Spherical coordinates Definition and 351 Threads

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.

View More On Wikipedia.org
  1. K

    The length of a path on a sphere (in spherical coordinates)

    So, I'm to show that in spherical coordinates, the length of a given path on a sphere of radius R is given by: L= R\int_{\theta_1}^{\theta_2} \sqrt{1+\sin^2(\theta) \phi'^2(\theta)}d\theta, where it is assumed \phi(\theta), and start coordinates are (\theta_1,\phi_1) and (\theta_2, \phi_2)...
  2. Hepth

    Massive Vector Polarizations in Spherical Coordinates

    I can't seem to find one, but does anyone have a reference to the fourvector polarizations for a massive vector particle in spherical coordinates where a momentum is defined as p = \{E, |\vec{p}| \sin \theta \sin \phi, |\vec{p}|\sin \theta \cos \phi , |\vec{p}| \cos \theta\} theta goes...
  3. I

    How to evaluate this nabla expression in spherical coordinates?

    I'm currently working out the Schrödinger equation for a proton in a constant magnetic field for a research project, and while computing the Hamiltonian I came across this expression: (\vec{A}\cdot\nabla)\Psi where \Psi is a scalar function of r, theta, and phi. How do you evaluate this...
  4. M

    How do spherical coordinates work for finding volume in a given region?

    Homework Statement Find VR_{z}^{2} = \int \!\!\! \int \!\!\! \int_{E} (x^{2} + y^{2})dV given a constant density lying above upper half of x^{2}+y^{2} = 3z^{2} and below x^{2}+y^{2}+z^{2} = 4z.Homework Equations The Attempt at a Solution Why does it say upper half of x^{2}+y^{2} = 3z^{2}? It's...
  5. E

    Triple Integration from Rectangular to Spherical Coordinates

    Homework Statement Convert the integral from rectangular coordinates to spherical coordinates 2 √(4-x^2) 4 ∫ ∫ ∫ x dz dy dx -2 -√(4-x^2) x^2+y^2 Homework Equations x=ρ sin∅ cosθ y=ρ sin∅ cosθ z=ρ cos∅ In case the above integrals cannot be understood: -2...
  6. T

    Patch of a surface in spherical coordinates?

    Homework Statement I am currently trying to prove: S = ∫∫a2sinΦdΦdθ Here is my work (note that in my work I use dS instead of S, this is an accident): I end up with: S = ∫∫a*da2sinΦdΦdθ Where da is the infinitesimal thickness of the surface. Why am I getting the wrong answer?
  7. A

    Surface Area of a Sphere in Spherical Coordinates; Concentric Rings

    Hey, folks. I'm trying to derive the surface area of a sphere using only spherical coordinates—that is, starting from spherical coordinates and ending in spherical coordinates; I don't want to convert Cartesian coordinates to spherical ones or any such thing, I want to work geometrically...
  8. M

    Kinetic Energy in Spherical Coordinates

    Homework Statement Derive the expression for kinetic energy of a classical particle in spherical coordinates. Homework Equations I believe the answer I am supposed to reach is: T=\frac{1}{2} m (\dot{r}^2 + r^2\dot{\theta^2} + r^2\dot{\phi ^2}sin^2\theta) The Attempt at a Solution...
  9. Coelum

    Inverse Jacobi Matrix in Spherical Coordinates

    Dear all, I am reading R.A. Sharipov's Quick Introduction to Tensor Analysis, and I am stuck on the following issue, on pages 38-39. The text is freely available here: http://arxiv.org/abs/math/0403252. If my understanding is correct, then the Jacobi matrices for the direct and inverse...
  10. A

    Rotation of Gridded Spherical Coordinates to the Same Grid

    I have a uniform grid of data in spherical coordinates. e.g. theta = 0, 1, 2, ... 180 and phi = 0, 1, 2, ... 359 which forms a 2D matrix. I wish to rotate these points around a cartesian axis (x, y, z-axis) by some angle alpha. To accomplish this I currently do the following: 1. Convert to...
  11. lonewolf219

    Find La Placian of a function in cartesian and Spherical Coordinates

    Homework Statement Prove the La Placian of V(x,y,z)=(zx^{2})/(x^{2}+y^{2}+z^{2}) in Cartesian coordinates is equal to that in Spherical coordinatesHomework Equations \nabla^{2}V=0 The Attempt at a Solution I have attempted to calculate all the terms out, and there were A LOT. I was hoping...
  12. M

    Volume of a cone using spherical coordinates with integration

    Find the volume of a cone with radius R and height H using spherical coordinates. so x^2 + y^2 = z^2 x = p cos theta sin phi y= p sin theta sin phi z= p cos phi I found theta to be between 0 and 2 pie and phi to be between 0 and pie / 4. i don't know how to find p though. how...
  13. E

    Cylindrical / Spherical Coordinates

    I'm trying to convert the below Cartesian coordinate system into cylindrical and spherical coordinate systems. For the cylindrical system, I had r,vector = er,hat + sint(e3,hat). While I do have a technically correct answer for the spherical coordinate system, I believe, I was wondering if there...
  14. J

    Graphing Covariant Spherical Coordinates

    I am studying Riemannian Geometry and General Relativity and feel like I don't have enough practice with covariant vectors. I can convert vector components and basis vectors between contravariant and covariant but I can't do anything else with them in the covariant form. I thought converting the...
  15. L

    Finding limits on spherical coordinates

    Homework Statement find the limits on spherical coordinates. where ε is the region between z²=y²+x² and z = 2(x²+y²) no matter what i try i can't seem to find the limits, especially for "ρ", so far i got 0<θ<2Pi and 0<φ<Pi.
  16. B

    How to graph spherical coordinates

    Homework Statement given I=∫∫∫ρ^3 sin^2(∅) dρ d∅ dθ the bounds of the integrals: left most integral: from 0 to pi middle integral: from 0 to pi/2 right most integral: from 1 to 3 i have no idea how to graph this, i was hoping someone would be able to recommend some techniques.
  17. T

    Solving Poisson-Boltzmann equation in Cylindrical and Spherical Coordinates

    Homework Statement I don't have a specific problem in mind, it's more that I forgot how to solve the particular equation from first principles. \nabla^{2} \Phi = k^{2}\Phi Places I've looked so far have just quoted the results but I would like the complete method or the appropriate...
  18. M

    Understanding Spherical Coordinates

    questioning what ρ does. What is the difference between the two equations? Let k be the angle from the positive z-axis and w be the angle from the pos x-axis parametric equation of a sphere with radius a paramet eq. 1: x = asin(k)cos(w) y = asin(k)sin(w) z= acos(k) 0≤w≤2pi 0≤k≤pi...
  19. M

    Stress Tensor in Spherical Coordinates

    Homework Statement Calculate the deformation of a sphere of radius R and density \rho under the influence of its own gravity. Assume Hooke's law holds for the material. Homework Equations Not applicable; my question is simply one of understanding. The Attempt at a Solution I want...
  20. F

    Evaluate integral by using spherical coordinates

    ∫03∫0sqrt(9-x2)∫sqrt(x2+y2)sqrt(18-x2-y2) (x2+y2+z2)dzdxdy x=\rhosin\varphicosθ y=\rhosin\varphisinθ z=\rhocos\varphi Change the integrand to \rho and integrate wrt d\rhodθd\varphi I don't know how to find the limits of integration. Normally I would draw a picture and reason it out...
  21. W

    Spherical Coordinates Question

    In spherical coordinates we have three axes namely r, θ, ∅ the ranges of these axes are 0≤r≤∞ 0≤θ≤∏ 0≤∅≤2∏ what will happen in a physical situation if we allow θ to change from zero to 2∏
  22. L

    Solid hemisphere center of mass in spherical coordinates

    Hello, I am struggling with what was supposed to be the simplest calc problem in spherical coordinates. I am trying to fid the center of mass of a solid hemisphere with a constant density, and I get a weird result. First, I compute the mass, then apply the center of mass formula. I divide...
  23. D

    Triple integral in spherical coordinates

    Homework Statement The problem is to calculate the volume of the region contained within a sphere and outside a cone in spherical coordinates. Sphere: x2+y2+z2=16 Cone: z=4-√(x2+y2) Homework Equations I am having difficulty converting the equation of the cone into spherical coordinates...
  24. P

    Laplacian in Spherical Coordinates

    Homework Statement Homework Equations All above. The Attempt at a Solution Tried the first few, couldn't get them to work. Any ideas, hopefully for each step?
  25. K

    Help with a triple integral in spherical coordinates

    Homework Statement Use spherical coordinates. Evaluate\int\int\int_{E}(x^{2}+y^{2}) dV where E lies between the spheres x2 + y2 + z2 = 9 and x2 + y2 + z2 = 25. The attempt at a solution I think my problem may be with my boundaries. From the given equations, I work them out to be...
  26. B

    Scalar product in spherical coordinates

    Hello! I seem to have a problem with spherical coordinates (they don't like me sadly) and I will try to explain it here. I need to calculate a scalar product of two vectors \vec{x},\vec{y} from real 3d Euclidean space. If we make the standard coordinate change to spherical coordinates we can...
  27. D

    MHB Boundary conditions spherical coordinates

    Laplace axisymmetric $u(a,\theta) = f(\theta)$ and $u(b,\theta) = 0$ where $a<\theta<b$. The general soln is $$ u(r,\theta) = \sum_{n=0}^{\infty}A_n r^n P_n(\cos\theta) + B_n\frac{1}{r^{n+1}}P_n(\cos\theta) $$ I am supposed to obtain $$ u(r,\theta) = \sum_{n =...
  28. S

    Hamiltonian in spherical coordinates

    Homework Statement The total energy may be given by the hamiltonian in terms of the coordinates and linear momenta in Cartesian coordinates (that is, the kinetic energy term is split into the familiar pi2/2m. When transformed to spherical coordinates, however, two terms are angular momentum...
  29. S

    Conversion of energy expression from Cartesian to spherical coordinates

    A text I am reading displays the attached image. Can someone explain the general method for obtaining the velocity analogues of those terms (in parentheses) in 1.5? I know the second and third terms in parentheses in 1.6 and 1.7 are the squares of angular velocities, but can a general procedure...
  30. T

    Cylindrical and Spherical Coordinates Changing

    Homework Statement Convert the following as indicated: 1. r = 3, θ = -π/6, φ = -1 to cylindrical 2. r = 3, θ = -π/6, φ = -1 to cartesian The Attempt at a Solution I just want to check if my answers are correct. 1. (2.52, -π/6, 1.62) 2. (-2.18, -1.26, 1.62)
  31. R

    Spherical coordinates, vector field and dot product

    Homework Statement Show that the vector fields A = ar(sin2θ)/r2+2aθ(sinθ)/r2 and B = rcosθar+raθ are everywhere parallel to each other. Homework Equations \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0) The Attempt at a Solution So, if the dot product equals 1. They should be...
  32. Z

    Trouble understanding meaning of triple integral in spherical coordinates

    Homework Statement Evaluate \iiint\limits_B e^{x^2 + y^2 + z ^2}dV where B is the unit ball. Homework Equations See above. The Attempt at a Solution Does this evaluate the volume of f(x, y, z) within the unit ball (i.e. anything falling outside the unit ball is discarded)...
  33. A

    Expressing Spherical coordinates in terms of cylindrical

    Homework Statement I'm trying to express spherical coordinates in terms of cylindrical and vice versa. I would appreciate it if someone could give me some feedback on my attempt at a solution. Thanks for the help! The Attempt at a Solution Spherical(cylindrical) r=(ρ^2+z^2)^(1/2)...
  34. E

    Volume integral of an ellipsoid with spherical coordinates.

    Homework Statement By making two successive simple changes of variables, evaluate: I =\int\int\int x^{2} dxdydz inside the volume of the ellipsoid: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=R^{2} Homework Equations dxdydz=r^2 Sin(phi) dphi dtheta dr The...
  35. G

    Mathematica [Mathematica] Solving Heat Equation in Spherical Coordinates

    Hello Folks, I have this equation to solve (expressed in LaTeX): \frac{\partial{h}}{\partial t} = \frac{1}{n} \left[ \frac{1}{r^2 \sin^2{\phi}} \frac{\partial}{\partial \theta} \left( K \frac{\partial h}{\partial \theta} \right) + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi}...
  36. G

    Metric tensor in spherical coordinates

    Hi all, In flat space-time the metric is ds^2=-dt^2+dr^2+r^2\Omega^2 The Schwarzschild metric is ds^2=-(1-\frac{2MG}{r})dt^2+\frac{dr^2}{(1-\frac{2MG}{r})}+r^2d\Omega^2 Very far from the planet, assuming it is symmetrical and non-spinning, the Schwarzschild metric reduces to the...
  37. G

    Integrating the metric in 3-D Spherical coordinates

    Guys, I read that integrating the ds gives the arc length along the curved manifold. So in this case, I have a unit sphere and its metric is ds^2=dθ^2+sin(θ)^2*dψ^2. So how to integrate it? What is the solution for S? Note, it also is known as ds^2=dΩ^2 Thanks!
  38. T

    Triple Integrals: Spherical Coordinates - Finding the Bounds for ρ

    Homework Statement Find the volume of the solid that lies above the cone z = root(x2 + y2) and below the sphere x2 + y2 + x2 = z. Homework Equations x2 + y2 + x2 = ρ2 The Attempt at a Solution The main issue I have with this question is finding what the boundary of integration is for ρ. I...
  39. E

    Converting to Spherical Coordinates then integrating? Am I doing this right?

    Converting to Spherical Coordinates...then integrating? Am I doing this right? Homework Statement Consider the integral ∫∫∫(x2z + y2z + z3) dz dy dx, where the left-most integral is from -2 to 2, the second -√(4-x2) to √(4-x2) and the right-most integral is from 2-√(4-x2-y2) to...
  40. E

    Derivation of heat transfer equation for spherical coordinates

    Homework Statement where λ= thermal conductivity \dot{q}= dissipation rate per volume Homework Equations qx=-kA\frac{dT}{dx} The Attempt at a Solution I don't know where to start from to be honest, so any help would be greatly appreciated
  41. M

    Evaluate the triple integral (with spherical coordinates)

    Homework Statement Firstly sorry for my bad english,i have a one question for you(İ try it but i didn't solve it ) Homework Equations The Attempt at a Solution i know problem will be solved spherical coordinates but i don't know how i get angles (interval) theta and fi ...
  42. R

    Volume in spherical coordinates

    Homework Statement Calculate volume of the solid region bounded by z = √(x^2 + Y^2) and the planes z = 1 and z =2 Homework Equations The Attempt at a Solution
  43. M

    Cross product in spherical coordinates.

    Homework Statement i am trying to solve for the magnetic torque a circular loop of radius R exerts on a square loop of side length b a distance r away. The circular loop has a normal vector towards the positive z axis, the square loop has a normal towards the +y axis. The current is I in both...
  44. H

    Which version of spherical coordinates is correct?

    ∅θ,θI've come across two distinct 'versions' of the spherical coordinates. Could someone tell me which is correct or if both are fine. Version 1: A spherical coordinate is (rho,θ,∅) x=rhocos(θ)sin(∅) ; y=rhosin(θ)sin(∅) ; z=rhocos(θ) Version 2: A...
  45. L

    Some expressions with Del (nabla) operator in spherical coordinates

    Reading through my electrodynamics textbook, I frequently get confused with the use of the del (nabla) operator. There is a whole list of vector identities with the del operator, but in some specific cases I cannot figure out what how the operation is exactly defined. Most of the problems...
  46. T

    Triple integral in spherical coordinates

    I want to check if I'm doing this problem correctly. Homework Statement Region bounded by x^2+y^2=4 and bounded by the surfaces z = 0, and z=\sqrt{9-x^2-y^2}. Set up triple integrals which represent the volume of the solid using spherical coordinates. Homework Equations...
  47. D

    Integral Bounds Determination in Spherical Coordinates

    Homework Statement How to determine the integral bounds of phi in spherical polar coordinates. Please see my exact question at the end of page 2 of 2 in attachments. Homework Equations Please see my attachments The Attempt at a Solution Please see my attachments.
  48. Z

    Divergence of Spherical Coordinates

    Homework Statement Compute the divergence of v = (1/(r^2)) r where r = sin(u)cos(v)i + sin(u)sin(v)j + cos(u)k, r^2 = x^2 + y^2 + z^2 The Attempt at a Solution I can only think to express r as a function of x,y,z and do it. I know there's a simpler way though, but it's driving me...
  49. C

    Volume in Spherical Coordinates

    Homework Statement express a volume element dV= dx*dy*dz in spherical cooridnates.
  50. J

    Magnetic Field Equation in Spherical Coordinates to Cartesian Coordinates

    Homework Statement The magnetic field around a long, straight wire carrying a steady current I is given in spherical coordinates by the expression \vec{B} = \frac{\mu_{o} I }{2∏ R} \hat{\phi} , where \mu_{o} is a constant and R is the perpendicular distance from the wire to...
Back
Top