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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How to derive curl in spherical coordinates

This is one of those questions where I know I am making a dumb mistake someplace and I am trying to check where it is. $$ \begin{vmatrix} \frac{\bf\hat r}{r^2\sin\theta} & ...
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3answers
103 views

Writing triple integrals in spherical coordinates over nonspherical/nonconical regions

Defining upper and lower limits of integration for $\rho$, $\theta$, and $\phi$ is relatively easy when writing a triple integral in spherical coordinates if the region of integration is defined by ...
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1answer
114 views

Angled Spherical Sector - formulas?

I am glad to be here. First off, please excuse my almost saddening lack of knowledge. Math (in general) isn't my strong point. I might ask some really basic questions, you have been warned :) The ...
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0answers
52 views

How to rotate point on a spherical coordinate

I am looking for formula that shows hot to find the spherical coordinates of a point after the axis rotated. Assume that I am on earth and I have the coordinate of a point in lan/lot (which I believe ...
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1answer
57 views

Energy of particles on sphere and uniform rotation

I have a computer program containing some particles on a unit sphere, characterized by their positions $\{(\theta_i,\phi_i)\}$. They have a total energy given by the arc distance between particle ...
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1answer
105 views

Project point onto line in Latitude/Longitude

Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D. Diagram:
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71 views

A problem on vector calculus on spherical coordinate systems

I'm wondering whether I could get some idea on a problem I've been working on, here it is: h and g are two functions defined on the surface of a unit sphere, and their values depend on the ...
2
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1answer
195 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
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1answer
55 views

Consider the region shared by ρ=8cos(φ) and ρ=4. Find the volume of the region.

Consider the region shared by ρ=8cos(φ) and ρ=4. Find the volume of the region. I know it involves a triple integral, but do not understand how to set up the integral.
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2answers
539 views

How to calculate the angle between two vectors, defined by 3 points on the earth?

I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values. ...
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1answer
131 views

Integration in n-spherical coordinates

I'd like to compute the following integral: $$I = \int_{\mathbb{R}^n} {\rm d}^n x \; \frac{e^{i \vec x \cdot \vec k}}{\vec x^2}$$ My first step is to use generalized spherical coordinates and then I ...
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31 views

New direction of propagation of light

\begin{align} \mu'_x & = \frac{\sin\theta(\mu_x \mu_z \cos\varphi - \mu_y \sin\varphi)}{\sqrt{1-\mu_z^2}}+ \mu_x \cos\theta \\ \mu'_y & = \frac{\sin\theta(\mu_y \mu_z \cos\varphi + \mu_x ...
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3answers
749 views

Solve the triple integral $\iiint_D (x^2 + y^2 + z^2)\, dxdydz$

How does one go about solving the integral: $$ \iiint_D (x^2 + y^2 + z^2)\, dxdydz, $$ where $$ D=\{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + z^2 \le 9\}. $$ I believe I am supposed to convert to ...
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1answer
127 views

Divergence in spherical coordinates

On the one hand there is an explicit formula for divergence in spherical coordinates, namely: $$ \nabla \cdot \vec{F} = \frac{1}{r^2} \partial_r (r^2 F^r) + \frac{1}{r \sin \theta} \partial_\theta ...
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2answers
96 views

can a great circle route be predicted from initial condition?

Assuming the world is a sphere with no wind, can the great circle route of a vessel be predicted from the current position $\{\phi_i,\lambda_i\}$ and the current true course $\theta_i$? Presently, ...
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111 views

Find a common point that three lines meet.

I have a base 2D triangle with 3 lines coming out of each vertex with their own coordinate point (xyz) and a set distance, is it possible to calculate the specific point that they should meet? I also ...
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1answer
68 views

Finding a 3D co-ordinate using triangluation

I'm trying to find a real world coordinate where 3 spheres collide and interact. At the moment I have been able to set up my triangulation equations so that I can work out the 2D position of where my ...
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2answers
139 views

Maximum Gravity Around a Unit Sphere

Most simulations that involve planetary gravity use Newton's law of universal gravitation and treat planets like point-masses. This is very accurate at large distances, and fairly accurate all the ...
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1answer
123 views

Interpolating geographic coordinates

I have two geographic coordinates. Let's call them $A$ and $B$: A = latitude 41.34759, longitude -75.77415 B = latitude 41.34769, longitude -75.77404 My unknown ...
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1answer
308 views

$dxdydz \to -r^2\sin(\theta)\sin(\phi+\theta)dr d\phi d\theta$?

So I got this answer $-r^2\sin \theta\sin(\phi+\theta)dr d(\phi)d(\theta)$ which I think is wrong because I googled it and it must be $-r^2\sin\theta dr d\phi d\theta,$ but $\sin(\phi+\theta$) clearly ...
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1answer
75 views

Why there is discontinuity at Zenith in Spherical-coordinate system?

I tried to plot the following function in spherical-coordinate system : $$ r(\phi,\theta)=\left(\frac{\sin\phi}{\phi}\frac{\sin\theta}{\theta}\right)^2$$ (definition/references for ...
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2answers
212 views

Proving that $\nabla \times (U(r) \hat{r} = 0 $

I was just checking to see if I wsa doing this right, as it isn't a formal proof. Just showing the identity. Let $U(r) \hat{r}$ b a vector in spherical coordinates. Given that the vector is only ...
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2answers
116 views

Deriving equations of motion in spherical coordinates

OK, we've been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} ...
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2answers
81 views

Finding the the radius of a sphere

I'm having a hard time to find the radius of this sphere equation. I got the center correct, but I can't get the correct answer for the radius. I'm completing the square, but my solution is off. I ...
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239 views

Finding out the nodes from spherical symmetric equation for mathematica and then plotting it in gnu-plot

What I want to do is to draw a curve for the spherical partial differential equation for S at rho=0: \begin{equation} \frac{\partial^2S}{\partial \rho^2}+\frac{d-1}{\rho}\,\frac{\partial S}{\partial ...
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1answer
609 views

Reverse use of Haversine formula

Alright the title is not the best. What I want to do is to change the given parameters in Haversine's formula. If we know the lat,lng of two points we can calculate their distance. I found the ...
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1answer
733 views

How to calculate a Vector Field in Spherical Coordinates

I am having trouble with the following problem. I keep on getting a long unmanagable result - so any suggestion as to where I've gone wrong/how to do this would be a lifesaver! Please? Consider a ...
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1answer
187 views

Are spherical coordinates unique orthogonal coordinates on sphere?

Spherical coordinates on unit sphere are defined by the following transformation: $$\begin{cases}x=\sin\theta\cos\varphi\\ y=\sin\theta\sin\varphi\\ z=\cos\theta\end{cases}$$ Are these coordinates ...
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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 ...
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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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477 views

Express spherical coordinates with different centers in terms of each other.

Imagine that you have two spheres with a distance $R$ from one center to the other one. Now, it is well known how one would get the cartesian position vector of each point in sphere 1 by using ...
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2answers
663 views

Probability density function for radius within part of a sphere

I would like to find the probability density function for radius within a given section of a sphere. For example, suppose I specify $\pi / 4 < \theta < \pi / 3$ and $\pi /7 < \phi < \pi ...
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99 views

Transforming an integral over $R^n$ to a radius and a directional vector (aka Spherical-Radial)

In several papers by John Monahan and Alan Genz there's mention of a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty ...
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2answers
331 views

How to verify a conversion to spherical coordinates?

Is it possible to verify if a conversion of an integral in Cartesian coordinates to spherical coordinates was done correctly other than revising it looking for mistakes? I mean, is there some kind of ...
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56 views

“Great Circle” distance [duplicate]

Given two points on a sphere, then the "great circle distance" between two points is the length of the smallest arc of a great circle containing both points. Assume that $\Sigma$ is a sphere of radius ...
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535 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\| ...
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1answer
80 views

Exterior Product $d\Phi_1\wedge d\Phi_2$ and spherical coordinates

One short question: If $\Phi\colon\mathbb{R}^3\to\mathbb{R}^3$, defined by $$ \begin{pmatrix}r\\\vartheta\\\phi\end{pmatrix}\mapsto\begin{pmatrix}r\sin \vartheta\cos \phi\\r\sin \vartheta ...
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59 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, ...
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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 $d_n(j)$ ...
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3k views

Volume of a Cylinder Using Cylindrical Coordinates and Triple Integration

Calculate the Volume $V$ of a right circular cylinder of radius $a$ and height $h$, using cylindrical coordinates and triple integration.
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Nodes in spherical equations and graph matching

The graph for this spherical equation is, (equation no 44) $$\frac{d^2 S}{d \rho^2}+ \frac{D-1}{\rho}\frac{dS}{d \rho}-S +S^3=0.$$ What I didn't understand here is $S_0, S_1,S_2$ and $S_3$. ...
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1answer
136 views

Laplacian on Sphere of Function Only Depending on Angle Between Points

Consider a function $f:S^2 \to \mathbb{R}$ , with $S^2$ the unit $2$-sphere in $\mathbb{R}^3$. Let's say that $f$ depends only on the polar angle $\theta$ from the north pole (e.g., $f(r,\theta,\phi) ...
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1answer
348 views

Triple Integral over a shifted sphere

I am interested in the following: Let $f(x,y,z)$ be a given (known) function in Cartesian coordinates, and let $B_d(p_0) = \{ y: ||y-p_0|| < d \}$ (i.e., a sphere centered at p_0). I want to ...
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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 ...
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1answer
184 views

Convert spherical coordinates to Cartesian coordinates for a vector

So let's say I have a normalized vector $N$ given in cartesian coordinates and I have another normalized vector $V$, defined in spherical coordinates relative to the vector $N$. So $\theta_V$ is the ...
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3answers
122 views

Doubt with bounds and integrand of $\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{2}\rho^2\sin{\phi}d\rho d\phi d\theta$

Question as follows. Find the volume of the solid enclosed between the spheres $x^2+y^2+z^2=4$ and $x^2+y^2+z^2=4z \Leftrightarrow x^2+y^2+(z-2)^2=4$. I constructed the following integral and after ...
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1answer
235 views

Spherical coordinates of a unit vector around a normal $N$

So if I have a unit normal for a surface $N(x,y,z)$ and an incident unit vector $V(x,y,z)$ to that surface, how would I represent the vector V in spherical coordinates relative to the normal?
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162 views

Directions in spherical coordinates

Say I have a system with standard spherical coordinates. There's a man on that sphere and he's standing on the equator facing east. He chooses a random angle $0°-360°$ and turns that much in the clock ...
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1answer
338 views

Laplace-Beltrami on a sphere

I'm trying to compute the Laplace-Beltrami of the function $u(r,\varphi,\theta) = 12\sin(3\varphi)\sin^3(\theta)$ on a unit sphere. Note that $\varphi$ is the azimuth, i.e. $\varphi \in [0,2\pi]$ and ...
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2answers
215 views

Vector Picking on the Unit Sphere

Imagine a vector from the center of a unit sphere to its surface: Now imagine a second vector generated in indentical fashion. Given the first vector, how can I generate vectors to uniformally ...