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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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
705 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 ...
2
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1answer
125 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 ...
2
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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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0answers
104 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
66 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
134 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
118 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
291 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
72 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
201 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
80 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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1answer
229 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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0answers
26 views

Cross product in spherical coordinates [duplicate]

Does anybody know how the cross product in spherical coordinates looks like? So I have a vector with $ae_r+be_{\theta} + ce_{\phi}$ with another vector $de_r+fe_{\theta} + ge_{\phi}$, how does there ...
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1answer
548 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 ...
4
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1answer
687 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 ...
4
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1answer
174 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 ...
2
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0answers
97 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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0answers
65 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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1answer
433 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
606 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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0answers
93 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
318 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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0answers
55 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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0answers
487 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
78 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 ...
2
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0answers
56 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
81 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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1answer
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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82 views

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) ...
2
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1answer
332 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 ...
2
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0answers
146 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
181 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
121 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
216 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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1answer
155 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
314 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 ...
2
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2answers
205 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 ...
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2answers
149 views

Spherically Symmetric Function

Suppose $f:\mathbb{R}^3\setminus B(0,1) \to \mathbb{R}$ is smooth and satisfies $f(S^2)=0$, i.e. the unit sphere is a level set of $f$. Does it necessarily follow that $f$ is a spherically symmetric ...
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0answers
60 views

Addition of spherical surface vectors

I'm making a planet simulator, which makes much use of a sphere. I'm trying to figure out how to represent and manipulate vectors on the surface of the sphere. Currently, my coordinates are all ...
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0answers
317 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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1answer
216 views

Transformation to spherical coordinate system

If I have a sphere $T: x^{2}+y^{2}+z^{2}\leqslant 10z$ by transformation to the spherical coordinate system by the: $ x=r\cos\theta\sin\varphi\\ y=r\sin\theta\sin\varphi\\ z=r\cos\varphi $ What is ...
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2answers
214 views

Numerical method to solve a trigonometric (cotangent) function - transient heat transfer problem

I was trying to develop a mathematical model for transient one-dimensional heat conduction of spheres using approximate analytical solution as mentioned in Cengel{refer page number 229 in that ...
3
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1answer
375 views

integrating a vector over a sphere

I have the following triple integral in spherical coordinates $(r,\theta,\phi)$: $$\int_0^{2\pi}\int_0^\pi\int_0^RCr^3\hat\theta\cdot r^2dr\sin{\theta}d\theta d\phi$$ How do I handle the ...
2
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4answers
817 views

Intersection of two arcs on sphere

I have two arcs on a sphere that are defined as pair of points: (θ₀, φ₀), (θ₁, φ₁). I need to find a point where they intersect, or some indication if they don't. What is important is that they are ...
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1answer
54 views

How to resolve this equation to another value?

Sorry guys, I don't know how to be more specific in the question without writing a way too long question... Anyway my problem: I have this formula to calculate the distance between two points on the ...
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2answers
2k views

Parametric Equation for Great Circle

So I've been doing a lot of searching and haven't found exactly what I'm looking for. My math skills are a bit rusty, so I haven't had luck deriving this on my own. What I'm looking for is an ...
2
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1answer
75 views

Determining new coordinates after a rotation of a sphere

Imagine that I am standing at a place on Earth, using coordinates of say N41 W74. Now the Earth's axis rolls 90 degrees, causing the N/S axis to become the equator, and rotation resumes as before. ...