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.

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  1. V

    PDE equation in spherical coordinates

    I am looking for ideas on how to solve this equation: \nabla \cdot \left( \vec{A} + F \hat{b} \right) = 0 where \vec{A} and \hat{b} are known vectors of (r,\theta,\phi) and F is the unknown scalar function to be determined. Also, \nabla \cdot \hat{b} = 0. So the equation can also be expressed...
  2. J

    Explaining Spherical Coordinates and Coordinate Vectors

    Homework Statement (a) For spherical coordinates, show that \hat{\theta} points along the negative z-axis if \theta = 90°. (b) If \phi also equals 90°, in what direction are \hat{r} and \hat{\phi}?Homework Equations The Attempt at a Solution can i just explain this in words.. like for a...
  3. Roodles01

    Vector addition; spherical coordinates

    Should be quite easy, really, given that it's just adding things together, hey ho. Problem a position vector of point (1), identified by sherical coordinates, is 5m away from point (2). I have a unit vector R1,2 identified by spherical coordinates [Aex - Bey +Cez], giving the direction to...
  4. N

    Finding surface area of cone in spherical coordinates

    Hello everyone, I recently tried to find the surface area of a hollow cone (there is no base, like an ice cream cone) using spherical coordinates. With cylindrical coordinates I was able to do this easily using the following integral: \int \int \frac{R}{h}z \sqrt{\frac{R^{2}}{h^{2}} + 1}...
  5. K

    Setting up a triple integral with spherical coordinates

    Homework Statement http://img28.imageshack.us/img28/7118/capturenbc.jpg Homework Equations x2 + y2 + z2 = p2 http://img684.imageshack.us/img684/3370/eq0006m.gif The Attempt at a Solution Using the relevant equations I converted the given equation to: ∫∫∫e(p3/2) * p2 *...
  6. K

    Triple integral spherical coordinates.

    Homework Statement Here is the question given: Homework Equations The Attempt at a Solution So i set p as x^2 + y^2 + z^2 so p lies in between b and a. But how do i find the restrictions on the two angles, theta and phi?
  7. H

    Finding work done in spherical coordinates

    [b]1. Find the work done by the force F=r3*cos2\varphi*sin\varphi*\hat{r} + r3*cos\varphi*cos(2\varphi) \hat{\varphi} from the point (0,0,0) to (2,0,0) Homework Equations Work=\int F*dr where dr= dr\hat{r} + rd\varphi\hat{\varphi}The Attempt at a Solution When muliplying the line element, dr...
  8. R

    Deriving the square angular momentum in spherical coordinates

    Homework Statement I want to derive the square of the total angular momentum as shown here: http://en.wikipedia.org/wiki/Angular_momentum_operator#Angular_momentum_computations_in_spherical_coordinates Homework Equations The x,y, and z components of angular momentum are shown in the...
  9. X

    Lagrangian of a Particle in Spherical Coordinates (Is this correct?)

    Homework Statement a.) Set up the Lagrange Equations of motion in spherical coordinates, ρ,θ, \phi for a particle of mass m subject to a force whose spherical components are F_{\rho},F_{\theta},F_{\phi}. This is just the first part of the problem but the other parts do not seem so bad...
  10. M

    What are the Spherical Coordinates for a Quarter Ball Volume?

    Homework Statement I am having so much trouble with this one problem ( and spherical coordinates in general ). Any help would be amazing: ∫∫∫ 1 / √(x2+y2+z2) Over -4≤x≤4, 0≤y≤√(16-x2), 0≤z≤√(16-x2-y2) Homework Equations The Attempt at a Solution I know that rho2 will...
  11. M

    What is the Triple Integral for the Given Solid in Spherical Coordinates?

    Homework Statement Set up the triple integral for the volume of the given solid using spherical coordinates: The solid bounded below by the sphere ρ=6cosθ and above by the cone z=sqrt(x2+y2) Homework Equations The Attempt at a Solution I thought i had this set up right where ρ...
  12. D

    Spherical Coordinates for a Sphere with Variable Radius

    Homework Statement f(x,y,z) = 1 x^{2} + y^{2} + z^{2} ≤ 4z z ≥ \sqrt{x^2 + y^2} Homework Equations The Attempt at a Solution How can I know ρ if there is z variable? Do I just square root both 4 and z? For θ, since it is sphere it would be 0 ≤ θ ≤ 2\pi right? for \phi, is...
  13. T

    Spherical Coordinates Integral

    Homework Statement Using spherical coordinates, find the volume of the solid that lies within the sphere x2+y2+z2=4, above the xy-plane and below the cone z=√(x2+y2)Homework Equations The Attempt at a Solution This is what I have so far...
  14. X

    Graphing in spherical coordinates

    Homework Statement The question involves a triple integral, but I can figure that out once I know what this looks like visually. It is the graph of ρ = 1 + cos(∅) How exactly would I graph this? Homework Equations x = ρ * sin(∅) * cos(θ) y = \rho * sin(∅) * sin(θ) z = ρ * cos(∅)...
  15. romsofia

    Laplace equation in spherical coordinates

    Homework Statement Verify by direct substitution in Laplace's equation that the functions (2.19) are harmonic in in appropriate domains in ℝ2 Homework Equations (2.19)= {u_n(r, \theta)= \lbrace{1,r^{n}cos(n \theta), r^{n}sin(n \theta), n= 1, 2...; log(r), r^{-n}cos(n \theta), r^{-n}sin(n...
  16. R

    Spherical coordinates vector question

    I've no idea where to put this question but here it is I am trying to work through the examples our lecture has given in class and I wasn't getting them at all the first thing that confused me was \nabla . \underline{r} = 3 I tried this myself with \nabla . \underline{r} =...
  17. P

    Kinetic Energy in Spherical Coordinates? (For the Lagrangian)

    I'm doing a Lagrangian problem in spherical coordinates, and I was unsure how to express the kinetic energy, so I looked it up and wiki states it should be this: http://en.wikipedia.org/wiki/Lagrangian#In_the_spherical_coordinate_system Which would give me the correct answer, but I'm...
  18. J

    Partial derivative in spherical coordinates

    I am facing some problem about derivatives in spherical coordinates in spherical coordinates: x=r sinθ cos\phi y=r sinθ sin\phi z=r cosθ and r=\sqrt{x^{2}+y^{2}+z^{2}} θ=tan^{-1}\frac{\sqrt{x^{2}+y{2}}}{z} \phi=tan^{-1}\frac{y}{x} \frac{\partial x}{\partial r}=sinθ cos\phi then \frac{\partial...
  19. J

    Quick question with spherical coordinates and vectors

    So here's the question: An ant crawls on the surface of a ball of radius b in such a manner that the ants motion is given in spherical coordinates by the equations: r = b, \phi = \omegat and \vartheta = \pi / 2 [1 + \frac{1}{4} cos (4\omegat). Find the speed as a function at time t and the...
  20. B

    Dot product in spherical coordinates

    Homework Statement What is the dot product of two unit vectors in spherical coordinates?Homework Equations A∙B = ||A|| ||B|| cos(\theta) = cos(\theta)The Attempt at a Solution The above equation is the only relevant form of the dot product in terms of the angle \theta that I can find. However...
  21. J

    Forgotten my maths Simple 1D ODE, spherical coordinates

    Hi, I seem to have forgotten some of my math how-to, as I haven't done this in a while. Looking through my notes, Bird, Stewart and Lightfoot, Greenberg, etc. don't really help. My equation is this, at steady state: 0 = 1/r^2 ∂/∂r (D*r^2 ∂C/∂r) + P Where P is some production rate...
  22. Y

    Any grapth utility program that can plot graph with spherical coordinates?

    I want to study antenna patterns of different arrangements. I am looking for a very cheap software ( free is even better) to plot graph if I provide the \;R,\theta,\phi. Even if 2D plot would be helpful like keeping either \;\theta\;\hbox { or }\; \phi\; constant and vary the other angle to...
  23. M

    Deriving Sphere Volume using Spherical Coordinates: Why 0°-360°?

    I wanted to derive the volume of a sphere using triple integration with spherical coordinates, but instead of taking the limits of θ as (0° ≤ θ ≤ 180°), I chose to take (0° ≤ θ ≤ 360°), and therefore, for φ as (0° ≤ φ < 180°), Now of course the integral of sin(θ) from 0° to 360° is zero, and...
  24. W

    Multivariable calculus, Integral using spherical coordinates

    Homework Statement Using spherical coordinates, set up but DO NOT EVALUATE the triple integral of f(x,y,z) = x(x^2+y^2+z^2)^(-3/2) over the ball x^2 + y^2 + z^2 ≤ 16 where 2 ≤ z. Homework Equations x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ ρ^2 = x^2 + y^2 + z^2 ∫∫∫w...
  25. Y

    Clarification on curl and divergence in cylindrical and spherical coordinates.

    Divergence and Curl in cylindrical and spherical co are: \nabla \cdot \vec E \;=\; \frac 1 r \frac {\partial r E_r}{\partial r} + \frac 1 r \frac {\partial E_{\phi}}{\partial \phi} + \frac {\partial E_z}{\partial z} \;=\; \frac 1 {R^2} \frac {\partial R^2 E_R}{\partial R} + \frac 1 {R\;sin...
  26. U

    Spherical coordinates: volume bound by z=r andz^2+y^2+x^2=4

    Homework Statement Using spherical coordinares, find the smaller volume bounded by the cone z=r and the sphere z^2+y^2+x^2=4 Homework Equations x^2+y^2+z^2=4 ; rho=2, z=rhocosphi The Attempt at a Solution Shot in the dark: Tried function integrating (rho squared - rhocosphi)...
  27. A

    How can the curl be calculated in polar or spherical coordinates?

    Can anyone show me how you get the curl in polar or spherical coordinates starting from the definitions in cartesian coordianates? I haven't been able to do this.
  28. B

    Spherical coordinates and partial derivatives

    Hello! My problem is that I want to find (\frac{\partial}{{\partial}x}, \frac{\partial}{{\partial}y}, \frac{\partial}{{\partial}z}) in spherical coordinates. The way I am thinking to do this is...
  29. T

    Calculating Centre of Mass for Northern Hemisphere Using Spherical Coordinates

    Homework Statement Calculate the z-component of the centre of mass for a northern hemisphere of radius R with constant density \rho_0 > 0 using spherical coordinates (r,\theta, \varphi ) defined by: x(r,\theta, \varphi) = r\sin\theta\cos\varphi \;\;\;\;\;\;0 \leq r < \infty y(r,\theta...
  30. X

    Spherical Coordinates of a Point with Rectangular Coordinates

    Homework Statement Find the spherical coordinates of the point with rectangular coordinates (2√2, -2√2, -4√3) Homework Equations ? The Attempt at a Solution The textbook gives the answer as (8, -pi/4, 5pi/6) No idea how to get to this. Any help appreciated.
  31. O

    Cartesian torque to Spherical Coordinates

    I'm writing a function for Matlab and I'm trying to figure out how to apply a torque matrix in cartesian coordinates to an object in spherical coordinates. The short story is this: For interest's sake, a friend and I have written a function with creates a tree which random branch...
  32. S

    Triple integral from cartesian to spherical coordinates

    Homework Statement evaluate the following triple integral in spherical coordinates:: INT(=B) = (x^2+y^2+z^2)^2 dz dy dx where the limits are: z = 0 to z = sqrt(1-x^2-y^2) y = 0 to z = sqrt(1-x^2) x = 0 to x = 1 Homework Equations The only thing I know for sure is how to set...
  33. Y

    Help with conversion from rectangular to spherical coordinates

    This is not a homework. Actual this is part of my own exercise on conversion where \vec A = \vec B \;X\; \vec C and I intentionally set up B and C so the \theta_B \hbox { and } \theta_C \;=\; 60^o \; respect to z-axis: \vec B_{(x,y,z)} = (2,4, 2\sqrt{(\frac 5 3)}) \;\;\hbox { and }\;\...
  34. T

    How to express a circle in spherical coordinates

    I've got a unit sphere sitting at the origin. This sphere is cut by an arbitrary plane. I'm looking to find the equation of the circle that results from the intersection in spherical coordinates. This is for a computer program I'm writing, and I've already set it up to approximate this by...
  35. F

    Position in Spherical Coordinates

    Homework Statement This is a bit hard to describe without a decent picture (or a decent brain) but try to bare with me. Picture below shows two spheres, if the origin is at centre of A, and a line d joins the centre of the two spheres, how do I describe the position of a point r from each...
  36. P

    Changing to spherical coordinates

    Homework Statement Evaluate by changing to spherical coordinates \int^1_0\int^{\sqrt{1-x^2}}_0\int^{\sqrt{2-x^{2}-y^2}}_{\sqrt{x^{2}+y^2}}xydzdydx Homework Equations dz dy dz = {\rho}^{2}sin{\phi} The Attempt at a Solution This problem is quite simple to do in cylindrical...
  37. atomqwerty

    Cylindrical and spherical coordinates

    Homework Statement Write the vector D_{p}=2\partial/ \partial x-5\partial/ \partial y+3\partial/ \partial z \in T_{p}\Re^{3} in cylindrical and spherical coordinates Homework Equations NA The Attempt at a Solution x=r cost y=r sint z=z ...
  38. S

    Triple Integral, Spherical Coordinates

    Homework Statement Calculate: integral B [ 1/sqrt(x^2+y^2+(z-a)^2) ] dx dy dz , when B is the sphere of radius R around (0,0,0), a>R. Homework Equations The Attempt at a Solution I tried spherical coordinates for the integrand: x=rsin(p)cos(t) y=rsin(p)sin(t) z=rcos(p)+a The problem is when I...
  39. M

    Rotations in spherical coordinates

    I have a few questions about rotations. First off if i have two vectors r_{a,b}=(1,\theta_{a,b},\phi_{a,b}) And i define \Delta\theta=\theta_b-\theta_a and \Delta\phi=\phi_b-\phi_a. Then take the map T(1,\theta,\phi)=(1,\theta+\Delta\theta,\phi+\Delta\phi). Is T a rotation? I would...
  40. C

    How do you determine angle phi in spherical coordinates?

    In general, how do I determine the angle phi when you have to use spherical coordinates for integration?
  41. B

    Triple Integral in Spherical Coordinates

    Evaluate the integral by changing to spherical coordinates. Not sure how to go about figuring out the limits of integration when changing to spherical coordinates.
  42. J

    Identifying Surfaces in Spherical Coordinates

    Homework Statement \rho = sin\theta * sin\phi Homework Equations I know that \rho^{2} = x^{2} + y^{2}+z^{2} The Attempt at a Solution I tried converting it to cartesian coordinates but I can't seem to get a workable answer that way. I know that the answer is the sphere with radius...
  43. Telemachus

    Triple integral doubt, spherical coordinates

    Hi there. I have some doubts about this exercise. It asks me to use the appropriated coordinates to find the volume of the region indicated below. The region is determined by the first octant under the sphere x^2+ y^2+ z^2=16 and inside the cylinder x^2 +y^2=4x The first thing I did was to...
  44. T

    Integration and cylindrical and spherical coordinates

    Homework Statement I have three problems and I could really use some help. 1. Integrate the function f(x,y,z) = y over the part of the elliptic cylinder x^2/4 +y^2/9 = 1 that is contained in the sphere of radius 4 centered at the origin and such that x≥0, y≥ 0, z≥0. 2. Find the total...
  45. M

    Triple integral in spherical coordinates

    Homework Statement use spherical coordinates to calculate the triple integral of f(x,y,z) over the given region. f(x,y,z)= sqrt(x^2+y^2+z^2); x^2+y^2+z^2<=2z The Attempt at a Solution Once I find the bounds, I can do the integral. But I'm having trouble with the bounds of rho. This...
  46. T

    Confused by velocity in spherical coordinates

    Hello, I am trying to work out how you derive velocity in terms of spherical coordinates, could anyone point me in the direction of a simple and quite explicit derivation. I keep getting confused! Thanks.
  47. A

    Vector fields in cylindrical and spherical coordinates

    I am reading the Wikipedia entry http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates" . There, in particular I see this: Time derivative of a vector field To find out how the vector field A changes in time we calculate the time derivatives. In cartesian...
  48. V

    Position vector in spherical coordinates.

    I have not done this in a while and I am having a brain fart. Given: A wheel of radius R rotates with angular velocity Ct2 k\hat{} (lies in x-y plane, rotating about z). A point P on the circles is P(x,y,z) = (0,R,0) Ques: What is the position vector of point P in spherical coordinates...
  49. L

    Exact solution to advection equation in spherical coordinates

    I've been trying to find the exact solution to the advection equation in spherical coordinates given below \frac{\partial{\phi}}{\partial{t}} + \frac{u}{r^2}\frac{\partial{}}{\partial{r}}r^2\phi = 0 Where the velocity u is a constant. First I tried to expand the second term using product...
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