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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6
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105 views

When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar coordinates or spherical are just plain ...
5
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522 views

3d-diffusion equation in spherical coordinates (numerical), boundary problem

There is one boundary problem $$\frac{\partial u}{\partial t}= \operatorname{div}\left(a^2 E \nabla u\left(r,\varphi,\psi \right) \right) $$ in a ball $$ B_{1}(0)=\left\{x \in \mathbb{R^3}: \left\| ...
4
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0answers
109 views

$\operatorname{div}$ and $\operatorname{grad}$ in spherical coordinates. Formula from general relativity goes crazy

I try to calculate the gradient of a function and the divergence of a vector field in spherical coordinates. Nothing special so far, but a formula that I learned in a general relativity lecture ...
4
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0answers
68 views

Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
3
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0answers
20 views

Converting a polar integral to spherical

$$\int_0^{2\pi} \int_0^{\sqrt{2}}\int_r^{\sqrt{4-r^2}}\mathrm{d}z \, r \, \mathrm{d}r \, \mathrm{d}\theta$$ So in spherical this would become: $$\int_0^{2\pi} \int_0^{\pi/4}\int_0^2 \rho^2\sin\phi \, ...
3
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0answers
803 views

Spherical coordinates grad and div.

Struggling with the following: Prove the identity $$ \nabla = e_{r}(e_{r} \cdot \nabla) + e_{\theta}(e_{\theta} \cdot \nabla) + e_{\phi}(e_{\phi} \cdot \nabla).$$ Given the vector fields ...
2
votes
0answers
63 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
2
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0answers
98 views

How do you set up the integral in spherical coordinates in the following problem?

Find the volume bounded by the surface $z = x^2 + y^2$ and $x^2+y^2 = 1$ in the first quadrant. The answer is $\pi/8$ using rectangular and cylindrical coordinates and that is the correct answer, but ...
2
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0answers
57 views

Finding the coordinates of the corners of an aligned pole-centered spherical square

Given a spherical square of radius $1$, with edge midpoints at $(1, x, 0)$, $(1, x, \pi/2$), $(1, x, \pi)$ and $(1, x,3 \pi/2)$ (in the spherical coordinate system of (radial distance, polar angle, ...
2
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0answers
85 views

expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
2
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0answers
148 views

Diffusion in Spherical Coordinates with mixed BC

I have been working through the book "A Guide to First-Passage Processes" and wanted to branch out on my own doing a calculation similar to what occurs in chapter 6. My basic problem comes from the ...
2
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0answers
518 views

Inverse Jacobian matrix of spherical coordinates

I found inverse transformation from spherical coordinates to cartesian coordinates (on $x>0$, $y>0$ and $z>0$). I have $$ r = w_1(x,y,z) = \sqrt{x^2+y^2+z^2} $$ $$ \theta = w_2(x,y,z) = ...
2
votes
0answers
644 views

Calculate the volume between two spheres

I have to find the volume between the two spheres. Their equations are $x^2+y^2+z^2=2$ and $x^2+y^2+(z-\frac1 2)^2=\frac 1 4$ for $z>0$ The first one has center $(0,0,0)$ with $r=\sqrt 2$ and the ...
2
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0answers
496 views

How do I find the inverse Fourier transform of a function that is separable into a radial and an angular part?

I need to take the inverse Fourier transform of a function that is initially specified in spherical coordinates: $f(r, \theta, \phi) = \int_{R^3}F(k, ...
1
vote
0answers
15 views

Looking for algorithm for spherical point in polygon that works across meridian and anti-meridian

I need to process millions of latitude/longitude points every day to see if they are located within a defined lat/lon bounded polygon. The polygon may be rectangular, or it may be some irregular 3.. ...
1
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0answers
23 views

Parametrization of a bounded solid.

So, I have a solid bounded by $z=\sqrt{x^2+y^2}, z=\sqrt{1-x^2-y^2}, z=2$ I had to parametrize it using spherical coordinates so I used $$\begin{cases} x(\rho, \theta, ...
1
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0answers
46 views

Calculate a point on a geodesic line on an ellipsoid

I have a problem which i don't understand how to achieve. Maybe someone could sheed some light on it. Have a look at this picture: What I try to achieve is to determine the point D on the geodesic ...
1
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0answers
40 views

How to find Latitudes and Longitudes of projections of the vertices of a rectangular plane below earth's surface?

I want to find out the latitudes and longitudes of projections of the vertices of a rectangular plane inside the earth's surface. I know dimensions of rectangle, angles of orientation and latitude and ...
1
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0answers
24 views

Projecting non-central Gaussian on a sphere?

Suppose I have an $n$-dimensional Gaussian random variable with mean $\mu\in \mathbb{R}^n$ and covariance matrix $\sigma^2I_n$, where $I_n$ is the identity matrix. If I condition on the distribution ...
1
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0answers
47 views

volume in spherical coordinates

I am trying to find the region bounded by the sphere $p = 2\cos\psi$ and hemisphere $p=1$, $z\geq 0$. Not quiet sure not to do this problem, so please help.
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0answers
29 views

Proof of $\vec(r) \times \nabla$ in spherical coordinates

My professor claims, that $\vec{r} \times \vec{\nabla} = \vec{e}_{\varphi} \frac{\partial}{\partial \vartheta} - \vec{e}_{\vartheta} \frac{1}{r\sin \vartheta} \frac{\partial}{\partial \varphi}$ in ...
1
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0answers
55 views

Integrating Over Truncated Sphere

Consider the volume bounded by the sphere of radius R (with center $r_0$ = R/2 z) and the plane z = 0. Evaluate $\int dS$ over the surface of the truncated sphere. I'm not sure how to integrate over ...
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0answers
129 views

Conversion from Spherical Coordinates to Cartesian Coordinates aligned along arbitrary polar axis

I have spherical coordinates $w = (\theta, \phi)$ such that $\theta$ is the angle between $w$ and the polar axis (let's assume $z$ is up). Assuming $w$ is a unit vector, the conversion to cartesian ...
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0answers
67 views

Joint PDF for spherical region

A sphere has a coordinate system (r, $\theta$, $\phi$) with the origin at the center of the sphere. What is the joint PDF of the r and $\phi$ coordinates, $f_{r,\phi}(r,\phi)$, for a randomly ...
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328 views

Distance of two points in spherical coordinates

Today I was trying to solve a physics exercise and ran into some mathematical problems. Consider two concentric spheres $K_{1,2}$ with radii $R_{1,2}$. I wanna solve the following integral: $$E_{ij} ...
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0answers
367 views

Generating evenly distributed points on a sphere

How could I write an algorithm to generate n points distributed 'evenly' on a sphere? I already wrote an algorithm to generate points distributed uniformly on the surface (here), but by 'evenly' ...
1
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0answers
163 views

Centre of a spherical triangle

Suppose I have a triangle defined by 3 unit vectors {$v_1, v_2, v_3$} in a 3 dimensional complex inner product space. What would be the centre of such a triangle? I guess it should be something like ...
1
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0answers
497 views

Rotating co-ordinates in 3D

Suppose I have 3 axes, $x$, $y$, and $z$ such that $x$ is horizontal, $y$ is vertical, and $z$ goes in/out of the computer screen where $+$ve values stick out and $-$ve values are sunken in. Suppose ...
1
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0answers
66 views

Knowing coordinates of a point having two coordinates and the distance.

I have the two geographic coordinates of the lower corners of a wall. So, for example, i want to know what is the coordinate that is for example 15cm on the right of the lower corner left. Is that ...
1
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0answers
158 views

Cross-section of a circle with a three-dimensional Gaussian

Suppose I have a three-dimensional Gaussian with mean $\bar{\mu}$, volume $A$ and covariance matrix $\Sigma$ $$G(X)=\frac{A}{\sqrt{(2\pi)^{3}\det(\Sigma)}}e^{-\frac{1}{2}(X-\mu)^{T}\cdot ...
1
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0answers
151 views

Relative spherical coordinates of 2 points.

I have 2 points in space, defined by their spherical coordinates. I'd like to know the spherical coordinates of the second point in a reference system centered on the first point (I know the unit ...
1
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0answers
135 views

What is the name for the axis that “inclination angle” is measured from in spherical coordinates?

What is the name for the axis that "inclination angle" is measured from in spherical coordinates? I keep trying to call it "polar axis", and the other axis from which the azimuth is measured, the ...
0
votes
0answers
8 views

2D histogram from points on spherical surface of spherical coordinate system

What are the options available for creating a 2D histogram of points distribution on a sphere? Both polar angle (phi) and azimuth angle (theta) are known, and radial distance is 1.
0
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0answers
18 views

Rotation invariant method to compare points on spherical surface

Is there any rotation invariant method that I can use to compare the similarity between the three groups of "A" points as shown below?
0
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0answers
10 views

Intermediate vector naming convention when converting 3D Vector to Spherical Coord

Given the "Red" arrow as the vector of interest in the image below, is there a standard (or practical) name for the "Blue" vector? This is for the purpose of naming variables/methods related to the ...
0
votes
0answers
20 views

Finding the intersection of 2 coordinates in spherical coordinate system

Sorry in advance for messing up any math term or being confusing. I have the following data: lat1, lon1, alt1, v1, h1 and ...
0
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0answers
22 views

Volume of a shape using spherical coordinates and integrals

A solid is described in spherical coordinates by the inequality ρ ≤ sin(φ). Find its volume. So, I took ρ from 0 to sin(φ), φ from 0 to pi, and theta from 0 to 2Pi and formed the integral: integral ...
0
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0answers
25 views

From points on a latitude-longitude map to direction on hemisphere

I've wrote a program in matlab that following a given algorithm individuate some points on a latitude-longitude map. This is an example output: Now I need to map these points (in blue) to ...
0
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0answers
17 views

Transforming uncertainties in spherical coordinates

I'm not looking for a very rigorous answer, it's a practical problem that I've encountered and whichever solution is best will be determined by trying it out on my data. I have a measurement of an ...
0
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0answers
17 views

Unit vectors in spherical co-ordinate system

I see that a vector can be described in spherical co-ordinates, with respect to it's cartesian co-ordinates as $$ ...
0
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0answers
17 views

Interesting question about a measurement using $J^2$

I really dont understand how to do part d)iv) on this question. This seems strange as it is only worth 2 marks? What step am I missing, I feel this may be rather obvious to others. for part d)i) I ...
0
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0answers
33 views

grouping geographical points in spherical bins

I want to create spherical bins in the Earth's surface which will encompass already distributed geographical points for subsequent purposes. The two requirements are that the spherical bins should ...
0
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0answers
11 views

New Axis calculation

I have a task whereby I use an accelerometer to calculate acceleration for a vehicle. The problem I am attempting to solve is to allow the accelerometer to be in any oriertaion. Basically I have a ...
0
votes
0answers
57 views

Volume element in spherical coordinates

In spherical coordinates, we have $ x = r \sin \theta \cos \phi $; $ y = r \sin \theta \sin \phi $; and $z = r \cos \theta $; so that $dx = \sin \theta \cos \phi\, dr + r \cos \phi \cos \theta ...
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0answers
30 views

Showing the uniqueness of the solution to some problem?

I have faced problems proving all kinds of uniqueness theorems but this one I've come across seems particularly tricky to me. Can you help me? The function g(x,y,z) is zero for r²>a² (where ...
0
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0answers
48 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
0
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0answers
61 views

A basic question on surface area of spherical cap of sphere

Consider a sphere of radius 1. Now chop a spherical cap with latitude line $\phi$ at the bottom of the cap is removed from top (say $0<\phi<\frac{\pi}{2}$). I want to know the surface area of ...
0
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0answers
57 views

Mean value of a function over the n-sphere's surface.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
0
votes
0answers
224 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
0
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0answers
246 views

Volume enclosed by two spheres (triple integral, cylindrical coordinates)

The question: Find the volume of the solid enclosed by the sphere $x^2 + y^2 + z^2 - 6z = 0$ , and the hemisphere $x^2 + y^2 + z^2 = 49 , z ≥ 0$ I set up the triple integral ...