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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2answers
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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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4answers
2k views

Latitude and longitude of points on a line

How could you get the latitude and longitude of four points (equal distance apart) on a line from $(27,-82)$ to $(28,-81)$? The four points should split the line into 5 parts.
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1answer
38 views

Integration of function $\sqrt{1 - \sum_{i}x_{i}^{a_{i}}}$ by using spherical coordinates in D dimensions

Let's have integral $$ I = \int \limits_{0}^{\text{T.P.}}dx_{1}...dx_{D}\sqrt{1 - \sum_{i}x_{i}^{a_{i}}}, \quad \text{T.P.}: \quad 1 - \sum_{i}x_{i}^{a_{i}} = 0 $$ By rewriting it in coordinates $x_{i}...
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1answer
50 views

Reference for differentiation of an integral over variable ball

I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$ \frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y), $$ where $dS$ is the Lebesgue surface ...
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1answer
538 views

What is the generalization of Parseval's theorem into spherical coordinates?

what is the relationship between the total power of a function given in spherical coordinates in the Fourier domain: $E_k=\int_{\mathbb{R}^3}|F(k,\Theta,\Phi)|^2k^2 \sin(\Theta)\,dk\,d\Theta\, d\Phi$...
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1answer
41 views

Spherical law of cosines

The spherical law of cosines states that $$\cos c = \cos a \cos b + \cos C \sin a \sin b,$$ where $a,b,c$ are sides of a spherical triangle, and $C$ the angle. Is there a proof for this theorem ...
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1answer
37 views

Triple integral in spherical / cylindrical coordinates - where's the error? Exercise check

I have done an exercise in two different ways but I have obtained two different results and I can't understand what's wrong. Please, help me. Given: $V=\{(x,y,z)\in R^3: x^2+y^2+z^2\leq 1, \frac{...
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1answer
79 views

Using the Dirac delta function to find the density of point masses/charges

Here is an example from a textbook: Suppose there is a unit charge or unit mass at the point $(x,y,z)=(-1,\sqrt{3},-2)$; then in rectangular coordinates, the ...
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1answer
38 views

Solve $I=\iiint\limits_\Omega z^2 dv$ using spherical coordinate system, $\Omega: x^2 +y^2 + z^2 \le R^2 \cap x^2 +y^2 + z^2 \le 2Rz$

Question: Solve $I=\iiint\limits_\Omega z^2 dv$ using spherical coordinate system. $\Omega$ is the common part of $x^2 +y^2 + z^2 \le R^2 $ and $ x^2 +y^2 + z^2 \le 2Rz$. My attempt:Because $ r^2 \le ...
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1answer
81 views

Applied Mathematics: Spherical Polar Coordinates and Newton's Second Law

I've been attempting this question but can't seem to find a solution. Question: A particle of mass $m$ moves under the influence of a force which, in spherical polar coordinates, only acts in the ...
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1answer
547 views

conversion of laplacian from cartesian to spherical coordinates

In cartesian coordinates, the Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\qquad(1)$$ If it's converted to spherical ...
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2answers
69 views

Understanding Spherical coordinates on ellipses.

I was given the following problem: $$\iiint\limits_D (4x^2+9y^2+36z^2)\,dV,$$ where $V$ is the interior of the ellipsoid $$\frac{x^2}{9}+\frac{y^2}{4}+z^2=1.$$ The problem gives what the new ...
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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: ...
2
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1answer
49 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 = R(\cos\phi_i\sin\theta_i,\sin\phi_i\sin\theta_i,\...
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1answer
48 views

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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2answers
2k views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
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2answers
128 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, I'...
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2answers
450 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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1answer
3k 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
215 views

Triple Integral in Spherical Co-ordinates

Find the volume bounded by the surface $(x^2 + y^2 + z^2)^2 = 2z(x^2 + y^2)$ I have $x = \rho \sin\phi \cos\theta$, $y = \rho \sin\phi \sin\theta$, $z = \rho \cos\phi$. Therefore, $(x^2 + y^2 + z^2)^...
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1answer
594 views

Equation for the sensitivity pattern of a bi-directional microphone?

Can anyone give me an equation that expresses the sensitivity pattern of a bi-directional microphone, as a function of azimuth and elevation angle? A bi-directional microphone pattern looks something ...
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1answer
456 views

Can you formulate a $ \phi, \theta $ restriction in spherical coordinates for a great circle?

Further to this question Quaternion rotation has a nice property that you can trace any great circle you like. You specify the axis of rotation, and you will automatically follow the great circle ...
2
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1answer
692 views

integral of a spherically symmetric 3-dimensional function over all space

I'm very sorry because it may be a very basic question but I'm not able whether to solve it for sure, nor to find an answer in stackexchange or elsewhere. I have to calculate $ \int \int n(\vec{r})u(...
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1answer
27 views

Finding the volume of Torus, Jacobian of spherical substitution.

I thought to find the volume of a Torus, like I would a sphere, where the spherical substitution was: $$x=r\cos\varphi\sin \theta , y= r\sin\varphi \sin \theta, z=r\cos \theta \\ g(r,\varphi,\theta)\...
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1answer
36 views

Triple integral in spherical coordinates in an example

I am not sure how to do this. I am given a function in spherical coordinates. $C$ is a normalization constant given by the triple integral. How can I find C and use that to do part (b)?
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2answers
41 views

Equation used to represent a disc galaxy

I'm trying to create a solid which looks something like a disc galaxy: Key features are: Bulge in the middle Tapered "width" as it extends to a disc shape The end goal would be to use Python to ...
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2answers
52 views

cylindrical coordinates - base vector integral

Hello I have a problem where $\hat{p}$ is a basevector in a cylindrical system $$\int_{0}^{\pi/4}\hat{p} d\theta$$ I know that; $\hat{p} = \hat{x}\cos\theta + \hat{y}\sin \theta$ $$ \hat{x}\int_{0}^...
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1answer
45 views

Conversion of spherical coordinates to cartesian

For the flow $A = \frac{c}{r}$ with $r=\sqrt{x^2+y^2+z^2}$ I wanted to calculate the velocity field with $\nabla A$ As a result I get $(-\frac{c}{r^2},0,0)$. So far so good. When I tried converting it ...
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1answer
77 views

Calculus 3 Spherical coordinates: I'm not sure how to set this up.

find the volume of the region enclosed by the sphere $x^2+y^2+z^2=324$ and the cylinder $(x-9)^2+y^2=81$ by using spherical coordinates. I'm just not seeing how to convert this into a form where ...
2
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1answer
43 views

Triple Integrals: Conversion

I'm currently in second year calculus and have come across a problem that I'm struggling badly to try and understand. The question is as follows: Sketch the region of integration of the following ...
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1answer
227 views

Volume of sphere with triple integral

Using the same notations as in this picture : The element of volume is: $r^2 \sin(\theta) \, dr \, d\theta \, d\phi$ If I try to create the volume visually, I begin with integrating $r$ between $0$...
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3answers
56 views

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 \sin\...
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3answers
2k 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 ...
2
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1answer
74 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 (it'...
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2answers
696 views

How to find all 3 orthogonal vectors to a 4D vector

For a program I'm writing, I need to find the vectors orthogonal to a given vector rotated at an arbitrary angle, and in 4D. It is a unit vector. For 3D, I found the two orthogonal vectors like this:...
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3answers
160 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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2answers
321 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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1answer
708 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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2answers
184 views

Are there derivations of equations of non-degenerate real quadric surfaces

Take the ellipsoid for example $$(x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1$$ in the x-y plane you have an ellipse described by $$(x^2/a^2)+(y^2/b^2)=1$$ (suppose z=constant) in the y-z plane you have an ellipse ...
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1answer
497 views

Change of coordinate system on a sphere

This might take a while to explain, so bear with me: I've got a perfect sphere. I've set up an arbitrary longitude/latitude ("angle") coordinate system on it (imagine an equator around the middle, ...
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2answers
607 views

Evaluating an integral in spherical coordinates over on odd shaped region.

I have to evaluate this integral: $$ \int_{0}^{\sqrt{2}}\int_{y}^{\sqrt{4-y^2}}\int_{0}^{\sqrt{4-x^2-y^2}} \sqrt{x^2+y^2+z^2}dzdxdy $$ in spherical coordinates. I see that the region in the xy ...
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1answer
4k views

Transforming from one spherical coordinate system to another

I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points ...
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2answers
564 views

How to use a Rhumb Line?

I am new to working with coordinate data and figured out the equation I am looking for is the Rhumb Line. I went to go research it and found a lot of equations and I still have no idea where to start. ...
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1answer
42 views

Cube in Spherical Coordinates not centred at the origin

I`ve seen that there are already a couple of questions about how to describe a cube in Spherical Coordinates. However they are all centred at the origin. I would like to describe a cube in Spherical ...
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1answer
29 views

Find a circle on sphere using spherical distance

I have a sphere with radius $R$. On this sphere I also have a point $P_1$ written in spherical coordinates, so I know $\theta_1$, $\phi_1$ and $R$ for this point (same as on this picture). I also ...
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0answers
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Calculate the Angle between two vectors in 3d Spherical Coordinates

I have two vectors in spherical coordinates, both originating at the origin and both with the same magnitude equal to one. One is vertical: {1,0,0} and the other undefined: {Ms,Mt,Mp}. The other one ...
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0answers
27 views

Pattern of collision of bouncy balls in a sphere?

Suppose that you have two infinitely bouncy golf balls that exist inside a perfect sphere in weightless suspension, and both golf balls start bouncing at a random angle and are 10 or 100 times ...
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0answers
67 views

Error in distance between points in spherical coordinates

I have two points with spherical coordinates: $a=(r_1,\theta_1,\phi_1)$ and $b=(r_2,\theta_2,\phi_2)$. The cartesian coordinates of the points are: $$ (r_i \cos\theta_i \cos\phi_i, r_i \cos\theta_i \...
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1answer
79 views

Spherical distance between two points in terms of latitude and longitude

I have seen the answer to this question - Great arc distance between two points on a unit sphere However in a fortran program that I have this is the code to calculate spherical distance between two ...
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1answer
64 views

How to integrate a part of a sphere limited by a rectangle?

I have a function which depends on the solid angle $\frac{df}{d \Omega} = \frac{df}{d\phi d\theta} $. I want to integrate it over a part of the sphere limited by a rectangle. How should I set the ...