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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Finding the Limits of the Triple Integral (Spherical Coordinates)

Let $D$ be the region in $\mathbb{R}^3$ below $z=-\sqrt{x^2 + y^2}$ and above $z=-\sqrt{4-x^2 -y^2}$. Rewrite \begin{align*}\iiint \limits_D z^2 dV\end{align*} using Spherical Coordinates. I ...
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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.
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triple integral and limits

$$\iiint\limits_H (x^2+y^2) \, dx \, dy \, dz \\ H=\{(x,y,z) \in R^3: 1 \le x^2+y^2+z^2 \le9, z \le 0 \}$$ I'm using Spherical coordinate system: $$x=r\cos \theta \cos\phi $$ $$y=r\cos \theta ...
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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?
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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 ...
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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.. ...
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1answer
37 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
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1answer
35 views

Longitude and latitude problem

I find this question challenging. I am trying to solve this question for my younger brother. So here it goes: An airplane leaves an airport $X$, 20.6$^0E$ and 36.8$^0N$, and flies due south along the ...
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1answer
33 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 ...
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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 ...
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1answer
71 views

Determine if one point lies between two other points on a sphere

My question is rather simple. Can I use the dot product to determine if a coordinate lies between two others? With coordinates I mean a Point P(latitude, longitude) on the surface of the sphere. I ...
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2answers
81 views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
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1answer
47 views

Flow of fluid through a really tricky closed surface S (divergence theorem)

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
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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, ...
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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 \, ...
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1answer
56 views

Find the volume inside

Find the volume inside the torus $\rho=\sin\phi$. First of all how can $\rho=\sin\phi$ represent a torus? I can't even visualise that. All Ideas are welcome, this looks like a 'food for thought ...
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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 ...
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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 ...
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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 ...
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2answers
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How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell

Solving a diffusion-type ODE across a spherical shell, the equation is: $$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$ with boundary conditions $f(r_1)=f_1$ and $f(r_2)=f_2$. The solution is: ...
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43 views

Granted I have NE and SW coordinates for a rectangle, how do I get the center point?

I've got the NE and SW coordinates/points for a minimum bounding rectangle. How do I calculate the center point of this rectangle? At first thought, I could calculate this using simple division. ...
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1answer
47 views

Minimum of a potential function

I'm looking for extremes (minimum) of $$V = \frac{\alpha}{|\vec{r}_1-\vec{r}_2|} + \beta (\vec{r}_1 + \vec{r}_2)\cdot \vec{e}_z$$ where $\vec{r}_i = ...
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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 ...
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1answer
22 views

Sampling on Axis-Aligned Spherical Quad

Given spherical coordinates on a unit sphere, imagine a spherical quad defined by two ranges $[\phi_0,\phi_1]$ and $[\theta_0,\theta_1]$. If you have a globe, for example, the grid formed by the ...
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64 views

How to compute triple integral in spherical coordinates

I need to compute: $\displaystyle\int \int \int z dxdydz$ over the domain: $\left\{x^2+y^2+z^2\leqslant 16,z\geqslant 0\right\}$ Im trying to use spherical coords as: \begin{equation} ...
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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 ...
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1answer
44 views

Spherical-Coordinate Reference Frame

my problem looks apparently easy but I can figure out a solution. I'm going through a paper about the Boltzmann equation and I got stuck with this change of coordinates. The original formula for Q ...
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359 views

Distance between two points in spherical coordinates

I want to find the distance between two points in spherical coordinates, so I want to express $||x-x'||$ where $x=(r,\theta, \phi)$ and $x' = (r', \theta',\phi')$ by the respective components. Is this ...
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1answer
30 views

Implicit partial derivative of a spherical cap

Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld) $A(a,h) = \pi(a^2 +h^2)$, ...
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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 $$ ...
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104 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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Surface area of the part of the sphere $x^2+y^2+z^2=a^2$ that is inside the cylinder $x^2+y^2=ax$

I've been solving some surface area problems lately, but I don't think that the same approach that I was using will work with this one (or at least will result in a lot work). So, I believe I should ...
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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 ...
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Coordinates of tilted circle.

The original question is as follows: Imagine a wire located at the intersection of $x^2+y^2+z^2=1$ and $x+y+z=0$, whose density depends on position according to $\rho({\bf x})=x^2$ per unit length. ...
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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 ...
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1answer
43 views

N points in a circle around a point on a sphere.

Consider a 3D sphere: $(x_{c}, y_{c}, z_{c})$ : cartesian coordinates of the center $r$ : the radius Consider a random point on the surface of this sphere of coordinates : $(x_{0}, y_{0}, ...
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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 ...
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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 ...
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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 ...
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1answer
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How do I find the limits for $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$?

Evaluate $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$ where $W$ is the solid bounded by the two spheres $x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ where $0 < b < a$. ...
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Integral using spherical coordinates

I am trying to compute the volume of the following set : intersection of cylinder $x^2 + y^2 \leq R$ and sphere $x^2 + y^2 + z^2 \leq 4R^2$. I am having trouble setting up the integral properly ...
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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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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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vector components and dot product with unit vector

$E_{0}\hat z=\vec E_{0}rcos\theta=E_{0}cos\theta\, \hat r$ and $E_{0}cos\theta\, \hat r\circ \hat r =E_{0}cos\theta$ This just doesn't look right to me for some reason...
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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 ...
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1answer
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Distortion in spherical coordinates

I'm trying to realized 3d models of stones. My idea was to create a 2D random angular distribution with opportune correlation, namely $R(\theta,\varphi)=rand(\theta,\varphi,c_l)$ where ...
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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 ...
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1answer
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Curl of a function with only angular dependence

Let a function in spherical coordinates $$\vec F(\vec r) = \int{ d^3\vec r\,' \vec j(\vec r\,') } \,e^{-ik\hat r \cdot \vec r \,'}$$ Where $\vec j$ is a vector function. So $\vec F$ only depends on ...
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Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
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Find a vector in cartesian coordinates given its relative location to another vector in spherical coordinates

Here is my problem: -I have an arbitrary normalized vector N in cartesian coordinates -I am trying to find normalized vector M, also in cartesian coordinates -I am given the azimuth and polar ...