What is Spherical coordinates: Definition and 351 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. nomadreid

    B Negative radius convention equivalent but not equal?

    In https://en.wikipedia.org/wiki/Spherical_coordinate_system under the heading "Unique coordinates" using the convention (r,P,A) =(radial distance, polar angle, azimuthal angle) ("physicist's convention") we have (r,P,A) is equivalent to (-r,-P, π-A). My three dimensional imagination is...
  2. milkism

    Separation of variables in spherical coordinates (electrostatics)

    Problem: Solution: When I looked at an example problem, they started writing the potential in terms of the Legendre polynomials. The example problem: This is what I did: $$V_0 \alpha P_2 (\cos(\theta)) \Rightarrow \frac{\alpha 3 \cos ^2 (\theta)}{2} - \frac{\alpha}{2} \Rightarrow \frac{\alpha...
  3. phos19

    Solving Curl A in Spherical Coordinates: Tips & Hints

    I've tried writing the curl A (in spherical coord.) and equating the components, but I end up with something that is beyond me: \begin{equation} {\displaystyle {\begin{aligned}{B_r = \dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) =\frac {1}{r\sin \theta }}\left({\frac {\partial...
  4. Addez123

    How to calculate a sink using spherical coordinates

    The issue is that the singularity is not in the center of the sphere. So how would I calculate it? I have a few questions: 1. Can I calculate the terms separately like so: $$A = grad(a+b) = grad(a) + grad(b)$$ 2. If I use a spherical coordinate system with the center being at the singularity I...
  5. 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...
  6. 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...
  7. A

    When to use the Jacobian in spherical coordinates?

    Greetings! here is the solution which I undertand very well: my question is: if we go the spherical coordinates shouldn't we use the jacobian r^2*sinv? thank you!
  8. Mayhem

    Deriving the Laplacian in spherical coordinates

    As a part of my self study, I am trying to derive the Laplacian in spherical coordinates to gain a deeper understanding of the mathematics of quantum mechanics. For reference, this the sphere I am using, where ##r## is constant and ##\theta = \theta (x,y, z), \phi = \phi(x,y)##. Given the...
  9. Danielle46

    I have to prove that vectors in spherical coordinates are clockwise

    I should use the cross product but I don´t know how. I tried to calculate it but it didn´t work out as expected. Please can you give me one example how to do it ?
  10. 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...
  11. K

    Help solving this Heat Equation please

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  12. 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...
  13. Another

    Velocity in spherical coordinates

    Why the velocity in spherical coordinates equal to ## v^2 = v \dot{} v = \dot{r}^2 + \dot{r}^2\dot{\theta}^2## maybe ## v^2 = v \dot{} v = (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} + \hat{ \phi } \dot{\phi } r \sin{ \theta}) \dot{} (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} +...
  14. J

    I Changing spherical coordinates in a Lagrangian

    In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where ##\theta## is the colatitud or polar angle and ##\phi## is the...
  15. 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 ##\theta##...
  16. T

    Vector Field Transformation to Spherical Coordinates

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  17. 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...
  18. E

    I Velocity Vector Transformation from Cartesian to Spherical Coordinates

    Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...
  19. L

    A Volume element in Spherical Coordinates

    For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then dV=4\pi r^2dr always?
  20. C

    Element of surface area in spherical coordinates

    r,θ,ϕ For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at...
  21. L

    I The free particle in spherical coordinates

    Hi! I'm studying Shankar's Principle of quantum mechanics I didn't get the last conclusion, can someone help me understand it, please. Where did the l over rho come from?
  22. K

    Setup for Spherical Astronomy Problem

    My apologies for not detailing my attempts at a solution; I'm not sure how to to digitally illustrate or describe the various setups I attempted before looking at the solution to this problem. I am also ONLY asking about the setup, though I included the full question for context. The solution to...
  23. L

    A Laplace transform in spherical coordinates

    Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...
  24. M

    Surface area of a shifted sphere in spherical coordinates

    Homework Statement find the surface area of a sphere shifted R in the z direction using spherical coordinate system. Homework Equations $$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$ $$x^2+y^2+(z-R)^2=R^2$$ The Attempt at a Solution I tried to use the sphere equation mentioned above and...
  25. 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...
  26. 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...
  27. PeroK

    Angular Momentum in Spherical Coordinates

    I've started on "Noether's Theorem" by Neuenschwander. This is page 35 of the 2011 edition. We have the Lagrangian for a central force: ##L = \frac12 m(\dot{r}^2 + r^2 \dot{\theta}^2 + r \dot{\phi}^2 \sin^2 \theta) - U(r)## Which gives the canonical momenta: ##p_{\theta} = mr^2...
  28. 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
  29. R

    B Confusion about the radius unit vector in spherical coordinates

    If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?
  30. Felipe Lincoln

    Conservative force in spherical coordinates

    Homework Statement Is ##F=(F_r, F_\theta, F_\varphi)## a conservative force? ##F_r=ar\sin\theta\sin\varphi## ##F_\theta=ar\cos\theta\sin\varphi## ##F_\varphi=ar\cos\varphi## Homework Equations ##\nabla\times F=0## The Attempt at a Solution In this case we have to use the curl for spherical...
  31. M

    B Triple integral in spherical coordinates.

    While deriving the volume of sphere formula, I noticed that almost everyone substitute the limits 0 to 360 for the angle (theta) i.e the angle between the positive x-axis and the projection of the radius on the xy plane.Why not 0to 360 for the angle fi (angle between the positive z axis and...
  32. 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...
  33. L

    I What is dx, dy and dz in spherical coordinates

    What is dx, dy and dz in spherical coordinates
  34. 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...
  35. Another

    Vector potential in spherical coordinates

    in this problem i can solve v = ω x r = <0, -ωrsinψ, 0> in cartesian coordinates but i don't understand A in sphericle coordinates why? (inside) A = ⅓μ0Rσ(ω x r) = ⅓μ0Rσωrsin(θ) θ^ how to convert coordinate ?
  36. R

    MHB Spherical coordinates and triple integrals

    Suppose $\displaystyle f = e^{(x^2+y^2+z^2)^{3/2}}$. We want to find the integral of $f$ in the region $R = \left\{x \ge 0, y \ge 0, z \ge 0, x^2+y^2+z^2 \le 1\right\}$. Could someone tell me how we quickly determine that $R$ can be written as: $R = \left\{\theta \in [0, \pi/2], \phi \in [0...
  37. M

    A PDE in Spherical Coordinates

    Given the PDE $$f_t=\frac{1}{r^2}\partial_r(r^2 f_r),\\ f(t=0)=0\\ f_r(r=0)=0\\ f(r=1)=1.$$ We let ##R(r)## be the basis function, and is determined by separation of variables: ##f = R(r)T(t)##, which reduces the PDE in ##R## to satisfy $$\frac{1}{r^2 R}d_r(r^2R'(r)) = -\lambda^2:\lambda^2 \in...
  38. 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...
  39. Philosophaie

    I Convert from rectangular to Spherical Coordinates

    How do you convert this to Spherical Components? Spherical Convention = (radial, azimuthal, polar) ##\vec r = |\vec r| * \cos{(\theta)} * \sin{(\phi)} * \hat x +|\vec r| * \sin{(\theta)} * \sin{(\phi)} * \hat y +|\vec r| * \cos{(\phi)} * \hat z## Is this correct? ##\vec r =\sqrt{(x^2 + y^2 +...
  40. karush

    MHB 15.5.63 - Rewrite triple integral in spherical coordinates

    Write interated integrals in spherical coordinates for the following region in the orders $dp \, d\theta \, d\phi$ and $d\theta \, dp \, d\phi$ Sketch the region of integration. Assume that $f$ is continuous on the region \begin{align*}\displaystyle...
  41. LyleJr

    I Derivation of the Laplacian in Spherical Coordinates

    Hi all, Sorry if this is the wrong section to post this. For some time, I have wanted to derive the Laplacian in spherical coordinates for myself using what some people call the "brute force" method. I knew it would take several sheets of paper and could quickly become disorganized, so I...
  42. C

    Improper integral with spherical coordinates

    Homework Statement I have a question. I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent. x^2y^2z^2/r^(17/2) * f(x,y,z)dV. Homework Equations...
  43. 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 $$\int_{\frac...
  44. K

    Equation for finding the gradient in spherical coordinates

    <Mentor note: moved from a technical forum and therefore without template>So I´m trying to understand how to use the equation for finding the gradient in spherical coordinates, just going from cartesian to spherical seemed crazy. Now I´m at a point where I want to try out what I have read and I...
  45. J

    Finding the curl of velocity in spherical coordinates

    Homework Statement The angular velocity vector of a rigid object rotating about the z-axis is given by ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point. a.) Assuming that ω is constant, evaluate v...
  46. 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
  47. Dopplershift

    Need Help With Gradient (Spherical Coordinates)

    Homework Statement Find te gradient of the following function f(r) = rcos(##\theta##) in spherical coordinates. Homework Equations \begin{equation} \nabla f = \frac{\partial f}{\partial r} \hat{r} + (\frac{1}{r}) \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{rsin\theta}...