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Questions tagged [spherical-coordinates]

Questions on spherical coordinates, a three-dimensional coordinate system where a point is represented in terms of its distance from the origin, and its latitude and longitude angles (or complements thereof).

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Converting Integral to Spherical Coordinates with Unit Sphere

Can someone please explain how to convert this integral to spherical coordinates? Here, $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$ and $y=x+tw\in$ $B(x, 2r)$ $\subset \mathbb{R}^n$ where $0<t&...
Sjaikisan's user avatar
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$\langle v'',v\times v'\rangle $ is constant. Can you prove $v$ lies on a plane?

I tried to make the title short but of course there are additional hypothesis. Let $v:I\to \mathbb{S}^2$ be a regular curve parametrized by arc-length. This is to say that the tangent $v'$ is also a ...
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3 answers
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How to define a line in 3d space in terms of $\rho, \theta$, and $\phi$?

I have been struggling recently to come up with a solution for parameterizing a line to be swept by a vector in spherical coordinates. More specifically, imagine a line segment defined in the form $\...
anonymous user's user avatar
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Vector field of rotation around an axis in spherical coordinates

This is a qualifying exam question in Differential Geometry. I'm new to the subject and am reading up from John Lee's Introduction to Smooth Manifolds. Let $M=S^2\subset\mathbb{R}^3$ be the unit ...
giraffe's user avatar
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How to define complex valued spherical coordinates?

I am currently tackling an optimization problem involving complex valued vectors. However the optimization is solely about finding the optimal "direction" of the vector. So any (complex-...
Henri's user avatar
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1 answer
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Why couldn't I correctly use the spherical coordinate system when calculating this surface intergral?

The problem I've been trying to solve is this: Consider sphere surface $\Sigma: x^2+y^2+z^2=2ax(a \gt 0)$, find surface integral: $\iint_{\Sigma}(x^2+y^2+z^2)dS$. What I tried to do is to first ...
Edward Xu's user avatar
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1 answer
63 views

Best Coordinate system - Lagrangian problem

In $\Bbb R^3$ consider an heavy point $P$ whose mass $m$ on a circumference $\Gamma$ of radius $R$, centered in the origin. Now consider that $\Gamma$ lives in the plane $$\Pi = \{( x,y,z) \in \Bbb R^...
Turquoise Tilt's user avatar
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Integral over a sphere in $R^n$

Let $\sigma$ be the Lebesgue measure on the unit sphere $\mathbb S^{n-1}$ of $\mathbb R^n$. Let $\Sigma$ be a semi-definite positive symmetric matrix in dimension $n$. Is it possible to get a closed-...
Aristodog's user avatar
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Azimuth and Elevation angle in Spherical coordinates w.r.t to a given point

I am given a point $A = (x_b, y_b,z_b)$ and $C =(x_c, y_c, z_c)$ in cartesian co-ordinates. I am trying to compute the azimuth and elevation angle point $C$ make w.r.t point $A$. Can I compute this ...
wanderer's user avatar
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4 votes
1 answer
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Integral of $\exp(\langle x, a+ib\rangle)$ over hypersphere

I am looking to compute the following integral over $S_{n-1}$ the unit hypersphere in $\mathbb{R}^n$ \begin{equation} I(a,b) = \frac{1}{|S_{n-1}|}\int_{S_{n-1}}e^{\langle x,a+ib \rangle}dx \end{...
QLoop's user avatar
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How to convert to the Cartesian coordinates from new spherical coordinates after the rotation of spherical coordinate axes?

In a spherical coordinate system, an arbitrary point on the unit sphere is represented by the coordinates $(\theta, \phi)$. Although the $\theta$ and $\phi$ values are defined with respect to the $x$-,...
kachigusa's user avatar
  • 101
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1 answer
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Identify and describe the surface described by the equation: φ = c where π/2 < c < π.

In this context, φ refers to the polar angle between the z-axis and the radial line ρ as described by the spherical coordinate system with the following equations: x=ρcos(θ)sin(φ) y=ρsin(θ)sin(φ) z=...
largecoconutballs06's user avatar
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Prove isometric embedding of $S^2$ into $\mathbf R^5$

I'm trying to solve Exercise $132$ on the last page of this pdf Let $S^2$ denote the unit sphere. The map $f: \mathbf{R}^3 \rightarrow \mathbf{R}^5$ is given by $$ f(x, y, z)=\left(y z, z x, x y, \...
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Series Solution of Laplace Equation in Spherical Coordinates

I am a physics student and this question was asked on the Physics stack exchange as well. I just want you to go through the derivation first. I was recently Studying Griffiths Electrodynamics after a ...
Charu _Bamble's user avatar
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Three dimensional spherical interpolation like trilinear interpolation

Say we have a 5x5x5 grid where a quaternion, q, exists at every point. The objective is to express the distribution of quaternions in a functional form in terms of the positions, i.e. q(x,y,z) = Ax + ...
Jesse Feng's user avatar
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0 answers
72 views

How to find the tangent when converting from 2D angles to Spherical to Cartesian coordinates?

I am making a spherical/ball-in-socket joint, and I want to limit the movement of the bodies relative to each other. The limit is defined as 2 angles $\alpha$ and $\beta$ which make a 2D rectangle. I ...
Liburia's user avatar
1 vote
1 answer
25 views

A 3D integral (Hylleraas wave function)

In quantum mechanical context I would like to evaluate this function (Hylleraas type wave function for Helium atom ground state): $$ I(\boldsymbol{r}) = \int_{R^3} d^3 r' e^{- a_1 r'} \Vert \...
Fefetltl's user avatar
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Find position vector in spherical coordinates given Fx, Fy, Fz, Mx, My, Mz, and r [closed]

I've run into a bit of a roadblock figuring out this problem. I need to find the position vector given all three dimensions of force and moment as well as the radius of the sphere. I've gotten the ...
Jake1234's user avatar
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1 answer
35 views

Volume integral of vector field in spherical and cartesian coordinates

I am trying to reconcile the different results obtained when integrating a vector field in either spherical or cartesian coordinates. Take for example the vector field in spherical coordinates (...
ZehDeckel's user avatar
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1 answer
40 views

Region between two spheres in spherical coordinates

Given the spheres $$s_1\colon x^2+y^2+z^2=-2z\Leftrightarrow x^2+y^2+(z+1)^2=1$$ and $$s_2\colon x^2+y^2+z^2=-z\Leftrightarrow x^2+y^2+\left(z+\frac{1}{2}\right)^2=\frac{1}{4}$$ I know they are below ...
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0 votes
1 answer
33 views

Change vector's basis from 3d spherical to cartesian coordinates [duplicate]

I am writing a simulation program. I have a vector field in spherical coordinates which I need to transform into Cartesian coordinates. I understand how this works in $2D$ case - simple enough, I just ...
Maciej's user avatar
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1 vote
1 answer
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Confusion about the units of divergence

For a 3-D vector field $V = (u \hat{\textbf{i}} + v\hat{\textbf{j}} + w\hat{\textbf{k}})$: $$\nabla \cdot V\Big[\frac{m}{s}\Big] = \frac{\partial u [\frac{m}{s}]}{\partial x [m]} + \frac{\partial v[\...
Researcher R's user avatar
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0 answers
8 views

Given a vector field in spherical coordinates, compute the flux through a disk at z = -d

I want to compute the flux of the magnetic field $B = \frac{\mu_0m}{4\pi r^3}(2cos(\theta)\vec{e_r}+sin(\theta)\vec{e_\theta})$ through the disk at $z=-d$ with radius b centered around the z-axis. ...
Merkel_Bot's user avatar
0 votes
2 answers
42 views

Orthographic Projection and Concentric Circles [closed]

Let $C$ be the bounded set of concentric circles centered at the origin. Let $r_n = \sqrt{\frac{n}{\pi}}$ for any integer $n \in [1, k]$ be the radius of the circle $C_n \in C$. Assume $C$ is a ...
Aphrontos's user avatar
  • 115
2 votes
1 answer
67 views

Integration by substitution in Selberg's Integral

I am reading the article "Hilbert--Schmidt volume of the set of mixed quantum states" (https://arxiv.org/abs/quant-ph/0302197). I do not understand the step in which we start from (4.2) and ...
Zsombor's user avatar
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2 votes
1 answer
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What coordinate substitution should I perform to evaluate this triple integral?

I am trying to evaluate the following triple integral: \begin{equation} \int_{-1}^1 \int_{-\sqrt{4-4x^2}}^{\sqrt{4-4x^2}} \int_{\sqrt{4x^2 + z^2}}^2 ye^{4x^2 + y^2 + z^2} \, dy\, dz\, dx \end{equation}...
Christopher Miller's user avatar
1 vote
0 answers
52 views

Is this the correct equation for divergence of a horizontal vector field in spherical coordinates?

There is a Horizontal vector field $\textbf{V} = <u\hat{\lambda}+v\hat{\theta_{lat}} + 0\hat{\textbf{r}}>$ which is gridded along Earth's surface. Physically speaking, the vector field's ...
Researcher R's user avatar
0 votes
1 answer
140 views

How do I solve for surface area in this case?

Okay, I have the parametric equation in spherical coordinates for a sphere, a cone tangent to that sphere and a circle inclined with an angle $\Omega$ to the $zy$ plane. ( Desmos graph link ). I need ...
Anonymous001's user avatar
0 votes
1 answer
57 views

How to change coordinates of a differential operator?

Say for example, I start with $\frac{\partial}{\partial x}$ and want to change x to the spherical coordinate $$x = \rho\sin(\phi)\cos(\theta)$$ I know this isn't correct, by my brain immediately goes $...
Researcher R's user avatar
4 votes
0 answers
53 views

evaluate the volume of solid

Consider the paraboloid $(\mathcal{P}): z=x^2+y^2$ and the plane $(\mathcal{Q}): 2x+2y+z=2$. Let $\mathcal{S}$ be the solid region bounded above by $(\mathcal{Q})$ and below by $(\mathcal{P})$. Find ...
Student's user avatar
  • 319
7 votes
1 answer
185 views

Mysterious Coordinates on $S^4$ involving Quaternions

Let $U$ and $U'$ be $S^4 - x_N$ and $S^4 - x_S$, respectively, where $x_N$ is the North pole and $x_S$ is the South pole. The usual stereographic projection maps $U$ into $R^4$ and $U'$ into $R^4$. If ...
User175a23's user avatar
0 votes
0 answers
64 views

Satellite look angle to observer's geographic longitude and latitude

Suppose a satellite is above the Earth at a given elevation (sat_elevation), hovering above a certain geographic longitude (...
Coto TheArcher's user avatar
1 vote
1 answer
68 views

Interpolating points on a sphere between two points

I managed to solve it using the following function: given a cartesian point A and point B. the geodesic path on a sphere is defined as: r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1] then normalize r(t)/||...
Jenia Golbstein's user avatar
0 votes
2 answers
39 views

set the limits of integration of the spherical coordinates between two paraboloids and a plane

Find the volume of the solid $\mathcal{S}$ enclosed laterally by the paraboloids $\mathcal{P}_1$ of equation $z = x^2 + y^2$ and $\mathcal{P}_2$ of equation $z = 3(x^2 + y^2)$ and from above by the ...
Student's user avatar
  • 319
1 vote
2 answers
159 views

Spherical Tractrix

A pantoon P initially at North pole NP heading to Greenwich meridian intersection with equator, $( 0^{\circ} E, 0^{\circ} N $ ) towards point G moving down and east. It is dragged by a ship S moving ...
Narasimham's user avatar
3 votes
3 answers
89 views

Integrating $e^{-(x^2+y^2+z^2)/a^2}$ over $\mathbb{R}^3$

I want to compute the following integral $$ \int_{\mathbb{R}^3}e^{-(x^2+y^2+z^2)/a^2}\,dxdydz $$ and I thought that using spherical coordinates could make it easier, since $r^2=x^2+y^2+z^2$. With this ...
sam wolfe's user avatar
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1 vote
1 answer
107 views

Finding the coordinates of a moving point along the surface of a sphere.

I am trying to do some simulation of the binomial point process, and I want some help with the following problem. I am trying to find a formula for the final coordinates of a moving point on the ...
SecretKeeper's user avatar
1 vote
1 answer
32 views

Radial Coulomb's integral equality

Introduction In quantum mechanics, we have some integrals like this one (spherically symmetric electron-electron electrostatic repulsion): $$ I = \int_0^{\infty} r^2 R(r)^2 V(r) d r $$ with $R(r)$ the ...
Fefetltl's user avatar
  • 191
0 votes
0 answers
38 views

How can I curve a rectangular plane to match the surface of a sphere?

I have a flat rectangular plane representing an area of the earth. I am trying to bend this plane on all three axes to fit the surface of a sphere (in this case I am modeling the earth as a perfect ...
John's user avatar
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0 votes
3 answers
365 views

Find the equation of the sphere with points P such that the distance from P to A(−2, 4, 4) is twice the distance from P to B(6, 3, −1).

As you can tell from the title, I need to find the equation of the sphere with points $P$ such that $2|PB| = |PA|$. The coordinates for $A$ are $(-2, 4, 4)$ and the coordinates for $B$ are $(6, 3, -1)$...
baron's user avatar
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0 votes
0 answers
78 views

Spherical Trig: Finding A Missing Angle w/ The Sine Law

I've been working on these spherical trig questions that I initially thought were rather straightforward. Instead, my attempts to finish this assignment have descended into madness because I cannot--...
upas's user avatar
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0 votes
1 answer
60 views

Given latitude and longitude and facing north, how can I calculate the rotation needed to face another latitude and longitude (namely 0,0)? [closed]

As the title says, given I'm somewhere on earth facing north, I can determine the magnitude of the distance to get to 0°,0°, but I am not sure how to calculate the rotation needed to be facing 0°,0°. ...
bclax5's user avatar
  • 11
1 vote
1 answer
116 views

How to plot spherical harmonics? [closed]

Let me start by saying that I am only interested in the mathematical aspect of the thing. I would like to plot just for the fun of it the spherical harmonics that are used to plot the electronic ...
Charlie's user avatar
2 votes
1 answer
129 views

Integrating general powers of the Mahalanobis norm with respect to spherical measure

Let $S^{d-1}$ denote the $d$-dimensional sphere, and $\sigma_{d-1}$ denote the corresponding spherical measure. I am wondering how to go about solving the following integral for $c>0$ $$ \int_{S^{d-...
WeakLearner's user avatar
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0 votes
0 answers
15 views

How to define pdf of the distance to the point of the spherical cap?

Suppose we have a sphere centered at $(0, 0, 0)$, with the radius of $R_b$. We cut the sphere with the tangent plane centered at $(0, 0, R_a)$, where a dude is fixed on. (Here $0 < R_a < R_b$, ...
user1224303's user avatar
0 votes
2 answers
85 views

What location on a hypersphere maximizes the Manhattan distance of the radius?

In two dimension picture a unit circle. While the distance of the radius is constantly one, the Manhattan distance of the radius at zero degrees is equal to 1 while the Manhattan distance at 45 ...
Chair's user avatar
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3 votes
0 answers
101 views

Polar Coordinate Calculation of Hypercomplex Fractals

I've been doing some reading on how hypercomplex fractals are calculated using cartesian to n-spherical coordinate conversion. I've specifically been looking at these two sources: https://archive....
squirem's user avatar
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0 votes
0 answers
23 views

How to calculate the center of the circumcircle of the triangle formed by three geographic coordinates

Where is the center of those 3 geographic coordinates as an example: ...
O. Durand's user avatar
  • 109
2 votes
1 answer
299 views

How to find expectation value $p_y$ from the Bloch sphere?

Consider an arbitrary state: $$|\psi\rangle = a|0\rangle+b|1\rangle,$$ where $a=cos\left(\frac{\theta}{2}\right), b=sin\left(\frac{\theta}{2}\right)e^{i\phi}$ (neglecting global phase), $\phi$ is the ...
Curious's user avatar
  • 105
0 votes
0 answers
25 views

Calculating average over angles in n dimension

I have to calculate this integral $\int \frac{d S_q d S_p}{1 + a^2 +(\overline{q}+\overline{p})^2}$ Where $\overline{p},\overline{q}$ unit n-dimensional vectors. Integration goes over n-1 unit ...
Xian-Zu's user avatar
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