm{d}r.</math>

The surface element in a surface of azimuth Template:Mvar constant (a vertical half-plane) is <math display="block">\mathrm{d}S_\varphi = r \,\mathrm{d}r \,\mathrm{d}\theta.</math>

The volume element spanning from Template:Mvar to Template:Math, Template:Mvar to Template:Math, and Template:Mvar to Template:Math is specified by the determinant of the Jacobian matrix of partial derivatives, <math display="block"> J =\frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}

 =\begin{pmatrix}
    \sin\theta\cos\varphi & r\cos\theta\cos\varphi & -r\sin\theta\sin\varphi\\
    \sin\theta\sin\varphi & r\cos\theta\sin\varphi & \hphantom{-}r\sin\theta\cos\varphi\\
    \cos\theta & -r\sin\theta & \hphantom{-}0
  \end{pmatrix},

</math> namely <math display="block"> \mathrm{d}V = \left|\frac{\partial(x, y, z)}{\partial(r, \theta, \varphi)}\right| \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi=

r^2 \sin\theta \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi =
r^2 \,\mathrm{d}r \,\mathrm{d}\Omega ~.

</math>

Thus, for example, a function Template:Math can be integrated over every point in Template:Math by the triple integral <math display="block">\int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^\infty f(r, \theta, \varphi) r^2 \sin\theta \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi ~.</math>

The del operator in this system leads to the following expressions for the gradient and Laplacian for scalar fields, <math display="block">\begin{align} \nabla f &= {\partial f \over \partial r}\hat{\mathbf r}

 + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol\theta}
 + {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol\varphi},  \\[8pt]

\nabla^2 f &= {1 \over r^2}{\partial \over \partial r} \left(r^2 {\partial f \over \partial r}\right) + {1 \over r^2 \sin\theta}{\partial \over \partial \theta} \left(\sin\theta {\partial f \over \partial \theta}\right) + {1 \over r^2 \sin^2\theta}{\partial^2 f \over \partial \varphi^2} \\[8pt] & = \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r}\right) f + {1 \over r^2 \sin\theta}{\partial \over \partial \theta} \left(\sin\theta \frac{\partial}{\partial \theta}\right) f + \frac{1}{r^2 \sin^2\theta}\frac{\partial^2}{\partial \varphi^2}f ~, \\[8pt] \end{align}</math>And it leads to the following expressions for the divergence and curl of vector fields,

<math display="block">\nabla \cdot \mathbf{A}

 = \frac{1}{r^2}{\partial \over \partial r}\left( r^2 A_r \right)
 + \frac{1}{r \sin\theta}{\partial \over \partial\theta} \left( \sin\theta A_\theta \right)
 + \frac{1}{r \sin \theta} {\partial A_\varphi \over \partial \varphi},</math><math display="block">\begin{align}

\nabla \times \mathbf{A} = {} & \frac{1}{r\sin\theta} \left[{\partial \over \partial \theta} \left( A_\varphi\sin\theta \right)

   - {\partial A_\theta \over \partial \varphi}\right] \hat{\mathbf r} \\[4pt]

& {} + \frac 1 r \left[{1 \over \sin\theta}{\partial A_r \over \partial \varphi}

   - {\partial \over \partial r} \left( r A_\varphi \right) \right] \hat{\boldsymbol\theta} \\[4pt]

& {} + \frac 1 r \left[{\partial \over \partial r} \left( r A_\theta \right)

   - {\partial A_r \over \partial \theta}\right] \hat{\boldsymbol\varphi},

\end{align}</math>

Further, the inverse Jacobian in Cartesian coordinates is <math display="block">J^{-1} = \begin{pmatrix}

 \dfrac{x}{r}&\dfrac{y}{r}&\dfrac{z}{r}\\\\
 \dfrac{xz}{r^2\sqrt{x^2+y^2}}&\dfrac{yz}{r^2\sqrt{x^2+y^2}}&\dfrac{-\left(x^2 + y^2\right)}{r^2\sqrt{x^2+y^2}}\\\\
 \dfrac{-y}{x^2+y^2}&\dfrac{x}{x^2+y^2}&0

\end{pmatrix}.</math> The metric tensor in the spherical coordinate system is <math>g = J^T J </math>.

Distance in spherical coordinates

In spherical coordinates, given two points with Template:Mvar being the azimuthal coordinate <math display="block">\begin{align}

{\mathbf r} &= (r,\theta,\varphi), \\
{\mathbf r'} &= (r',\theta',\varphi')

\end{align}</math> The distance between the two points can be expressed as<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\begin{align}

{\mathbf D} &= \sqrt{r^2+r'^2-2rr'(\sin{\theta}\sin{\theta'}\cos{(\varphi-\varphi')} + \cos{\theta}\cos{\theta'})}

\end{align}</math>

Kinematics

In spherical coordinates, the position of a point or particle (although better written as a triple<math>(r,\theta, \varphi)</math>) can be written as<ref name="Cameron2019">Template:Cite book</ref> <math display="block">\mathbf{r} = r \mathbf{\hat r} .</math> Its velocity is then<ref name="Cameron2019" /> <math display="block">\mathbf{v} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \dot{r} \mathbf{\hat r} + r\,\dot\theta\,\hat{\boldsymbol\theta } + r\,\dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\varphi}}</math> and its acceleration is<ref name="Cameron2019" /> <math display="block"> \begin{align} \mathbf{a} = {} & \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} \\[1ex] = {} & \hphantom{+}\; \left( \ddot{r} - r\,\dot\theta^2 - r\,\dot\varphi^2\sin^2\theta \right)\mathbf{\hat r} \\ & {} + \left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right) \hat{\boldsymbol\theta } \\ & {} + \left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \hat{\boldsymbol\varphi} \end{align} </math>

The angular momentum is <math display="block"> \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times m\mathbf{v} = m r^2 \left(- \dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\theta}} + \dot\theta\,\hat{\boldsymbol\varphi }\right) </math> Where <math>m</math> is mass. In the case of a constant Template:Mvar or else Template:Math, this reduces to vector calculus in polar coordinates.

The corresponding angular momentum operator then follows from the phase-space reformulation of the above, <math display="block"> \mathbf{L}= -i\hbar ~\mathbf{r} \times \nabla =i \hbar \left(\frac{\hat{\boldsymbol{\theta}}}{\sin(\theta)} \frac{\partial}{\partial\phi} - \hat{\boldsymbol{\phi}} \frac{\partial}{\partial\theta}\right). </math>

The torque is given as<ref name="Cameron2019" /> <math display="block"> \mathbf{\tau} = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \mathbf{r} \times \mathbf{F} = -m \left(2r\dot{r}\dot{\varphi}\sin\theta + r^2\ddot{\varphi}\sin{\theta} + 2r^2\dot{\theta}\dot{\varphi}\cos{\theta} \right)\hat{\boldsymbol\theta} + m \left(r^2\ddot{\theta} + 2r\dot{r}\dot{\theta} - r^2\dot{\varphi}^2\sin\theta\cos\theta \right) \hat{\boldsymbol\varphi} </math>

The kinetic energy is given as<ref name="Cameron2019" /> <math display="block"> E_k = \frac{1}{2}m \left[ \left(\dot{r}\right)^2 + \left(r\dot{\theta}\right)^2 + \left(r\dot{\varphi}\sin\theta\right)^2 \right] </math>

See also

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Bibliography

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External links

• Template:Springer
• MathWorld description of spherical coordinates
• Coordinate Converter – converts between polar, Cartesian and spherical coordinates