# spherical coordinates Definition and Topics - 39 Discussions

In 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 direction, and the 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 among sources and disciplines. This article will use the ISO convention frequently encountered in physics:

(
r
,
θ
,
φ
)

{\displaystyle (r,\theta ,\varphi )}
gives the radial distance, polar angle, and azimuthal angle. In many mathematics books,

(
ρ
,
θ
,
φ
)

{\displaystyle (\rho ,\theta ,\varphi )}
or

(
r
,
θ
,
φ
)

{\displaystyle (r,\theta ,\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 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 and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system. The polar angle is often replaced by the elevation angle measured from the reference plane, so that the 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 hyperspherical coordinate system.

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1. ### I Fourier transform of a function in spherical coordinates

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2. ### Scale factors in spherical coordinates

how they got that value for the scale factors h?
3. ### Help solving this Heat Equation please

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4. ### I Dot product in spherical coordinates

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5. ### How to find the curl of a vector field which points in the theta direction?

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6. ### Vector Field Transformation to Spherical Coordinates

I am trying to solve the following problem from my textbook: Formulate the vector field $$\mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}}$$ in spherical coordinates. My solution is the following: For the unit vectors I use...
7. ### Divergence of a position vector in spherical coordinates

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8. ### Setup for Spherical Astronomy Problem

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9. ### I Converting from spherical to cylindrical coordinates

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10. ### A How to find the displacement vector in Spherical coordinate

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11. ### Self adjoint operators in spherical polar coordinates

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12. ### Question about Spherical Metric and Approximations

Homework Statement This is Problem 2 from Chapter 1, Section V of A. Zee's Einstein Gravity in a Nutshell. Zee asks us to imagine a colony of "eskimo mites" that live at the north pole. The geometers of the colony have measured the following metric of their world to second order (with the...
13. ### Laplacian in spherical coordinates

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14. L

### How to calculate the dipole moment of the spherical shell?

Homework Statement A spherical shell of radius R has a surface charge distribution σ = k sinφ. Calculate the dipole moment of the spherical shell. Homework Equations P[/B]' = ∫r' σ(r') da' The Attempt at a Solution So I believe my dipole will be directed along the y axis, as the function...
15. ### Spherical Integral with abs value in limits

Homework Statement This has been driving me crazy I can't for the life of me figure out how to convert the limits of this integral into spherical coordinates because there is an absolute value in the limits and i'm absolutely clueless as to what to do with with. Homework Equations...
16. ### Finding the Electric Field given the potential in spherical

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17. ### Position vector in spherical coordinates

Is the position vector r=xi+yj+zk just r=rerin spherical coordinates?
18. ### I How to write the unit vector for the spherical coordinates

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19. ### A Transforming Spin Matrices (Sx, Sy, Sz) to a Spherical Basis

Say I have {S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array}\right)} Right now, this spin operator is in the Cartesian basis. I want to transform it into the spherical basis. Since, {\vec{S}} acts like a vector I think that I only need to...
20. ### I Spherical coordinates via a rotation matrix

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21. ### Volume enclosed by two spheres using spherical coordinates

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22. ### A sphere with a hole through it (a triple integral).

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23. ### A Ellipse of transformation from spherical to cartesian

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24. ### Defining rho in spherical coordinates for strange shapes?

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25. ### Variable separation - Schrödinger equation

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26. ### A Separating the Dirac Delta function in spherical coordinates

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27. ### Coordinate transformation from spherical to rectangular

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28. ### Velocity in spherical polar coordinates

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29. ### Laplace equation in spherical coordinates

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30. ### A circle in a non-euclidean geometry

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