Spherical coordinates Definition and 41 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:


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


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


{\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. josephsanders

    B Method of images and spherical coordinates

    I am finding the potential everywhere in space due to a point charge a distance 'a' on the z-axis above an infinite xy-plane held at zero potential. This problem is fairly straight forward; place an image charge q' = -q at position -a on the z-axis. I have the solution in cartesian coordinates...
  2. Peter-

    I Calculating an increasing angle in Spherical Coordinates for a curve

    I'm making a program that generates lines in 3D space. One feature that I need is to have an incrementally increasing angle on a line (a bending line / curve). The problem is simple if the line exists in the xy-plane, then it would be a case of stepping say 1m, increase the azimuthal angle φ...
  3. redtree

    I Fourier transform of a function in spherical coordinates

    I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...
  4. E

    Scale factors in spherical coordinates

    how they got that value for the scale factors h?
  5. K

    Help solving this Heat Equation please

    I want to solve the heat equation below: I don't understand where the expression for ##2/R\cdot\int_0^R q\cdot sin(k_nr)\cdot r \, dr## came from. The r dependent function is calculated as ##sin(k_nr)/r## not ##sin(k_nr)\cdot r##. I don't even know if ##sin(k_nr)/r## are orthogonal for...
  6. F

    I Dot product in spherical coordinates

    I'm learing about antennas in a course, and we are using Jin's Electromagnetic text. This isn't a homework problem, I'm just trying to understand what I'm supposed to do in this situation. This part of the text discusses how to evaluate a radiation pattern. One of the steps to evaluate the...
  7. Adesh

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

    I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$ (##r## is the distance from origin, ##\phi## is azimuthal angle and...
  8. T

    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...
  9. Terrycho

    Divergence of a position vector in spherical coordinates

    I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me. The vector field of A is written as follows, , and the divergence of a vector field A in spherical coordinates are written as...
  10. K

    Setup for Spherical Astronomy Problem

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  11. M

    I Converting from spherical to cylindrical coordinates

    I have the coordinates of a hurricane at a particular point defined on the surface of a sphere i.e. longitude and latitude. Now I want to transform these coordinates into a axisymmetric representation cylindrical coordinate i.e. radial and azimuth angle. Is there a way to do the mathematical...
  12. A

    A How to find the displacement vector in Spherical coordinate

    Is there a way of subtracting two vectors in spherical coordinate system without first having to convert them to Cartesian or other forms? Since I have already searched and found the difference between Two Vectors in Spherical Coordinates as...
  13. J

    Self adjoint operators in spherical polar coordinates

    Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is self adjoint ? e.g. suppose i have the operator i ∂/∂ϕ. If the operator was a function of x I know exactly what to do, just check <ψ|Qψ>=<Qψ|ψ> But what about dr, dphi and d theta
  14. W

    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...
  15. B

    Laplacian in spherical coordinates

    Homework Statement Hello at all! I have to calculate total energy for a nucleons system by equation: ##E_{tot}=\frac{1}{2}\sum_j(t_{jj}+\epsilon_j)## with ##\epsilon_j## eigenvalues and: ##t_{jj}=\int \psi_j^*(\frac{\hbar^2}{2m}\triangledown^2)\psi_j dr## My question is: if I'm in...
  16. 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...
  17. E

    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...
  18. J

    Finding the Electric Field given the potential in spherical

    Homework Statement The problem statement is in the attachment Homework Equations E[/B] = -∇φ ∇ = (∂φ/∂r)er The Attempt at a Solution I am confused about how to do the derivative apparently because the way I do it gives E = - (∂[p*r/4πε0r3]/∂r)er = 3*(p*r)/4πε0r4er
  19. X

    Position vector in spherical coordinates

    Is the position vector r=xi+yj+zk just r=rerin spherical coordinates?
  20. U

    I How to write the unit vector for the spherical coordinates

    So I'm reading the Schaum's outlines while trying to prepare for a big test I have in September. And I'm trying to understand something here that maybe someone can offer some clarification and guidance. So, using Coulomb's Law, we can find the electric field as follows: \begin{equation} dE...
  21. chi_rho

    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...
  22. U

    I Spherical coordinates via a rotation matrix

    First, I'd like to say I apologize if my formatting is off! I am trying to figure out how to do all of this on here, so please bear with me! So I was watching this video on spherical coordinates via a rotation matrix: and in the end, he gets: x = \rho * sin(\theta) * sin(\phi) y = \rho*...
  23. F

    Volume enclosed by two spheres using spherical coordinates

    Homework Statement Use spherical coordinates to find the volume of the solid enclosed between the spheres $$x^2+y^2+z^2=4$$ and $$x^2+y^2+z^2=4z$$ Homework Equations $$z=\rho cos\phi$$ $$\rho^2=x^2+y^2+z^2$$ $$dxdydz = \rho^2sin\phi d\rho d\phi d\theta$$ The Attempt at a Solution The first...
  24. TheSodesa

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

    Homework Statement A sphere has a diameter of ##D = 2\rho = 4cm##. A cylindrical hole with a diameter of ##d = 2R = 2 cm## is bored through the center of the sphere. Calculate the volume of the remaining solid. (Spherical or cylindrical coordinates?) hint: Place the shape into a convenient...
  25. O

    A Ellipse of transformation from spherical to cartesian

    Hi, I have to resample images taken from camera, whose target is a spherical object, onto a regular grid of 2 spherical coordinates: the polar and azimutal angles (θ, Φ). For best accuracy, I need to be aware of, and visualise, the "footprints" of the small angle differences onto the original...
  26. S

    Defining rho in spherical coordinates for strange shapes?

    Homework Statement The problem asks for a single triple integral (the integrand may be a sum but there must be a single definition for the bounds of the integral) representing the volume (in the first octant) of the shell defined by a sphere of radius 2 centered around the origin and a sphere...
  27. Konte

    Variable separation - Schrödinger equation

    Hello everybody, My question is about variable separation applied in the solution of general time-independent Schrodinger equation, expressed with spherical coordinates as: \hat{H} \psi (r,\theta,\phi) = E \psi (r,\theta,\phi) Is it always possible (theoretically) to seek a solution such as...
  28. J

    A Separating the Dirac Delta function in spherical coordinates

    The following integral arises in the calculation of the new density of a non-uniform elastic medium under stress: ∫dx ρ(r,θ)δ(x+u(x)-x') where ρ is a known mass density and u = ru_r+θu_θ a known vector function of spherical coordinates (r,θ) (no azimuthal dependence). How should the Dirac...
  29. T

    Coordinate transformation from spherical to rectangular

    Iam having trouble understanding how one arrives at the transformation matrix for spherical to rectangular coordinates. I understand till getting the (x,y,z) from (r,th ie., z = rcos@ y = rsin@sin# x = rsin@cos# Note: @ - theta (vertical angle) # - phi (horizontal angle) Please show me how...
  30. M

    Velocity in spherical polar coordinates

    I am looking at this derivation of velocity in spherical polar coordinates and I am confused by the definition of r, theta and phi. http://www.usna.edu/Users/math/rmm/SphericalCoordinates.pdf [Broken] I thought phi was the co latitude in the r,θ,∅ system and not the latitude. Of course the two...
  31. F

    Laplace equation in spherical coordinates

    Homework Statement Solve the Laplace equation inside a sphere, with the boundary condition: \begin{equation} u(3,\theta,\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi) \end{equation} Homework Equations \begin{equation} \sum^{\infty}_{l=0} \sum^{m}_{m=0} (A_lr^l + B_lr^{-l -1})P_l^m(\cos...
  32. J

    A circle in a non-euclidean geometry

    Homework Statement Consider a universe described by the Friedmann-Robertson-Walker metric which describes an open, closed, or at universe, depending on the value of k: $$ds^2=a^2(t)[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+sin^2\theta d\phi^2)]$$ This problem will involve only the geometry of space at...
  33. B

    Spherical coordinates path integral and stokes theorem

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  34. T

    Transform from Magnitude of P to R

    Hi everyone! How do I transform Momentum to Position in spherical coordinates? Sincerely, Thinker301
  35. H

    Solving Laplace's equation in spherical coordinates

    The angular equation: ##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta## Right now, ##l## can be any number. The solutions are Legendre polynomials in the variable ##\cos\theta##: ##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer...
  36. J

    Maxwell-Boltzmann Spherical Cap

    Homework Statement An ideal gas satisfying the Maxwell-Boltzmann distribution is leaking from a container of the volume V through a circular hole of area A'. The gas is kept in the container under pressure P and temperature T. The initial number density (concentration) is given by n0=N/V...
  37. K

    Spiralling around the Earth

    Homework Statement An airplane flies from the North Pole to the South Pole, following a winding trajectory. Place the center of the Earth at the origin of your coordinate system, and align the south-to-north axis of the Earth with your z axis. The pilot’s trajectory can then be described as...
  38. H

    Derive grad T in spherical coordinates

    Homework Statement ##x=r\sin\theta\cos\phi,\,\,\,\,\,y=r\sin\theta\sin\phi,\,\,\,\,\,z=r\cos\theta## ##\hat{x}=\sin\theta\cos\phi\,\hat{r}+\cos\theta\cos\phi\,\hat{\theta}-\sin\phi\,\hat{\phi}## ##\hat{y}=\sin\theta\sin\phi\,\hat{r}+\cos\theta\sin\phi\,\hat{\theta}+\cos\phi\,\hat{\phi}##...
  39. S

    Spherical trilateration

    Hi I am looking for an equation of intersection of 3 circles or 3 spheres, on the surface of the fourth (central) sphere, in a spherical coordinate circle. This should really be just a simple trilateration problem. I know this is usually done by transforming the spherical coordinate system to...
  40. R

    U(0)=0 for real expectation values of momentum

    Homework Statement The position-space representation of the radial component of the momentum operator is given by ## p_r \rightarrow \frac{\hbar}{i}\left ( \frac{\partial }{\partial r} + \frac{1}{r}\right ) ## Show that for its expectation value to be real:## \left \langle \psi|p_r|\psi \right...
  41. C

    Integrating a delta function with a spherical volume integral

    Homework Statement Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity. Homework Equations The Attempt at a Solution This integral actually came up in a homework problem for my E&M class and...