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{{Use American English|date = March 2019}}
{{Short description|3-dimensional coordinate system}}
{{more footnotes|date=June 2012}}
[[File:3D Spherical.svg|thumb|240px|right|Spherical coordinates {{math|(''r'', ''θ'', ''φ'')}} as commonly used in '''''physics''''' ([[International Organisation for Standardisation|ISO]] convention): radial distance {{mvar|r}}, polar angle {{mvar|θ}} ([[theta]]), and azimuthal angle {{mvar|φ}} ([[phi]]). The symbol {{mvar|ρ}} ([[rho]]) is often used instead of {{mvar|r}}.]]

[[File:3D Spherical 2.svg|thumb|240px|right|Spherical coordinates {{math|(''r'', ''θ'', ''φ'')}} as often used in '''''mathematics''''': radial distance {{mvar|r}}, azimuthal angle {{mvar|θ}}, and polar angle {{mvar|φ}}. The meanings of {{mvar|θ}} and {{mvar|φ}} have been swapped compared to the physics convention.]]

[[File:Spherical coordinate system.svg|thumb|240px|right|A globe showing the radial distance, polar angle and azimuthal angle of a point {{mvar|P}} with respect to a [[unit sphere]]. In this image, {{mvar|r}} equals 4/6, {{mvar|θ}} equals 90&deg;, and {{mvar|φ}} equals 30&deg;.]]

In [[mathematics]], a '''spherical coordinate system''' is a [[coordinate system]] for [[dimension|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|azimuthal angle]]'' of its [[orthogonal projection]] on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the [[polar coordinate system]]. 

The radial distance is also called the ''radius'' or ''radial coordinate''. The polar angle may be called ''colatitude'', ''zenith angle'', ''normal angle'', or ''inclination angle''.

The use of symbols and the order of the coordinates differs between sources. In one system frequently encountered in physics ({{mvar|r}}, [[theta|{{mvar|θ}}]], [[phi|{{mvar|φ}}]]) gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books (''r'', ''θ'', ''φ'') gives the radial distance, azimuthal angle, and polar angle. In both systems [[rho|{{mvar|ρ}}]] is often used instead of {{mvar|r}}. Other conventions are also used, so great care needs to be taken to check which one is being used. <!-- Please maintain a consistent convention in this article. -->

A number of different spherical coordinate systems following other conventions are used outside mathematics. In a [[geographical coordinate system]] positions are measured in latitude, longitude and height or altitude. There are a number of different [[celestial coordinate system]]s based on different [[Fundamental plane (spherical coordinates)|fundamental planes]] and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use [[radian]]s rather than [[degree (angle)|degrees]] and measure the azimuthal angle counterclockwise from the {{mvar|x}}-axis to the {{mvar|y}}-axis rather than clockwise from north (0°) to east (+90°) like the [[horizontal coordinate system]].<ref>Duffett-Smith, P and Zwart, J, p. 34.</ref> The polar angle is often replaced by the ''elevation angle'' measured from the reference plane. Elevation angle of zero is at the horizon.

The spherical coordinate system generalizes the two-dimensional polar coordinate system. It can also be extended to higher-dimensional spaces and is then referred to as a [[N-sphere#Spherical coordinates|hyperspherical coordinate system]].

== Definition ==

To define a spherical coordinate system, one must choose two orthogonal directions, the ''zenith'' and the ''azimuth reference'', and an ''origin'' point in space. These choices determine a reference plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point {{mvar|P}} are then defined as follows:

* The ''radius'' or ''radial distance'' is the [[Euclidean distance]] from the origin {{mvar|O}} to {{mvar|P}}.
* The ''inclination'' (or ''polar angle'') is the angle between the zenith direction and the line segment {{mvar|OP}}.
* The ''azimuth'' (or ''azimuthal angle'') is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment {{mvar|OP}} on the reference plane.

The sign of the azimuth is determined by choosing what is a ''positive'' sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system's definition.

The ''elevation'' angle is 90 degrees ({{sfrac|{{pi}}|2}} radians) minus the inclination angle.

If the inclination is zero or 180 degrees ({{pi}} radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.

In [[linear algebra]], the [[Euclidean vector|vector]] from the origin {{mvar|O}} to the point {{mvar|P}} is often called the ''[[position vector]]'' of ''P''.

=== Conventions ===

Several different conventions exist for representing the three coordinates, and for the order in which they should be written. The use of {{math|(''r'', ''θ'', ''φ'')}} to denote radial distance, inclination (or elevation), and azimuth, respectively, is common practice in physics, and is specified by [[International Organisation for Standardisation|ISO]] standard [[ISO 80000-2|80000-2:2009]], and earlier in [[ISO 31-11]] (1992).

However, some authors (including mathematicians) use {{mvar|φ}} for inclination (or elevation) and {{mvar|θ}} for azimuth, which "provides a logical extension of the usual polar coordinates notation".<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html">{{cite web
| url = http://mathworld.wolfram.com/SphericalCoordinates.html
| title = Spherical Coordinates
| author = [[Eric W. Weisstein]]
| publisher = [[MathWorld]]
| date = 2005-10-26
| accessdate = 2010-01-15
}}</ref> Some authors may also list the azimuth before the inclination (or elevation), and/or use {{mvar|ρ}} (rho) instead of {{mvar|r}} for radial distance. Some combinations of these choices result in a [[right-hand rule|left-handed]] coordinate system. The standard convention {{math|(''r'', ''θ'', ''φ'')}} conflicts with the usual notation for the two-dimensional [[polar coordinate system|polar coordinates]], where {{mvar|θ}} is often used for the azimuth. It may also conflict with the notation used for three-dimensional [[cylindrical coordinate system|cylindrical coordinates]].
<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html" />

The angles are typically measured in [[Degree (angle)|degrees]] (°) or [[radian]]s (rad), where 360°&nbsp;=&nbsp;2{{pi}} rad. Degrees are most common in geography, astronomy, and engineering, whereas radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context.

When the system is used for physical three-space, it is customary to use positive sign for azimuth angles that are measured in the counter-clockwise sense from the reference direction on the reference plane, as seen from the zenith side of the plane. This convention is used, in particular, for geographical coordinates, where the "zenith" direction is [[north]] and positive azimuth (longitude) angles are measured eastwards from some [[prime meridian]].

:{| class="wikitable" style="text-align:center"
|+ Major conventions
|-
! coordinates !! corresponding local geographical directions <br/> {{math|(''Z'', ''X'', ''Y'')}} !! right/left-handed
|-
| {{math|(''r'', ''θ''<sub>inc</sub>, ''φ''<sub>az,right</sub>)}} || {{math|(''U'', ''S'', ''E'')}} || right
|-
| {{math|(''r'', ''φ''<sub>az,right</sub>, ''θ''<sub>el</sub>)}}|| {{math|(''U'', ''E'', ''N'')}} || right
|-
| {{math|(''r'', ''θ''<sub>el</sub>, ''φ''<sub>az,right</sub>)}}|| {{math|(''U'', ''N'', ''E'')}} || left
|}
::'''Note:''' easting ({{mvar|E}}), northing ({{mvar|N}}), upwardness ({{mvar|U}}). Local [[azimuth]] angle would be measured, e.g., [[counterclockwise]] from {{mvar|S}} to {{mvar|E}} in the case of {{math|(''U'', ''S'', ''E'')}}.

=== Unique coordinates ===

Any spherical coordinate triplet {{math|(''r'', ''θ'', ''φ'')}} specifies a single point of three-dimensional space. On the other hand, every point has infinitely many equivalent spherical coordinates. One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. It is also convenient, in many contexts, to allow negative radial distances, with the convention that {{math|(−''r'', ''θ'', ''φ'')}} is equivalent to {{math|(''r'', ''θ'' + 180°, ''φ'')}} for any {{mvar|r}}, {{mvar|θ}}, and {{mvar|φ}}. Moreover, {{math|(''r'', −''θ'', ''φ'')}} is equivalent to {{math|(''r'', ''θ'', ''φ'' + 180°)}}.

If it is necessary to define a unique set of spherical coordinates for each point, one must restrict their ranges. A common choice is:

: {{math|''r'' ≥ 0}}
: {{math|0° ≤ ''θ'' ≤ 180° (π rad)}}
: {{math|0° ≤ ''φ'' < 360° (2π rad)}}

However, the azimuth {{mvar|φ}} is often restricted to the [[interval (mathematics)|interval]] {{nowrap|(−180°, +180°]}}, or {{nowrap|(−{{pi}}, +{{pi}}]}} in radians, instead of {{nowrap|[0, 360°)}}. This is the standard convention for geographic longitude.

The range {{nowrap|[0°, 180°]}} for inclination is equivalent to {{nowrap|[−90°, +90°]}} for elevation (latitude).

Even with these restrictions, if {{mvar|θ}} is 0° or 180° (elevation is 90° or −90°) then the azimuth angle is arbitrary; and if {{mvar|r}} is zero, both azimuth and inclination/elevation are arbitrary. To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are zero.

=== Plotting ===

To plot a dot from its spherical coordinates {{math|(''r'', ''θ'', ''φ'')}}, where {{mvar|θ}} is inclination, move {{mvar|r}} units from the origin in the zenith direction, rotate by {{mvar|θ}} about the origin towards the azimuth reference direction, and rotate by {{mvar|φ}} about the zenith in the proper direction.

== Applications ==

The [[geographic coordinate system]] uses the azimuth and elevation of the spherical coordinate system to express locations on Earth, calling them respectively [[longitude]] and [[latitude]]. Just as the two-dimensional [[Cartesian coordinate system]] is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of a sphere. In this system, the sphere is taken as a unit sphere, so the radius is unity and can generally be ignored. This simplification can also be very useful when dealing with objects such as [[Rotation matrix|rotational matrices]].

Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as [[multiple integral|volume integrals]] inside a sphere, the potential energy field surrounding a concentrated mass or charge, or global weather simulation in a planet's atmosphere. A sphere that has the Cartesian equation {{math|''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> {{=}} ''c''<sup>2</sup>}} has the simple equation {{math|''r'' {{=}} ''c''}} in spherical coordinates.

Two important [[partial differential equations]] that arise in many physical problems, [[Laplace's equation]] and the [[Helmholtz equation]], allow a [[separation of variables]] in spherical coordinates. The angular portions of the solutions to such equations take the form of [[spherical harmonics]].

Another application is ergonomic design, where {{mvar|r}} is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.

[[Image:Bosch 36W column loudspeaker polar pattern.png|thumb|The output pattern of an industrial [[loudspeaker]] shown using 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.

The spherical coordinate system is also commonly used in 3D [[game development]] to rotate the camera around the player's position.

=== In geography ===

To a first approximation, the [[geographic coordinate system]] uses elevation angle (latitude) in degrees north of the [[equator]] plane, in the range {{math|−90° ≤ ''φ'' ≤ 90°}}, instead of inclination. Latitude is either [[Latitude#Geocentric latitude|geocentric latitude]], measured at the Earth's center and designated variously by {{math|''ψ'', ''q'', ''φ''′, ''φ''<sub>c</sub>, ''φ''<sub>g</sub>}} or [[Latitude#Geodetic coordinates|geodetic latitude]], measured by the observer's local vertical, and commonly designated {{mvar|φ}}. The azimuth angle (longitude), commonly denoted by {{mvar|λ}}, is measured in degrees east or west from some conventional reference [[meridian (geography)|meridian]] (most commonly the [[IERS Reference Meridian]]), so its domain is {{math|−180° ≤ ''λ'' ≤ 180°}}. 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]]. <!--Must explain the conventions for positive latitude and longitude-->

The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called [[colatitude]] in geography.

Instead of the radial distance, geographers commonly use [[altitude]] above or below some reference surface, which may be the [[sea level]] or "mean" surface level for planets without liquid oceans. The radial distance {{mvar|r}} can be computed from the altitude by adding the mean radius of the planet's reference surface, which is approximately {{convert|6,360|±|11|km|mi|0|abbr=in}} for Earth.

However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be wrong 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 {{convert|21|km|mi|abbr=in|disp=or}}) and many other details.

=== In astronomy ===
In astronomy there are a series of [[Celestial coordinate system#Coordinate systems|spherical coordinate systems]] that measure the elevation angle from different [[Fundamental plane (spherical coordinates)|fundamental planes]]. These reference planes are the observer's [[Horizontal coordinate system|horizon]], the [[Equatorial coordinate system#Spherical coordinates|celestial equator]] (defined by Earth's rotation), the plane of the [[Ecliptic coordinate system|ecliptic]] (defined by Earth's orbit around the [[Sun]]), the plane of the earth terminator (normal to the instantaneous direction to the [[Sun]]), and the [[Galactic coordinate system|galactic equator]] (defined by the rotation of the [[Milky Way]]).

== Coordinate system conversions ==

{{also|List of common coordinate transformations#To spherical coordinates}}

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'' {{mvar|r}}, ''inclination'' {{mvar|θ}}, ''azimuth'' {{mvar|φ}}) can be obtained from its [[Cartesian coordinate system|Cartesian coordinates]] {{math|(''x'', ''y'', ''z'')}} by the formulae
: <math>\begin{align} r&=\sqrt{x^2 + y^2 + z^2} \\ \varphi &= \arctan \frac{y}{x} \\ \theta &= \arccos\frac{z}{\sqrt{x^2 + y^2 + z^2}} = \arccos\frac{z}{r} \end{align}</math>

The inverse tangent denoted in {{math|''φ'' {{=}} arctan {{sfrac|''y''|''x''}}}} must be suitably defined, taking into account the correct quadrant of {{math|(''x'',''y'')}}. See the article on [[atan2]].

Alternatively, the conversion can be considered as two sequential [[Polar coordinate system#Converting between polar and Cartesian coordinates|rectangular to polar conversions]]: the first in the Cartesian {{mvar|xy}}-plane from {{math|(''x'',''y'')}} to {{math|(''R'',''φ'')}}, where {{mvar|R}} is the projection of {{mvar|r}} onto the {{mvar|xy}}-plane, and the second in the Cartesian {{mvar|zR}}-plane from {{math|(''z'',''R'')}} to {{math|(''r'',''θ'')}}. The correct quadrants for {{mvar|φ}} and {{mvar|θ}} 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 {{mvar|xy}}-plane, that {{mvar|θ}} is inclination from the {{mvar|z}} direction, and that the azimuth angles are measured from the Cartesian {{mvar|x}}-axis (so that the {{mvar|y}}-axis has {{math|''φ'' {{=}} +90°}}). If ''θ'' measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the {{math|cos ''θ''}} and {{math|sin ''θ''}} below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (''radius'' {{mvar|r}}, ''inclination'' {{mvar|θ}}, ''azimuth'' {{mvar|φ}}), where {{math|''r'' ∈ {{closed-open|0, ∞}}}}, {{math|''θ'' ∈ {{closed-closed|0, π}}}}, {{math|''φ'' ∈ {{closed-open|0, 2π}}}}, by:

<!--***PLEASE do not swap theta and phi - check the conventions***-->:<math>\begin{align} x&=r \, \sin\theta \, \cos\varphi \\ y&=r \, \sin\theta \, \sin\varphi \\ z&=r \, \cos\theta\end{align}</math><!--***PLEASE do not swap theta and phi - check the conventions***-->

=== Cylindrical coordinates ===

{{main|Cylindrical coordinate system}}
<!-- if you notice--rho and r are described opposite of convention here--someone with extra time could remedy this -->

[[cylindrical coordinate system|Cylindrical coordinates]] (''radius'' {{mvar|ρ}}, ''azimuth'' <!--radians--> {{mvar|φ}}, ''elevation'' {{mvar|z}}) may be converted into spherical coordinates (''radius'' {{mvar|r}}, ''inclination'' {{mvar|θ}}, ''azimuth'' {{mvar|φ}}), by the formulas

: <math>\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>\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 {{mvar|φ}} in the same senses from the same axis, and that the spherical angle {{mvar|θ}} is inclination from the cylindrical {{mvar|z}}-axis.

== Integration and differentiation in spherical coordinates ==

The following equations assume that {{mvar|θ}} is inclination from the {{mvar|z}} (polar) axis (ambiguous since {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} are mutually normal):

The [[line element]] for an infinitesimal displacement from {{math|(''r'', ''θ'', ''φ'')}} to {{math|(''r'' + d''r'', ''θ'' + d''θ'', ''φ'' + d''φ'')}} is

: <math>
\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>\begin{align}
\hat{\mathbf r}
&=\sin \theta \cos \varphi \,\hat{\mathbf x} +
\sin \theta \sin \varphi \,\hat{\mathbf y} +
\cos \theta \,\hat{\mathbf z}
\\[5px]
\hat{\boldsymbol\theta}
&=\cos \theta \cos \varphi \,\hat{\mathbf x} +
\cos \theta \sin \varphi \,\hat{\mathbf y}
-\sin \theta \,\hat{\mathbf z}
\\[5px]
\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 {{mvar|r}}, {{mvar|θ}}, and {{mvar|φ}}, respectively,
and {{math|'''x̂'''}}, {{math|'''ŷ'''}}, and {{math|'''ẑ'''}} are the unit vectors in Cartesian coordinates.

The general form of the formula to prove the differential line element, is<ref name="q74503">{{cite web |title=Line element (dl) in spherical coordinates derivation/diagram |date=October 21, 2011 |work=[[Stack Exchange]] |url=https://math.stackexchange.com/q/74503 }}</ref>

: <math>
\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, you need to calculate how <math>{\mathbf r}</math> changes with each of the coordinates. With the conventions being used, we have

: <math>
\mathbf{r}=\begin{bmatrix}r\sin\theta\cos\varphi\\ r\sin\theta\sin\varphi\\ r\cos\theta\end{bmatrix}
</math>

Thus

: <math>
\frac{\partial\mathbf r}{\partial r}=\begin{bmatrix}\sin\theta\cos\varphi \\ \sin\theta\sin\varphi \\ \cos\theta\end{bmatrix} \,,
\frac{\partial\mathbf r}{\partial \theta}=\begin{bmatrix}r\cos\theta\cos\varphi \\ r\cos\theta\sin\varphi \\ -r\sin\theta\end{bmatrix} \,,
\frac{\partial\mathbf r}{\partial \varphi}=\begin{bmatrix}-r\sin\theta\sin\varphi \\ r\sin\theta\cos\varphi \\ 0 \end{bmatrix} \,
</math>

Then the desired coefficients are the magnitudes of these vectors:

: <math>
\begin{align}
& \left| \frac{\partial\mathbf r}{\partial r}\right| = 1 \\[5pt]
& \left| \frac{\partial\mathbf r}{\partial \theta}\right| = r \\[5pt]
& \left| \frac{\partial\mathbf r}{\partial \varphi}\right| = r\sin\theta
\end{align}
</math> <ref name="q74503" />

The [[Surface integral|surface element]] spanning from {{mvar|θ}} to {{math|''θ'' + d''θ''}} and {{mvar|φ}} to {{math|''φ'' + d''φ''}} on a spherical surface at (constant) radius {{mvar|r}} is

:<math>\mathrm{d}S_r=\left\|{\partial r\hat{\mathbf r} \over \partial \theta}\times {\partial r\hat{\mathbf r} \over \partial \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>\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 {{mvar|θ}} constant (a cone with vertex the origin) is

: <math>\mathrm{d}S_\theta=r \sin\theta\,\mathrm{d}\varphi\,\mathrm{d}r.</math>

The surface element in a surface of azimuth {{mvar|φ}} constant (a vertical half-plane) is

: <math>\mathrm{d}S_\varphi=r\,\mathrm{d}r\,\mathrm{d}\theta.</math>

The [[volume element]] spanning from {{mvar|r}} to {{math|''r'' + d''r''}}, {{mvar|θ}} to {{math|''θ'' + d''θ''}}, and {{mvar|φ}} to {{math|''φ'' + d''φ''}} is  ([[determinant]] of the [[Jacobian matrix]] of [[Partial derivative|partial derivatives]]):

:<math>\mathrm{d}V =  \left |\frac{\partial(x,y,z)}{\partial (r,\theta,\varphi)}\right| 
= 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 {{math|''f''(''r'', ''θ'', ''φ'')}} can be integrated over every point in ℝ<sup>3</sup> by the [[Multiple integral#Spherical coordinates|triple integral]]

: <math>\ \int\limits_{\varphi=0}^{2 \pi}\ \int\limits_{\theta=0}^{\pi}\ \int\limits_{r=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 [[gradient]], [[divergence]], [[curl (mathematics)|curl]] and [[Laplacian]]:

: <math>
\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\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}, \\[8pt]
\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} \\[8pt]
& {} + \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} \\[8pt]
& {} + \frac 1 r \left({\partial \over \partial r} \left( r A_\theta \right)
    - {\partial A_r \over \partial \theta}\right) \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.
\end{align}
</math>

== Kinematics ==

In spherical coordinates the position of a point is written

: <math>\mathbf{r} = r \mathbf{\hat r}.</math>

Its velocity is then

: <math>\mathbf{v} = \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

: <math>
\begin{align}
\mathbf{a} = {} & \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>

In the case of a constant {{mvar|φ}} or {{math|''θ'' {{=}} {{sfrac|π|2}}}}, this reduces to [[Polar coordinate system#Vector calculus|vector calculus in polar coordinates]].

== See also ==

* [[Celestial coordinate system]]
* [[Coordinate system]]
* [[Del in cylindrical and spherical coordinates]]
* [[Elevation (ballistics)]]
* [[Euler angles]]
* [[Gimbal lock]]
* [[Hypersphere]]
* [[Jacobian matrix and determinant]]
* [[List of canonical coordinate transformations]]
* [[Sphere]]
* [[Spherical harmonic]]
* [[Theodolite]]
* [[Vector fields in cylindrical and spherical coordinates]]
* [[Yaw, pitch, and roll]]

== Notes ==

{{reflist}}

== Bibliography ==

* {{cite book | author = [[Philip M. Morse|Morse PM]], [[Herman Feshbach|Feshbach H]] | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X | pages = 658 | lccn = 52011515}}
* {{cite book | author = [[Henry Margenau|Margenau H]], Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | url = https://archive.org/details/mathematicsofphy0002marg| url-access = registration| publisher = D. van Nostrand | location = New York | pages = [https://archive.org/details/mathematicsofphy0002marg/page/177 177–178] | lccn = 55010911 }}
* {{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | pages = 174–175 | lccn = 59014456}}
* {{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 95–96 | lccn = 67025285}}
* {{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Spherical Coordinates (r, θ, ψ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | pages = 24–27 (Table 1.05) | isbn = 978-0-387-18430-2}}
* {{cite book | author = Duffett-Smith P, Zwart J | year = 2011 | title = Practical Astronomy with your Calculator or Spreadsheet, 4th Edition | publisher = Cambridge University Press | location = New York | pages = 34 | isbn = 978-0521146548}}

== External links ==

* {{springer|title=Spherical coordinates|id=p/s086660}}
* [http://mathworld.wolfram.com/SphericalCoordinates.html MathWorld description of spherical coordinates]
* [http://www.random-science-tools.com/maths/coordinate-converter.htm Coordinate Converter &mdash; converts between polar, Cartesian and spherical coordinates]
* [http://www.physics.oregonstate.edu/bridge/papers/spherical.pdf Conventions for Spherical Coordinates] Description of the different conventions in use for naming components of spherical coordinates, along with a proposal for standardizing this.
{{Orthogonal coordinate systems}}

{{DEFAULTSORT:Spherical Coordinate System}}
[[Category:Coordinate systems]]

[[fi:Koordinaatisto#Pallokoordinaatisto]]