In

MathWorld description of spherical coordinates

{{Orthogonal coordinate systems Three-dimensional coordinate systems Orthogonal coordinate systems fi:Koordinaatisto#Pallokoordinaatisto

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a spherical coordinate system is 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
The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere
In astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics
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direction, and the '' azimuthal angle'' of its orthogonal projection
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Linear algebra is the branch of mathematics
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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''.
When radius is fixed, the two angular coordinates make a coordinate system on the sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalen ...

sometimes called spherical polar coordinates.
The use of symbols and the order of the coordinates differs among sources and disciplines. This article will use the ISO convention frequently encountered in physics: ''$(r,\backslash theta,\backslash varphi)$'' gives the radial distance, polar angle, and azimuthal angle. By contrast, in many mathematics books, $(\backslash rho,\backslash theta,\backslash varphi)$ or $(r,\backslash theta,\backslash varphi)$ gives the radial distance, azimuthal angle, and polar angle, switching the meanings of ''θ'' and ''φ''. Other conventions are also used, such as ''r'' for radius from the ''z-''axis, so great care needs to be taken to check the meaning of the symbols.
According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). There are a number of celestial coordinate system
Astronomical coordinate systems are organized arrangements for specifying positions of satellites, planet
A planet is a large, rounded astronomical body that is neither a star
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s 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 and measure the azimuthal angle counterclockwise from the -axis to the -axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system
The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles: altitude and azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the dir ...

. The polar angle is often replaced by the ''elevation angle'' measured from the reference plane towards the positive Z axis, so that the elevation angle of zero is at the horizon; the ''depression angle'' is the negative of the elevation angle.
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 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 are then defined as follows: * The ''radius'' or ''radial distance'' is theEuclidean distance
In mathematics
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from the origin to .
* The ''azimuth'' (or ''azimuthal angle'') is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment on the reference plane.
* The ''inclination'' (or ''polar angle'') is the angle between the zenith direction and the line segment .
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 the signed angle between the reference plane and the line segment , where positive angles are oriented towards the zenith. Equivalently, it is 90 degrees ( radians) minus the inclination angle.
If the inclination is zero or 180 degrees ( radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.
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Linear algebra is the branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. Thes ...

, the vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
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from the origin to the point is often called the ''position vector
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Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics
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'' 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 $(r,\backslash theta,\backslash varphi)$ to denote radial distance, inclination (or elevation), and azimuth, respectively, is common practice in physics, and is specified by ISO standard 80000-2:2019, and earlier in ISO 31-11 (1992). However, some authors (including mathematicians) use ''ρ'' for radial distance, ''φ'' for inclination (or elevation) and ''θ'' for azimuth, and ''r'' for radius from the ''z-''axis, which "provides a logical extension of the usual polar coordinates notation". Some authors may also list the azimuth before the inclination (or elevation). Some combinations of these choices result in aleft-handed
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjec ...

coordinate system. The standard convention $(r,\backslash theta,\backslash varphi)$ conflicts with the usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates, where is often used for the azimuth.
The angles are typically measured in degrees (°) or radians (rad), where 360° = 2 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
North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective
In linguistics, an adjective ( abbreviated ) is a word that generally ...

and positive azimuth (longitude) angles are measured eastwards from some prime meridian.
:
::Note: easting (), northing (), upwardness (). Local azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematica ...

angle would be measured, e.g., counterclockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...

from to in the case of .
Unique coordinates

Any spherical coordinate triplet $(r,\backslash theta,\backslash varphi)$ 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 $(-r,-\backslash theta,\backslash varphi180^\backslash circ)$ is equivalent to $(r,\backslash theta,\backslash varphi)$ for any , , and . Moreover, $(r,-\backslash theta,\backslash varphi)$ is equivalent to $(r,\backslash theta,\backslash varphi180^\backslash circ)$. If it is necessary to define a unique set of spherical coordinates for each point, one must restrict their ranges. A common choice is : : : However, the azimuth is often restricted to the interval , or in radians, instead of . This is the standard convention for geographic longitude. For , the range for inclination is equivalent to for elevation. In geography, the latitude is the elevation. Even with these restrictions, if is 0° or 180° (elevation is 90° or −90°) then the azimuth angle is arbitrary; and if 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 , where is inclination, move units from the origin in the zenith direction, rotate by about the origin towards the azimuth reference direction, and rotate by about the zenith in the proper direction.Applications

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 rotational matrices. Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as 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 has the simple equation in spherical coordinates. Two importantpartial differential equations
In mathematics
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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 is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.
Three dimensional modeling of loudspeaker
A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or ...

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 geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science
Science is a systematic endeavor t ...

'') in degrees north of the equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term ca ...

plane, in the range , instead of inclination. Latitude is either '' geocentric latitude'', measured at the Earth's center and designated variously by or '' geodetic latitude'', measured by the observer's local vertical, and commonly designated .
The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called '' colatitude'' in geography.
The azimuth angle (''longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek l ...

''), commonly denoted by , is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian), so its domain is . For positions on the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...

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, geographers commonly use ''altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

'' above or below some reference surface ('' vertical datum''), which may be the mean sea level
There are several kinds of mean in mathematics
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. The radial distance can be computed from the altitude by adding the radius of Earth, which is approximately .
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
The World Geodetic System (WGS) is a standard used in cartography
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Sci ...

(WGS), and take into account the flattening of the Earth at the poles (about ) 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 different fundamental planes. These reference planes are the observer'shorizon
The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether ...

, the celestial equator (defined by Earth's rotation), the plane of the 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 equator (defined by the rotation of the Milky Way
The Milky Way is the galaxy
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A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth
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).
Coordinate system conversions

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 , inclination , azimuth ) can be obtained from its Cartesian coordinates by the formulae : $\backslash begin\; r\; \&=\; \backslash sqrt\; \backslash \backslash \; \backslash theta\; \&=\; \backslash arccos\backslash frac\; =\; \backslash arccos\backslash frac=\; \backslash begin\; \backslash arctan\backslash frac\; \&\backslash text\; z\; >\; 0\; \backslash \backslash \; \backslash pi\; +\backslash arctan\backslash frac\; \&\backslash text\; z\; <\; 0\; \backslash \backslash \; +\backslash frac\; \&\backslash text\; z\; =\; 0\; \backslash text\; xy\; \backslash neq\; 0\; \backslash \backslash \; \backslash text\; \&\backslash text\; x=y=z\; =\; 0\; \backslash \backslash \; \backslash end\; \backslash \backslash \; \backslash varphi\; \&=\; \backslash sgn(y)\backslash arccos\backslash frac\; =\; \backslash begin\; \backslash arctan(\backslash frac)\; \&\backslash text\; x\; >\; 0,\; \backslash \backslash \; \backslash arctan(\backslash frac)\; +\; \backslash pi\; \&\backslash text\; x\; <\; 0\; \backslash text\; y\; \backslash geq\; 0,\; \backslash \backslash \; \backslash arctan(\backslash frac)\; -\; \backslash pi\; \&\backslash text\; x\; <\; 0\; \backslash text\; y\; <\; 0,\; \backslash \backslash \; +\backslash frac\; \&\backslash text\; x\; =\; 0\; \backslash text\; y\; >\; 0,\; \backslash \backslash \; -\backslash frac\; \&\backslash text\; x\; =\; 0\; \backslash text\; y\; <\; 0,\; \backslash \backslash \; \backslash text\; \&\backslash text\; x\; =\; 0\; \backslash text\; y\; =\; 0.\; \backslash end\; \backslash end$ The inverse tangent denoted in must be suitably defined, taking into account the correct quadrant of . See the article on atan2. Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian plane from to , where is the projection of onto the -plane, and the second in the Cartesian -plane from to . The correct quadrants for and 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 plane, that is inclination from the direction, and that the azimuth angles are measured from the Cartesian axis (so that the axis has ). If ''θ'' measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the and below become switched. Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (''radius'' , ''inclination'' , ''azimuth'' ), where , , , by : $\backslash begin\; x\; \&=\; r\; \backslash sin\backslash theta\; \backslash ,\; \backslash cos\backslash varphi,\; \backslash \backslash \; y\; \&=\; r\; \backslash sin\backslash theta\; \backslash ,\; \backslash sin\backslash varphi,\; \backslash \backslash \; z\; \&=\; r\; \backslash cos\backslash theta.\; \backslash end$Cylindrical coordinates

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 : $\backslash begin\; r\; \&=\; \backslash sqrt,\; \backslash \backslash \; \backslash theta\; \&=\; \backslash arctan\backslash frac\; =\; \backslash arccos\backslash frac,\; \backslash \backslash \; \backslash varphi\; \&=\; \backslash varphi.\; \backslash end$ Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae : $\backslash begin\; \backslash rho\; \&=\; r\; \backslash sin\; \backslash theta,\; \backslash \backslash \; \backslash varphi\; \&=\; \backslash varphi,\; \backslash \backslash \; z\; \&=\; r\; \backslash cos\; \backslash theta.\; \backslash end$ These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle in the same senses from the same axis, and that the spherical angle is inclination from the cylindrical axis.Generalization

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 : $ax^2\; +\; by^2\; +\; cz^2\; =\; d.$ The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: ''radius'' , ''inclination'' , ''azimuth'' ) can be obtained from its Cartesian coordinates by the formulae : $\backslash begin\; x\; \&=\; \backslash frac\; r\; \backslash sin\backslash theta\; \backslash ,\; \backslash cos\backslash varphi,\; \backslash \backslash \; y\; \&=\; \backslash frac\; r\; \backslash sin\backslash theta\; \backslash ,\; \backslash sin\backslash varphi,\; \backslash \backslash \; z\; \&=\; \backslash frac\; r\; \backslash cos\backslash theta,\; \backslash \backslash \; r^\; \&=\; ax^2\; +\; by^2\; +\; cz^2.\; \backslash end$ An infinitesimal volume element is given by : $\backslash mathrmV\; =\; \backslash left,\; \backslash frac\backslash \; \backslash ,\; dr\backslash ,d\backslash theta\backslash ,d\backslash varphi\; =\; \backslash frac\; r^2\; \backslash sin\; \backslash theta\; \backslash ,\backslash mathrmr\; \backslash ,\backslash mathrm\backslash theta\; \backslash ,\backslash mathrm\backslash varphi\; =\; \backslash frac\; r^2\; \backslash ,\backslash mathrmr\; \backslash ,\backslash mathrm\backslash Omega.$ The square-root factor comes from the property of thedeterminant
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that allows a constant to be pulled out from a column:
: $\backslash begin\; ka\; \&\; b\; \&\; c\; \backslash \backslash \; kd\; \&\; e\; \&\; f\; \backslash \backslash \; kg\; \&\; h\; \&\; i\; \backslash end\; =\; k\; \backslash begin\; a\; \&\; b\; \&\; c\; \backslash \backslash \; d\; \&\; e\; \&\; f\; \backslash \backslash \; g\; \&\; h\; \&\; i\; \backslash end.$
Integration and differentiation in spherical coordinates

The following equations (Iyanaga 1977) assume that the colatitude is the inclination from the (polar) axis (ambiguous since , , and are mutually normal), as in the ''physics convention'' discussed. The line element for an infinitesimal displacement from to is $$\backslash mathrm\backslash mathbf\; =\; \backslash mathrmr\backslash ,\backslash hat\; +\; r\backslash ,\backslash mathrm\backslash theta\; \backslash ,\backslash hat\; +\; r\; \backslash sin\; \backslash ,\; \backslash mathrm\backslash varphi\backslash ,\backslash mathbf,$$ where $$\backslash begin\; \backslash hat\; \&=\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash varphi\; \backslash ,\backslash hat\; +\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash varphi\; \backslash ,\backslash hat\; +\; \backslash cos\; \backslash theta\; \backslash ,\backslash hat,\; \backslash \backslash \; \backslash hat\; \&=\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash varphi\; \backslash ,\backslash hat\; +\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash varphi\; \backslash ,\backslash hat\; -\; \backslash sin\; \backslash theta\; \backslash ,\backslash hat,\; \backslash \backslash \; \backslash hat\; \&=\; -\; \backslash sin\; \backslash varphi\; \backslash ,\backslash hat\; +\; \backslash cos\; \backslash varphi\; \backslash ,\backslash hat\; \backslash end$$ are the local orthogonalunit vectors
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in the directions of increasing , , and , respectively,
and , , and are the unit vectors in Cartesian coordinates. The linear transformation to this right-handed coordinate triplet is a rotation matrix,
$$R\; =\; \backslash begin\; \backslash sin\backslash theta\backslash cos\backslash varphi\&\backslash sin\backslash theta\backslash sin\backslash varphi\&\; \backslash cos\backslash theta\backslash \backslash \; \backslash cos\backslash theta\backslash cos\backslash varphi\&\backslash cos\backslash theta\backslash sin\backslash varphi\&-\backslash sin\backslash theta\backslash \backslash \; -\backslash sin\backslash varphi\&\backslash cos\backslash varphi\; \&0\; \backslash end.$$
This gives the transformation from the spherical to the cartesian, the other way around is given by its inverse.
Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose
In linear algebra
Linear algebra is the branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities a ...

.
The Cartesian unit vectors are thus related to the spherical unit vectors by:
$$\backslash begin\backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash end\; =\; \backslash begin\; \backslash sin\backslash theta\backslash cos\backslash varphi\; \&\; \backslash cos\backslash theta\backslash cos\backslash varphi\; \&\; -\backslash sin\backslash varphi\; \backslash \backslash \; \backslash sin\backslash theta\backslash sin\backslash varphi\; \&\; \backslash cos\backslash theta\backslash sin\backslash varphi\; \&\; \backslash cos\backslash varphi\; \backslash \backslash \; \backslash cos\backslash theta\; \&\; -\backslash sin\backslash theta\; \&\; 0\; \backslash end\; \backslash begin\; \backslash boldsymbol\; \backslash \backslash \; \backslash boldsymbol\; \backslash \backslash \; \backslash boldsymbol\; \backslash end$$
The general form of the formula to prove the differential line element, is
$$\backslash mathrm\backslash mathbf\; =\; \backslash sum\_i\; \backslash frac\; \backslash ,\backslash mathrmx\_i\; =\; \backslash sum\_i\; \backslash left,\; \backslash frac\backslash \; \backslash frac\; \backslash ,\; \backslash mathrmx\_i\; =\; \backslash sum\_i\; \backslash left,\; \backslash frac\backslash \; \backslash ,\backslash mathrmx\_i\; \backslash ,\; \backslash hat\_i,$$
that is, the change in $\backslash mathbf\; r$ is decomposed into individual changes corresponding to changes in the individual coordinates.
To apply this to the present case, one needs to calculate how $\backslash mathbf\; r$ changes with each of the coordinates. In the conventions used,
$$\backslash mathbf\; =\; \backslash begin\; r\; \backslash sin\backslash theta\; \backslash ,\; \backslash cos\backslash varphi\; \backslash \backslash \; r\; \backslash sin\backslash theta\; \backslash ,\; \backslash sin\backslash varphi\; \backslash \backslash \; r\; \backslash cos\backslash theta\; \backslash end.$$
Thus,
$$\backslash frac\; =\; \backslash begin\; \backslash sin\backslash theta\; \backslash ,\; \backslash cos\backslash varphi\; \backslash \backslash \; \backslash sin\backslash theta\; \backslash ,\; \backslash sin\backslash varphi\; \backslash \backslash \; \backslash cos\backslash theta\; \backslash end=\backslash mathbf,\; \backslash quad\; \backslash frac\; =\; \backslash begin\; r\; \backslash cos\backslash theta\; \backslash ,\; \backslash cos\backslash varphi\; \backslash \backslash \; r\; \backslash cos\backslash theta\; \backslash ,\; \backslash sin\backslash varphi\; \backslash \backslash \; -r\; \backslash sin\backslash theta\; \backslash end=r\backslash ,\backslash hat,\; \backslash quad\; \backslash frac\; =\; \backslash begin\; -r\; \backslash sin\backslash theta\; \backslash ,\; \backslash sin\backslash varphi\; \backslash \backslash \; r\; \backslash sin\backslash theta\; \backslash ,\; \backslash cos\backslash varphi\; \backslash \backslash \; 0\; \backslash end\; =\; r\; \backslash sin\backslash theta\backslash ,\backslash mathbf\; .$$
The desired coefficients are the magnitudes of these vectors:
$$\backslash left,\; \backslash frac\backslash \; =\; 1,\; \backslash quad\; \backslash left,\; \backslash frac\backslash \; =\; r,\; \backslash quad\; \backslash left,\; \backslash frac\backslash \; =\; r\; \backslash sin\backslash theta.$$
The surface element spanning from to and to on a spherical surface at (constant) radius is then
$$\backslash mathrmS\_r\; =\; \backslash left\backslash ,\; \backslash frac\; \backslash times\; \backslash frac\backslash right\backslash ,\; \backslash mathrm\backslash theta\; \backslash ,\backslash mathrm\backslash varphi\; =\; \backslash left,\; r\; \backslash times\; r\; \backslash sin\; \backslash theta\; \backslash =\; r^2\; \backslash sin\backslash theta\; \backslash ,\backslash mathrm\backslash theta\; \backslash ,\backslash mathrm\backslash varphi\; ~.$$
Thus the differential solid angle
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they a ...

is
$$\backslash mathrm\backslash Omega\; =\; \backslash frac\; =\; \backslash sin\backslash theta\; \backslash ,\backslash mathrm\backslash theta\; \backslash ,\backslash mathrm\backslash varphi.$$
The surface element in a surface of polar angle constant (a cone with vertex the origin) is
$$\backslash mathrmS\_\backslash theta\; =\; r\; \backslash sin\backslash theta\; \backslash ,\backslash mathrm\backslash varphi\; \backslash ,\backslash mathrmr.$$
The surface element in a surface of azimuth constant (a vertical half-plane) is
$$\backslash mathrmS\_\backslash varphi\; =\; r\; \backslash ,\backslash mathrmr\; \backslash ,\backslash mathrm\backslash theta.$$
The volume element spanning from to , to , and to is specified by the determinant
In mathematics
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of the Jacobian matrix of partial derivatives,
$$J\; =\backslash frac\; =\backslash begin\; \backslash sin\backslash theta\backslash cos\backslash varphi\&r\backslash cos\backslash theta\backslash cos\backslash varphi\&-r\backslash sin\backslash theta\backslash sin\backslash varphi\backslash \backslash \; \backslash sin\backslash theta\; \backslash sin\backslash varphi\&r\backslash cos\backslash theta\backslash sin\backslash varphi\&r\backslash sin\backslash theta\backslash cos\backslash varphi\backslash \backslash \; \backslash cos\backslash theta\&-r\backslash sin\backslash theta\&0\; \backslash end,$$
namely
$$\backslash mathrmV\; =\; \backslash left,\; \backslash frac\backslash \; \backslash ,\backslash mathrmr\; \backslash ,\backslash mathrm\backslash theta\; \backslash ,\backslash mathrm\backslash varphi=\; r^2\; \backslash sin\backslash theta\; \backslash ,\backslash mathrmr\; \backslash ,\backslash mathrm\backslash theta\; \backslash ,\backslash mathrm\backslash varphi\; =\; r^2\; \backslash ,\backslash mathrmr\; \backslash ,\backslash mathrm\backslash Omega\; ~.$$
Thus, for example, a function can be integrated over every point in by the triple integral
$$\backslash int\backslash limits\_0^\; \backslash int\backslash limits\_0^\backslash pi\; \backslash int\backslash limits\_0^\backslash infty\; f(r,\; \backslash theta,\; \backslash varphi)\; r^2\; \backslash sin\backslash theta\; \backslash ,\backslash mathrmr\; \backslash ,\backslash mathrm\backslash theta\; \backslash ,\backslash mathrm\backslash varphi\; ~.$$
The del operator in this system leads to the following expressions for the gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

, divergence, curl and (scalar) Laplacian,
$$\backslash begin\; \backslash nabla\; f\; =\; \&\backslash hat\; +\; \backslash hat\; +\; \backslash hat,\; \backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$$
Further, the inverse Jacobian in Cartesian coordinates is
$$J^\; =\; \backslash begin\; \backslash dfrac\&\backslash dfrac\&\backslash dfrac\backslash \backslash \backslash \backslash \; \backslash dfrac\&\backslash dfrac\&\backslash dfrac\backslash \backslash \backslash \backslash \; \backslash dfrac\&\backslash dfrac\&0\; \backslash end.$$
The metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space
...

in the spherical coordinate system is $g\; =\; J^T\; J$.
Distance in spherical coordinates

In spherical coordinates, given two points with being the azimuthal coordinate :$\backslash begin\; \&=\; (r,\backslash theta,\backslash varphi),\; \backslash \backslash \; \&=\; (r\text{'},\backslash theta\text{'},\backslash varphi\text{'})\; \backslash end$ The distance between the two points can be expressed as :$\backslash begin\; \&=\; \backslash sqrt\; \backslash end$Kinematics

In spherical coordinates, the position of a point or particle (although better written as a triple$(r,\backslash theta,\; \backslash varphi)$) can be written as : $\backslash mathbf\; =\; r\; \backslash mathbf\; .$ Its velocity is then : $\backslash mathbf\; =\; \backslash frac\; =\; \backslash dot\; \backslash mathbf\; +\; r\backslash ,\backslash dot\backslash theta\backslash ,\backslash hat\; +\; r\backslash ,\backslash dot\backslash varphi\; \backslash sin\backslash theta\backslash ,\backslash mathbf$ and its acceleration is : $\backslash begin\; \backslash mathbf\; =\; \backslash frac\; =\; \&\; \backslash left(\; \backslash ddot\; -\; r\backslash ,\backslash dot\backslash theta^2\; -\; r\backslash ,\backslash dot\backslash varphi^2\backslash sin^2\backslash theta\; \backslash right)\backslash mathbf\; \backslash \backslash \; \&\; +\; \backslash left(\; r\backslash ,\backslash ddot\backslash theta\; +\; 2\backslash dot\backslash ,\backslash dot\backslash theta\; -\; r\backslash ,\backslash dot\backslash varphi^2\backslash sin\backslash theta\backslash cos\backslash theta\; \backslash right)\; \backslash hat\; \backslash \backslash \; \&\; +\; \backslash left(\; r\backslash ddot\backslash varphi\backslash ,\backslash sin\backslash theta\; +\; 2\backslash dot\backslash ,\backslash dot\backslash varphi\backslash ,\backslash sin\backslash theta\; +\; 2\; r\backslash ,\backslash dot\backslash theta\backslash ,\backslash dot\backslash varphi\backslash ,\backslash cos\backslash theta\; \backslash right)\; \backslash hat\; \backslash end$ The angular momentum is : $\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf\; =\; \backslash mathbf\; \backslash times\; m\backslash mathbf\; =\; m\; r^2\; (-\; \backslash dot\backslash varphi\; \backslash sin\backslash theta\backslash ,\backslash mathbf\; +\; \backslash dot\backslash theta\backslash ,\backslash hat)$ Where $m$ is mass. In the case of a constant or else , this reduces to vector calculus in polar coordinates. The corresponding angular momentum operator then follows from the phase-space reformulation of the above, :$\backslash mathbf=\; -i\backslash hbar\; ~\backslash mathbf\; \backslash times\; \backslash nabla\; =i\; \backslash hbar\; \backslash left(\backslash frac\; \backslash frac\; -\; \backslash hat\; \backslash frac\backslash right).$ The torque is given as : $\backslash mathbf\; =\; \backslash frac\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf\; =\; -m\; \backslash left(2r\backslash dot\backslash dot\backslash sin\backslash theta\; +\; r^2\backslash ddot\backslash sin\; +\; 2r^2\backslash dot\backslash dot\backslash cos\; \backslash right)\backslash hat\; +\; m\; \backslash left(r^2\backslash ddot\; +\; 2r\backslash dot\backslash dot\; -\; r^2\backslash dot^2\backslash sin\backslash theta\backslash cos\backslash theta\; \backslash right)\; \backslash hat$ The kinetic energy is given as : $E\_k\; =\; \backslash fracm\; \backslash left;\; href="/html/ALL/s/\backslash left(\backslash dot^2\backslash right)\_+\_\backslash left(r\backslash dot\backslash right)^2\_+\_\backslash left(r\backslash dot\backslash sin\backslash theta\backslash right)^2\_\backslash right.html"\; ;"title="\backslash left(\backslash dot^2\backslash right)\; +\; \backslash left(r\backslash dot\backslash right)^2\; +\; \backslash left(r\backslash dot\backslash sin\backslash theta\backslash right)^2\; \backslash right">\backslash left(\backslash dot^2\backslash right)\; +\; \backslash left(r\backslash dot\backslash right)^2\; +\; \backslash left(r\backslash dot\backslash sin\backslash theta\backslash right)^2\; \backslash right$See also

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Bibliography

* * * * * * *External links

*MathWorld description of spherical coordinates

{{Orthogonal coordinate systems Three-dimensional coordinate systems Orthogonal coordinate systems fi:Koordinaatisto#Pallokoordinaatisto