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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2
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16 views

How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R^d}\mid a<\|x\|_2<b\},$$ ...
-1
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0answers
12 views

Great circle distance on an ellipsoid [on hold]

Let's say I have a set of latitude and longitude (B,L on a reference ellipsoid WGS-84) and I also know the great circle distance (both in radians and meters) from my point to some point X on a sphere ...
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0answers
12 views

Create 3d rotation matrix given spherical coordinates? [on hold]

Is it possible to form a 3d rotation matrix given a spherical coordinates $\theta$ and $\phi$ ?
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1answer
28 views

Plotting Spherical Coordinates

I'm trying to plot the Poisson Kernel, where a = 1, so the resulting equation would be $$P(r,\theta) = \frac{1-r^2}{1-2r\cos(\theta)+r^2}$$ $0\leq r <a = 1 ,\textrm{ } -\pi \leq \theta \leq \pi$ ...
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1answer
26 views

Sperical coordinates and the divergence theorem

Use the divergence theorem to calculate $\int \int F\cdot dS$ where $F=<x^3,y^3,4z^3>$ and $S$ is the sphere $x^2+y^2+z^2=25$ oriented by the outward normal. I have found that ...
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0answers
25 views

Laplacian in spherical coordinates - brownian motion

Consider the Laplacian equation on the unit sphere, for a vector $f$. $\theta$ is polar angle, and $\phi$ is azimuthal angle. The Laplacian in spherical coordinate is : $$ \Delta f = {1 \over r^2} ...
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1answer
17 views

Point along great circle line (aka arc) closest to a target point on the ground

Given: an arc (aka a great circle line, not a straight-line) defined by two arbitrary end points (which I can express in lat/lon/altitude or earth-centered fixed (ECF) 3D 'cartesian' space). Think ...
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0answers
61 views

Computing the vector Laplacian in spherical coordinates using metric tensor

I want to compute the Laplacian of a vector in spherical coordinates using metric tensor. I started only with the first component but I have obtained a different formula! Let be $\vec{v} $ a vector ...
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0answers
22 views

Getting topological objects from the “cube” of $T^3$

One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$. To get ...
1
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1answer
65 views

Equation of a circle in spherical coordinates

What would be the equation of an arbitrary circle rotated along some angle theta around the X-axis in spherical coordinates? For simplicity we may assume that it is a circle with constant radius r. ...
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0answers
26 views

Find the volume of a shape using spherical coordinates.

Find the volume of the shape formed inside $x^2+y^2=z$ and $x^2+y^2=2$ using spherical coordinates. I know I need to do a triple integral but I can't quite get the integral limits.
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21 views

Minimizing arc length on unit sphere (geodesics)

I just completed a Calculus IV course and taught myself basic Calculus of Variations, and wanted to extend some of the basic principles of optimization from planes to surfaces. The arc length ...
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0answers
26 views

Let $r(s)$ be a regular closed curve that is in the sphere $S^2$. Prove that $\int_{\gamma}\tau(s)ds=0$ $\gamma$-map of this curve, $\tau$ torsion

Let $r(s)$ be a regular closed curve that is in the sphere $S^2$. Prove that $$\int_{\gamma}\tau(s)ds=0$$ $\gamma$-map of this curve, $\tau$ the function of torsion of this curve. Every year this ...
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0answers
50 views

Trajectory on a sphere

I've asked a question before concerning a parallel problem, and I read a wikipedia page on spherical caps (Nominal Animal), which gave me an idea to do the following: I have the Cartesian coordinates ...
1
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1answer
40 views

Is there a solution to the equation $tan({\phi})=\frac{0}{0}$

I've been reading about conversion from Cartesian ($x,y,z$) to Spherical (r, $\theta$, $\phi$) coordinates. The formula to find the value of ${\phi}$ is given as: $\tan({\phi})=\frac{y}{x}$ My ...
1
vote
1answer
271 views

How to solve this problem using spherical coordinates system?

The question is very simple: Volume inside the solid limited by:$ (X^2+Y^2+Z^2=16), (X^2+Y^2=4)$ using SPHERICAL coordinates system. The final answer however can be checked making a "cylindrical ...
1
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0answers
48 views

Estimating the distance between two coordinates but without using Euclidean distance

Bill opens up "Café Finder" on his phone, and it tells him that it will take him 10 minutes to get to his nearest Starbucks to grab a triple-shot frapa-crapa-flat-white, so he decides to walk. 20 ...
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0answers
13 views

Find the Radius of Sphere using TDOA

My goal is to calculate Position of impact using Trilateration. I followed this guide on wikipedia : Trilateration Wikipedia I don't know how to find the Radius R1,R2,R3.(Normally it is ...
9
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1answer
3k views

Rotating a point in spherical coordinates around Cartesian axis

If I have a point in spherical coordinates, and I rotate it around one of the Cartesian axes, what will be the new spherical coordinates for the point? Both spherical and Cartesian coordinate systems ...
0
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0answers
9 views

The smallest bounding sphere of a prolate spheroid domain

Let $\Omega\subset \mathbb{R}^3$ be a prolate spheroid domain. Denote by $d$ its interfocal distance and by $b$ the surface of the region occupied by $\Omega$. The question is how to prove that the ...
1
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1answer
26 views

Angle between arc of two points on a unit sphere and $xy$-plane [closed]

Suppose I have two points on a unit sphere whose spherical coordinates are $A(\theta_1,\phi_1)$ and $B(\theta_2,\phi_2)$, what is the angle between $xy$-plane and arc $AB$? Maybe I can draw a ...
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0answers
20 views

Conversion of laplacian form cartesian to spherical coordinates 2nd part

Using the same method of conversion of laplacian from cartesian to spherical coordinates and changing $\psi$ to $u$, I am trying to finish the demostration of the spherical laplacian. However, I have ...
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vote
2answers
29 views

Can someone please help me convert this triple integral to spherical coordinates?

I'm not sure how to approach the problem, though I know it needs to be broken up into two integrals before it can be evaluated based on the way the answer input is set up. I do not know how to start ...
1
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1answer
36 views

Write the equation $4x^{2}+4z^{2}=5$ in spherical coordinates.

Write the equation $4x^{2}+4z^{2}=5$ in spherical coordinates. I used the facts that $$ \begin{align} x&=ρ\sin\theta\cos\phi\;,\\ z&=ρ\cos\phi\;, \end{align} $$ And ended up with: $ 4 (ρ^2 ...
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1answer
29 views

sphere arc intersection

Given: an arc defined by two end points (which I can express in lat/lon/altitude or earth-centered fixed (ECF) 3D 'cartesian' space) a sphere defined by a center (lat/lon/alt or ECF) and a radius ...
5
votes
1answer
81 views

The centre of the earth

I'm a real beginner here (first post and first foray into math since high school, trying to catch up), so I'm going to try my best to explain my problem in mathematical terms then follow up with an ...
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2answers
764 views

converting kph and heading to xyz velocity vector

I am writing software (in C++) that is required to send out messages from our simulation system to another simulation system. Problem is we track the simulation object's current speed (kph) and ...
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0answers
36 views

Integration in spherical coordinates

I needed to solve the following integral on one of my exercise sheets, which seemed not too difficult: $ \phi(\vec{r}) = \dfrac{1}{4\pi\epsilon_0} \int\limits_0^{\infty} dr' \int\limits_0^{\pi} ...
2
votes
1answer
29 views

Find a circle on sphere using spherical distance

I have a sphere with radius $R$. On this sphere I also have a point $P_1$ written in spherical coordinates, so I know $\theta_1$, $\phi_1$ and $R$ for this point (same as on this picture). I also ...
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0answers
11 views

spherical radial transformation

several papers mentioned a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty \int_{\mathbf{z}'\mathbf{z} = 1} f(r\mathbf{z}) ~r^{n-1} ...
3
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2answers
5k views

building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $ x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta $ $$ \begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin ...
1
vote
1answer
20 views

Rewrite equation using cylindrical and spherical coordinates.

I want to rewrite the equation $z=x^2-y^2$ using cylindrical and spherical coordinates. The cartesian coordinates are of the form $(x,y,z)$. The spherical coordinates are of the form $(\rho, \theta, ...
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0answers
16 views

How to setup/evaluate a triple integral to show an interesting result in physics?

I know this isn't the physics forum, but the task i'm struggling with is purely mathematical. My task is as follows; Let $A$ be a sphere centered at origin with radius $R$ and assume $a \geq R$. ...
1
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1answer
56 views

Convergence of $\iiint \frac{dxdydz}{(x^{2}+y^{2}+z^{2})^\alpha}$

I'm studying the convergence of the following triple integral $$I(\alpha) = \iiint \limits_{\Omega_{3}}\frac{dxdydz}{(x^{2}+y^{2}+z^{2})^\alpha}\tag{*}$$ Where $$\Omega_{3} = \{(x, y, z) \in ...
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0answers
31 views

Marginals/conditionals of a normalized Gaussian vector

It is well - known that if $x=(x_1,...,x_n)^T\sim{N(0, \sigma^2I)}$, then its normalized version is uniformly distributed on the unit $n-1$ - sphere: $$ ...
1
vote
1answer
209 views

Triple integral to find the mass of the intersection between two spheres

I've got two unit spheres, one is centered at $(0,0,0)$ and the other at $(0,0,1)$, the intersection of these two spheres is my region $R$. I would like to find the integral: $$\iiint\limits_R ...
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0answers
20 views

Vectors and dot product in spherical coordinate system [duplicate]

Let $\vec{v_1}$ and $\vec{v_2}$ given: $\overrightarrow{V_1} = r_1\hat{u_r} + \theta_1\hat{u_\theta} + \phi_1\hat{u_\phi} \\ \overrightarrow{V_2} = r_2\hat{u_r} + \theta_2\hat{u_\theta} + ...
2
votes
1answer
66 views

Using the Dirac delta function to find the density of point masses/charges

Here is an example from a textbook: Suppose there is a unit charge or unit mass at the point $(x,y,z)=(-1,\sqrt{3},-2)$; then in rectangular coordinates, the ...
0
votes
1answer
18 views

What does this triple integral in spherical coordinate represent?

$$\int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{2\pi} r(b+r \cos\phi) d\phi d\theta dr$$ What does the above integral represent? I don't think it's volume of a figure, since either $\phi$ or $\theta$ should ...
2
votes
1answer
36 views

Triple integral in spherical coordinates in an example

I am not sure how to do this. I am given a function in spherical coordinates. $C$ is a normalization constant given by the triple integral. How can I find C and use that to do part (b)?
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1answer
68 views

Expressing regions in cylindrical and spherical coordinates

I need to express the following regions in both cylindrical and spherical coordinates. I am not sure what to do here exactly, in (a) for example, should I substitute the Cartesian equation of the ...
3
votes
0answers
115 views

Volume of $n$-dimensional spherical orthant in upper diagonal halfspace

Consider an $n$-dimensional Euclidean Space. Consider orthants in that space. Each orthant occupies $\frac{1}{2^n}$ of the volume of an $n$-dimensional unit sphere. Let's call that a spherical ...
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1answer
40 views

How to calculate distance between two points on the sphere?

I have two azimuth and altitude angles for two points on a sphere. How can I calculate the spherical distance between those points? And how can I deduce this formula from first equations?
2
votes
1answer
35 views

Solve $I=\iiint\limits_\Omega z^2 dv$ using spherical coordinate system, $\Omega: x^2 +y^2 + z^2 \le R^2 \cap x^2 +y^2 + z^2 \le 2Rz$

Question: Solve $I=\iiint\limits_\Omega z^2 dv$ using spherical coordinate system. $\Omega$ is the common part of $x^2 +y^2 + z^2 \le R^2 $ and $ x^2 +y^2 + z^2 \le 2Rz$. My attempt:Because $ r^2 \le ...
0
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0answers
25 views

Relationship between Normal coordinates and Spherical Coordinates

I am using the following coordinates on $S^3: (\psi, \theta, \phi)$ where $$\begin{cases}x_0 = \sin\psi,\\ x_1 = \sin\psi \cos\theta,\\ x_2 = \sin\psi \sin\theta \cos\phi,\\ x_3 = \sin\psi ...
0
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0answers
19 views

Composite Spherical Harmonics expansion and error propagation

Let's assume that $f$, $g$ and $t$ are three functions defined over the surface of a sphere. In particular $f$ is defined using $g$, $t$ and the integral operator as follows: ...
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0answers
17 views

angle between hrizontal and a line connecting the center of an oblate ellipse to a point in space

I would like to know how I can calculate the angle $\alpha$ in an oblate ellipse similarly to the sphere.
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1answer
40 views

Dot product in integral over spherical coordinates

Midway through solving a question, I have an intermediate integral: $$ G(\vec x,t)=\frac{c}{16 \pi^3} \iiint _{\mathbb{R}^3} \frac{\sin (tck)}{k} e^{i \vec{x} . \vec{k}} d^3 \vec k $$ Now what I ...
2
votes
0answers
44 views

Calculate the Angle between two vectors in 3d Spherical Coordinates

I have two vectors in spherical coordinates, both originating at the origin and both with the same magnitude equal to one. One is vertical: {1,0,0} and the other undefined: {Ms,Mt,Mp}. The other one ...