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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1answer
65 views

Find volume of region lying above $z=0$, below $z=4-x^2-y^2$ and inside extruded disc $x^2+y^2=2^2$

I am working on the following homework problem: Find the volume of the region that lies above the plane $z=0$, below the surface $z=4-x^2-y^2$ and inside the extruded disc $x^2+y^2=2^2$. I think ...
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2answers
203 views

Evaluate the integral by changing to spherical coordinates.

$$\int_{0}^{6} \int_0^{\sqrt{36-x^2}} \int_{\sqrt{x^2+y^2}}^\sqrt{72-x^2-y^2} xy~ dzdydx $$ I tried converting it and I ended up with $$\int_0^{2\pi}\int_0^{\pi}\int_0^{\sqrt{72}}\left[p\sin(\phi)\...
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0answers
20 views

Spherical to Cartesian coordinate ellipsoid overlap

I have two geographic coordinates; latitude and longitude , separated by few meters. I need to draw an ellipsoid of same major and minor axes ,centered around the geographic co-ordinates I used ...
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0answers
49 views

How to get parallels of tilted Equator?

I have a Great Circle on Earth, which is not an Equator nor Meridian, and it's not parallel to these. I have four geographical coordinate pairs for it, separated by 90 degrees, so I can use these in ...
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2answers
42 views

$\iiint_Ez(x^2+y^2+z^2)^{-3/2}dV$ where $E=\{(x,y,z):x^2+y^2+z^2\leq16, z\geq2\}$

Convert to spherical coordinates and evaluate: $$\iiint_Ez(x^2+y^2+z^2)^{-3/2}dV$$ where $E$ is the region satisfying the following inequalities: $x^2+y^2+z^2\leq16$, $z\geq2$. When I drew out the ...
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0answers
10 views

Scale factors and metric in cylindrical and spherical coordinates - isotropy of space [duplicate]

In cylindrical (polar) coordinates, the scale factors are $$h_r=1$$ $$h_{\theta}=r$$ $$h_z=1$$ Would it be correct to say that $h_i$ do not depend on $\theta$ because space is isotropic (has the same ...
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0answers
15 views

Shortest distance (in km) between a lat/long and a line between two lat/longs?

I know that Haversine can be used to calculate the absolute distance between two lat/long, but is there any implementation of the shortest distance (in km) between a lat/long and the line that is ...
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1answer
35 views

dot-product spherical

I want to calculate dotproduct of $ e_r *e_\phi $ they are unit vectors in spherical. where my spherical coordinates is $(r,\phi,\theta)$ My attempt is first to convert them to cartesian: which ...
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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
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
39 views

What is the equivalent of <x,y,z> in spherical coordinates?

I know this is a very newbie question, but what is the equivalent of $\langle x,y,z \rangle$ in spherical coordinates? I'd think it would be $\langle r, \theta, \phi \rangle$ but the divergences ...
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2answers
67 views

Calculating the percent of area “covered” by a vector pointing on a sphere

The question is inspired by rotational dynamics and how much of the sky could a camera "cover" when it rotates in a specific way. Let's say that we have solved the equations for rotational motion of a ...
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2answers
30 views

Converting from cylindrical to spherical coordinates?

I am supposed to convert the point $(100, -\dfrac{\pi}{6}, 50)$ from cylindrical to spherical. The $\rho$ and $\theta$ are easy ( $\sqrt{100^2+50^2}, \dfrac{\pi}{6}$ respectively) but the $\phi$ is ...
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1answer
32 views

Convert a density function $\rho(r)$ of sphere to ellipsoid.

Note: This is not a homework problem. As the title somewhat eludes to, I have a density function for a sphere as a function of radius $\rho(r)$. I would like to then flatten the sphere slightly into ...
2
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0answers
68 views

Integration over n-sphere

I am trying to integrate squares of sum of coordinates in n-dimensional sphere with radius bounded by r. $f(x,y)=x^2+y^2$. In spherical coordinates when $N=2,\;y=a\cos\theta$ and $x=a\sin\theta$ So ...
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0answers
59 views

Coordinates of an n-sphere [duplicate]

I'm a little embarrassed to ask this question because it should be easy but it's stumped me for over a week now. The answer will determine how I write some code, so it matters. According to Wikipedia ...
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2answers
103 views

Create a rectangle with coordinates (latitude and longitude)

I have two points on a map, I want to create a rectangle where the two points are the line that intersect the rectangle. I understand that there is no true rectangle on a sphere, but the areas I am ...
1
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1answer
60 views

Formula to convert Cartesian coordinates to spherical coordinates? [closed]

I have this formula: x, y, z = cos(vertical)*sin(horizontal), sin(vertical), cos(vertical)*cos(horizontal) Which maps a spherical coordinates (horizontal and ...
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0answers
35 views

Numerical evaluation of the Laplace operator near the singularity points in spherical coordinates

If one had to evaluate $\Delta Y_l^m$ numerically everywhere on the unit sphere, including the singularity points $\theta = 0,\pi$, how would they do it? Let's say $Y_l^m$ is a spherical harmonic. I'm ...
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0answers
25 views

Given how long it took the sound to reach two points, find how long it takes to reach the third point

I am having some issues trying to solve this puzzle. I am given 3 gps locations as starting points on a map. I am given the time it takes for the sound of a item being placed on the group in number of ...
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0answers
26 views

Spherical coordinate system remain the same if the origin is changed and each point make the same translation?

Now I have a spherical coordinate system whose origin is located at (a,b,c)[cartesian], and I have another point whose location is (r, theta, tho) in this spherical system, and P's cartesian ...
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2answers
90 views

Computing a double integral over a surface S, where S is the unit sphere,

$$ \int \int_S (x^2+y^2)d\sigma$$ Where S is the unit sphere centered at (0,0,0), and $\sigma$ is surface area. I arrived at the correct answer of $\large \frac{8\pi}{3}$, but I took an (educated?) ...
2
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1answer
554 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 ...
2
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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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0answers
37 views

How can I do a longitude/latitude tilt transformation?

I am trying to find a way to express the shortest path between two random points on a globe as a function expressed in longitude/latitude without using the geodesic equation (because it's messy and I ...
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3answers
61 views

Polar Co-ordinate proofs

The expression for acceleration in spherical polars is $$ \ddot{\mathbf r} =( \ddot r -r\dot\theta^2-r\dot\phi^2\sin^2\theta) \mathbf e_r + (r\ddot\theta+2\dot r \dot\theta-r\dot\phi^2\sin\theta\cos\...
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2answers
100 views

Find the maximum radius for given theta and phi (spherical coordinates) that will fall within a cuboidal boundary

I have a cuboid with measurements (width, depth, height) which is my boundary. The origin is the center of the cuboid. Given a theta(Azimuth) and phi(elevation), how do I find the highest radius that ...
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2answers
95 views

Evaluating $\iiint_v(3x^2+3y^2+3z^2) \, dv$ using Spherical Coordinates

I'm having issues solving $\iiint_v(3x^2+3y^2+3z^2) \, dv$ using Spherical Coordinates I made the ffg substitutions: $x=r\sin\theta\sin\phi, y=r\sin\theta \cos\phi, z=r\cos\theta$ Thus $3x^2+3y^2+3z^...
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0answers
32 views

How can I obtain a unit vector of a shifted spherical system?

I hope that I can explain myself clear enough, Assuming I have a sphere that has been moved down in the $Z$-axis. I know that r unit vector when the sphere is not shifted can be expressed as: $x\,\sin\...
2
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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
309 views

How to solve this problem using spherical coordinates system?

The question is very simple: Volume inside the solid limited by:$ (X^2+Y^2+Z^2=16), (X^2+Y^2=4)$ using SPHERICAL coordinates system. The final answer however can be checked making a "cylindrical ...
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1answer
31 views

Compute Surface Integral

Integrate $x^2+y^2$ over the upper hemisphere of radius $a>0$ centered at $(0,0,0)$. $\textbf{Edit}$ Consider the parametrization of the upper hemisphere given by $$X(\phi, \theta) = (a \sin \...
3
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2answers
235 views

Volume of a cube in spherical polars

Let us calculate the volume of the cube using spherical coordinates. The cube has side-length $a$, and we will centre it on the origin of the coordinates. Denote elevation angle by $\theta$, and the ...
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1answer
76 views

Correct order of taking dot product and derivatives in spherical coordinates

I tried to derive definition of divergence in spherical coordinates from gradient and got: $${\vec \nabla \cdot \vec A=\bigg (\frac{\partial}{\partial r}\hat r+\frac{1}{r}\frac{\partial}{\partial \...
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0answers
52 views

Closure for Legendre Polynomials

Suppose I have a PDE which can be solved by an expansion on the Legendre Polynomial basis, for an axisymmetric problem in spherical coordinates. For the derivation for such a problem, see these ...
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1answer
36 views

Proof of change in position vector in spherical coordinates

I have found it hard to proof that ${d\vec r=dr\hat r+rd\theta\hat \theta}$ in spherical coordinates. Also it would be great if somebody can explain what ${d\vec r}$ is because I read different things ...
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1answer
83 views

Projection of a 3D spherical function to a carteasian axis

I have a 3D function defined in a spherical coordinate system $(r,\theta,\phi)$, which is written as a product of a radial function $R_{nl}(r)$ and a spherical harmonic $Y_{lm}(\theta,\phi)$ I.e $$ \...
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3answers
742 views

Cross product spherical coordinates

I can't wrap my head around the result of the cross product of two vectors in spherical coordinates. Is it a vector or something that I can represent geometrically? For example, given two vectors in ...
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1answer
113 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system $(r, \theta, \phi)$, $\theta$ can range from $0$ to $2\pi$, but $\phi$ only varies from $0$ to $\pi$. Why is that?
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2answers
50 views

How are arc components of a spherical system derived?

I am studying a flight dynamics book (see Flight Dynamics by Stengel) and am rusty on spherical coordinates. Commonly, aerospace coordinates use a North/East/Down right-hand system. So $z=-h$, ...
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2answers
142 views

Rotate a unit sphere such as to align it two orthogonal unit vectors

I have two orthogonal vectors $a$, $b$, which lie on a unit sphere (i.e. unit vectors). I want to apply one or more rotations to the sphere such that $a$ is transformed to $c$, and $b$ is transformed ...
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3answers
487 views

Volume of the region outside of a cylinder and inside a sphere

The cylinder is $x^2 +y^2 = 1$ and the sphere is $x^2 + y^2 + z^2 = 4$. I have to find the volume of the region outside the cylinder and inside the sphere. The triple spherical integral for this ...
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1answer
124 views

analytic solution poisson equation spherical coordinates

I'm trying to analitically solve a poisson equation $\nabla^{2}p(r,\theta,\phi)=f(r,\theta)$ in spherical coordinates; the boundary condition is $p(\infty,\theta,\phi)$=0. I'm quite used to the ...
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0answers
33 views

Integrating over particular grids to obtain Spherical Harmonic coefficients

Theoretically the spherical harmonic expansion coefficients of a function $f$ should be calculated via a continuous integration: $$F_{lm} = \int_{0}^{2\pi}\int_{0}^{\pi} f(\theta,\phi)Y_{lm}^*(\theta,\...
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0answers
38 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function $...
2
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3answers
72 views

2-Sphere surface coordinate dimension

Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...
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2answers
71 views

How can I evaluate this integral over a sphere, with surface area element instead of volume element?

$$\int_S (x^2 + y^2)d\sigma,$$ where S is the sphere of radius 1 centered at (0,0,0) and $\sigma$ is surface area. I would like some hints on how to proceed. This is tricky, since I am not being ...
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1answer
100 views

Help with limits of integration in spherical coordinates

$$\int_\Omega F(\theta,\phi) \sin(\phi) \, d\Omega$$ where $\Omega$ represents the (outer) surface of a spherical cap on a unit sphere. Specifically, let the center of the spherical cap be in some ...
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1answer
66 views

Problem in deducing gradient in spherical coordinates.

I know the differential displacement in spherical coordinate as $$dr \cdot \widehat{r}+ r d\theta\cdot\widehat{\theta} + r\sin\theta d\phi\cdot \widehat{\phi}$$. But I can't figure out how the ...
2
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1answer
70 views

Spherical Parametrization of a Cholesky Decomposition

[Background: Trying to build up my math base knowledge (self-taught) to follow an explanation on page 291 of this document, dealing with spherical parametrization to estimate unknown variance-...