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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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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129 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
418 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
59 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
771 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
137 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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61 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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117 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
4k 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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88 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
145 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
555 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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0answers
199 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
232 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
130 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
328 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
245 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
575 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
350 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
235 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
366 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
438 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
312 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 ...
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1answer
614 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 ...
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4answers
2k 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
57 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
3k 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 ...
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1answer
114 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. ...
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1answer
2k views

Spherical harmonics expansion for a particular function

On the unit sphere, each square-integrable function can be expanded as a linear combination of spherical harmonics : $$ f(\theta,\phi) = \Sigma_{l=0}^\infty \Sigma_{m=-l}^{+l} f_{lm} Y_{lm} ...
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1answer
174 views

Changing from rectangular coordinates to spherical coordinates (integration)

I am taking calculus 3 and I have problems understanding how to change from rectangular coordinates to spherial ones (integration). For example, I have this problem: Find the volume of the solid $T$ ...
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1answer
520 views

Triple Integral Spherical Coordinates

So I have to compute the triple integral of this: $\int\int\int \frac{1}{1+x^2+y^2+z^2}$ and it says the equation of the sphere is $ x^2 + y^2 + z^2 = z$ which is just an elongated sphere running ...
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1answer
1k views

How to integrate a vector function in spherical coordinates?

How to integrate a vector function in spherical coordinates? In my specific case, it's an electric field on the axis of charged ring (see image below), the integral is pretty easy, but I don't ...
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1answer
145 views

Volume of a 3D sphere of radius $R$ using Riemannian metric in stereographic coordinates

The question is pretty much in the title. We were also given the hint that it could be useful to use spherical coordinates when calculating the integral (the actual answer is not required, just its ...
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2answers
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derivatives transformation

I'm currently doing a calculation for the connection coefficients using the standard space-time coordinates, namely x[0],x[1],x[2],x[3]. The setup is a spherically symmetric problem. In my ...
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3answers
249 views

Integration with Spherical Coordinates

Use spherical coordinates to find the volume of the solid inside both $x^2+y^2+z^2=16$ and $z=(x^2+y^2)^{1/2}$.
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1answer
482 views

Finding a third coordinate on a sphere that is equidistant from two known coordinates

Here is my problem that I'm having some trouble with: I have the coordinates (latitude and longitude) of two points on Earth. I have no problem finding the great circle distance between the two ...
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1answer
2k views

Find volume of a cone $z=k\sqrt{x^2+y^2}$ bounded by $z=h$ using spherical coordinates

We were given this exercise in class to take home but I am a bit confused with it. If anyone could help I would appreciate it. Let $C$ be a conical solid bounded above by $z=h$ and below by the cone ...
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1answer
160 views

Law sines in Spherical Triangle $\rightarrow$ Law sines in plane triangle

Could any one tell me how to estimate or get law of sines in Spherical Triangle to The Law of Sines in Plane Triangle? i.e $\frac{\sin a}{\sin A}=\frac{sin b}{\sin B}=\frac{\sin c}{\sin C}$ to ...
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1answer
149 views

problem or doubt regarding visualizing angles of spherical triangle

I must confess that I am not able to visualize or understand what is the angle of a spherical triangle say $ABC$ where $A,B,C$ are vertices of the triangle which is formed by intersection of three ...
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1answer
84 views

How to minimize the length of a curve on $S^2$

The length of a curve $\gamma$ starting from a point $p$ and ending at another point $q$ on $S^2$ is given by the formula $$l_{\gamma}(S^2)=\int_{0}^{1}\sqrt{(d\phi/dt)^2+ \sin^2\phi ...
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0answers
127 views

two points on a unit sphere

Consider the two vectors to the points on the unit sphere, $${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$ with $i=1,2$. Use the dot product to get the angle $\psi$ ...
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2answers
762 views

Discretize a circle on a sphere with a given center and radius

I would like to draw a discretized circle on the surface of a sphere (the Earth in that case). The input of the algorithm would be the center of the circle (expressed as longitude and latitude), the ...
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2answers
228 views

Simple approach for geo distance

I wanted to show my nephew(16) a simple approach to calculate the distance between two geo-locations. The mathematical knowledge of a 16-year old boy is limited to simple geometrical shapes like ...
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0answers
708 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) = ...
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2answers
226 views

Calculating longitude degrees from distance?

I need to calculate how many longitude degrees a certain distance from a point are, with the latitude held constant. Here's an illustration: Here x represents the longitude degrees, the new point ...
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1answer
324 views

Given an arc length and an angle, how do I get a sphere coordinate?

Assuming I start at the top of a sphere and am given the radius of the sphere, an angle to turn, and a distance to walk along the sphere, how could I find my destination in the sphere coordinate ...
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518 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' ...
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1answer
290 views

Distance measurement between latitude/longiture pairs.

I need to calculate the distance between two lat/lng coordinate pairs. In addition, If given an initial lat/lng coordinate, angle of travel, and distance, I need to calculate the resulting lat/lng ...
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217 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 ...
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1answer
347 views

full hessian, spherical coordinates

The question itself is pretty simple. I am running into confusion. Seems like there is a typo in the book. I wanna check myself. Maybe I am doing something wrong. Suppose we have the function (which ...