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:
(
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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
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...
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...
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...
lrcarr
Thread
dipole
sphere
sphericalcoordinates
surface charge density
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...
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
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...
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...
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*...
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...
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...
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...
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...
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...
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...
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...
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...
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...
Homework Statement
Homework Equations
The path integral equation, Stokes Theorem, the curl
The Attempt at a Solution
[/B]
sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First...
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...
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...
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...
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...
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...
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...