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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5
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1answer
84 views

The centre of the earth

I'm a real beginner here (first post and first foray into math since high school, trying to catch up), so I'm going to try my best to explain my problem in mathematical terms then follow up with an ...
0
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1answer
31 views

sphere arc intersection

Given: an arc defined by two end points (which I can express in lat/lon/altitude or earth-centered fixed (ECF) 3D 'cartesian' space) a sphere defined by a center (lat/lon/alt or ECF) and a radius (...
1
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0answers
41 views

Integration in spherical coordinates

I needed to solve the following integral on one of my exercise sheets, which seemed not too difficult: $ \phi(\vec{r}) = \dfrac{1}{4\pi\epsilon_0} \int\limits_0^{\infty} dr' \int\limits_0^{\pi} d\...
2
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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 ...
0
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0answers
14 views

spherical radial transformation

several papers mentioned a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty \int_{\mathbf{z}'\mathbf{z} = 1} f(r\mathbf{z}) ~r^{n-1} ~\mathrm{...
1
vote
1answer
23 views

Rewrite equation using cylindrical and spherical coordinates.

I want to rewrite the equation $z=x^2-y^2$ using cylindrical and spherical coordinates. The cartesian coordinates are of the form $(x,y,z)$. The spherical coordinates are of the form $(\rho, \theta, ...
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0answers
16 views

How to setup/evaluate a triple integral to show an interesting result in physics?

I know this isn't the physics forum, but the task i'm struggling with is purely mathematical. My task is as follows; Let $A$ be a sphere centered at origin with radius $R$ and assume $a \geq R$. ...
5
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3answers
163 views

Show that $\nabla\cdot\left(\dfrac{\mathbf{e}_r}{r^2}\right)=4\pi\delta(\mathbf{r})$ using the divergence theorem.

The book answer goes as follows: By the divergence theorem, in spherical coordinates we find $$\color{red}{\iiint_\limits{\large\text{volume}\,\tau}\nabla\cdot\...
1
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0answers
33 views

Marginals/conditionals of a normalized Gaussian vector

It is well - known that if $x=(x_1,...,x_n)^T\sim{N(0, \sigma^2I)}$, then its normalized version is uniformly distributed on the unit $n-1$ - sphere: $$ y:=\frac{x}{||x||_2}\sim{\text{Uniform}}(S_{n-...
1
vote
1answer
228 views

Triple integral to find the mass of the intersection between two spheres

I've got two unit spheres, one is centered at $(0,0,0)$ and the other at $(0,0,1)$, the intersection of these two spheres is my region $R$. I would like to find the integral: $$\iiint\limits_R z\;dV$...
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0answers
20 views

Vectors and dot product in spherical coordinate system [duplicate]

Let $\vec{v_1}$ and $\vec{v_2}$ given: $\overrightarrow{V_1} = r_1\hat{u_r} + \theta_1\hat{u_\theta} + \phi_1\hat{u_\phi} \\ \overrightarrow{V_2} = r_2\hat{u_r} + \theta_2\hat{u_\theta} + \phi_2\hat{...
2
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1answer
73 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 ...
0
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1answer
18 views

What does this triple integral in spherical coordinate represent?

$$\int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{2\pi} r(b+r \cos\phi) d\phi d\theta dr$$ What does the above integral represent? I don't think it's volume of a figure, since either $\phi$ or $\theta$ should ...
1
vote
1answer
57 views

Convergence of $\iiint \frac{dxdydz}{(x^{2}+y^{2}+z^{2})^\alpha}$

I'm studying the convergence of the following triple integral $$I(\alpha) = \iiint \limits_{\Omega_{3}}\frac{dxdydz}{(x^{2}+y^{2}+z^{2})^\alpha}\tag{*}$$ Where $$\Omega_{3} = \{(x, y, z) \in \mathbb{...
2
votes
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)?
0
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1answer
78 views

Expressing regions in cylindrical and spherical coordinates

I need to express the following regions in both cylindrical and spherical coordinates. I am not sure what to do here exactly, in (a) for example, should I substitute the Cartesian equation of the ...
3
votes
0answers
134 views

Volume of $n$-dimensional spherical orthant in upper diagonal halfspace

Consider an $n$-dimensional Euclidean Space. Consider orthants in that space. Each orthant occupies $\frac{1}{2^n}$ of the volume of an $n$-dimensional unit sphere. Let's call that a spherical ...
0
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1answer
45 views

How to calculate distance between two points on the sphere?

I have two azimuth and altitude angles for two points on a sphere. How can I calculate the spherical distance between those points? And how can I deduce this formula from first equations?
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0answers
26 views

Relationship between Normal coordinates and Spherical Coordinates

I am using the following coordinates on $S^3: (\psi, \theta, \phi)$ where $$\begin{cases}x_0 = \sin\psi,\\ x_1 = \sin\psi \cos\theta,\\ x_2 = \sin\psi \sin\theta \cos\phi,\\ x_3 = \sin\psi \sin\...
2
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1answer
37 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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0answers
20 views

Composite Spherical Harmonics expansion and error propagation

Let's assume that $f$, $g$ and $t$ are three functions defined over the surface of a sphere. In particular $f$ is defined using $g$, $t$ and the integral operator as follows: $$f(\omega)=\int_{\...
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0answers
18 views

angle between hrizontal and a line connecting the center of an oblate ellipse to a point in space

I would like to know how I can calculate the angle $\alpha$ in an oblate ellipse similarly to the sphere.
1
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1answer
44 views

Dot product in integral over spherical coordinates

Midway through solving a question, I have an intermediate integral: $$ G(\vec x,t)=\frac{c}{16 \pi^3} \iiint _{\mathbb{R}^3} \frac{\sin (tck)}{k} e^{i \vec{x} . \vec{k}} d^3 \vec k $$ Now what I ...
2
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0answers
48 views

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 ...
1
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3answers
37 views

How to solve this equation in spherical coordinates

I am trying to find the angles $\phi$ that satisfy the following equation: $$ \cos\phi + \sqrt{\cos^2\phi+15}=\frac{2}{\sin\phi}, $$ where $\phi \in [0,\pi ]$. The geometric interpretation of this ...
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3answers
149 views

How to minimize $ab + bc + ca$ given $a^2 + b^2 + c^2 = 1$?

The question is to prove that $ab + bc + ca$ lies in between $-1$ and $1$, given that $a^2 + b^2 + c^2 = 1$. I could prove the maxima by the following approach. I changed the coordinates to spherical ...
0
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0answers
23 views

Calculating the sun position fails

could you help me find the mistake(s) in my calculation of the sun position today on hawaii at 16:00? I'm following this Wikipedia article. Number of days since 2000/01/01 (2016/01/29): $$n =5873$...
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0answers
24 views

If a sphere is rotated so that $90°$N $0°$E is transformed to $50°$N $6°$E, to which point is $18°$N $77°$E transformed?

If a sphere is rotated so that point $90°$N $0°$E is moved along a great circle to point $50°$N $6°$E, to which point is point $18°$N $77°$E moved? $90°$N $0°$E $\implies 50°$N $6°$E $18°$N $77°$E $...
1
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1answer
55 views

Cartesian to Spherical coordinate conversion specific case when Φ is zero and θ is indeterminant

Following is the conversion for spherical to cartesian coordinate \begin{align} x &= r \cos\theta \sin\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\varphi \end{align} and we are ...
0
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0answers
25 views

Understanding unit normal curvilinear vectors to the surface of an octant of a sphere

I'm supposed to test divergence theorem on an octant of a sphere for a given vector field. The triple integral part was easy. However, I'm stuck with the double integral part. Now, there are four ...
4
votes
3answers
56 views

Evaluate the integral using spherical coordinates

Given the integral $\int^{1}_{0}\int^{\sqrt{1-x^{2}}}_{0}\int^{\sqrt{1-x^{2}-y^{2}}}_{0} \dfrac{1}{x^{2}+y^{2}+z^{2}}dzdxdy$ I need to evaluate this using spherical coordinates. So far I have that $...
0
votes
1answer
27 views

Change of Variable theorem and spherical coordinate transformation

If $V = \{(x,y,z) \text{ such that } x^2 + y^2 + z^2 < a^2\text{ and }z>0\}$, use the spherical coordinate transformation to express $\int_V{z}$ as an integral over an appropriate set in $(\rho, ...
2
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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 ...
2
votes
2answers
117 views

How to show $\DeclareMathOperator{curl}{curl}\curl\curl(e_r) = 0$

I want to figure out how to calculate $\text{curl}(e_r$). Where $e_r$ is a base vector for the Spherical co-ordinate system. Taking $e_r = (\sin\theta \cos\phi)i+(\sin\theta \sin\phi)j+(\cos\theta)...
2
votes
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 ...
1
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0answers
38 views

rotation of spherical surface in spherical coordinates

I need to plot a spherical surface in computer (like the surface of a lens). I know the normal vector (as an example, say $\ n=(1,2,3) $) of this surface and it originates from the centre of the ...
1
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1answer
19 views

rotate a scalar valued spherical function

I want to rotate a function $f(\theta,\phi)$ around an arbitrary angle in 3D space. (Assuming $\phi$ is in the $xy$ plane and goes from $0$ to $2\pi$, and $\theta$ starts from $+z$ and goes from $0$ ...
0
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1answer
14 views

Set up triple integral's boundary for $x^{2} + (y-a)^{2} + z^{2}=a^{2}$ in spherical coordinates.

I have trouble with setting up triple integral's boundary for $\rho$. Solid object's equation is $x^{2} + (y-a)^{2} + z^{2}=a^{2}$,which is a sphere centered at (0,a,0), in spherical coordinates. Note:...
1
vote
3answers
78 views

Volume Between Spheres – Spherical Coordinates

I'm trying to find the volume between the spheres: $x^2 + y^2 + z^2 = 9$ and: $x^2 + y^2 + (z-2)^2 = 9$ I have calculated this, but have a strong feeling that little of what I did was actually ...
0
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0answers
26 views

Stereographic projection $S^3 \to \mathbb{P}^2(\mathbb{C})$

I think I can find a stereographic projection $S^2\setminus\{(0,0,1)\} \to \mathbb{P}^1(\mathbb{C})\setminus\{[0,1]\}$ using spherical coordinates: it should be something like this $$(\theta,\phi)\to ...
1
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1answer
33 views

find ANY point of tangency from a point to a sphere using spherical coordinates

I have a point $B$ in 3d space. I also have sphere with centre $C$ and radius $R$. I'm trying to find ANY point of tangency $T$ from that point $B$ to that sphere using spherical coordinates. So ...
2
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0answers
64 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
40 views

University level triple integration problem help. [closed]

Use spherical coordinates to find the volume of the solid enclosed by the sphere $x^2+y^2+z^2=4a^2$ and the planes $z=0$ and $z=a$.
1
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1answer
51 views

Sum of two vectors in spherical coordinates. [duplicate]

What is the sum of two vectors in spherical coordinates? The coordinate system: Assume we have vectors $(r_1,\theta_1,\phi_1)$ and $(r_2,\theta_2,\phi_2)$ in spherical coordinates. I know the sum ...
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0answers
16 views

find a solution for an integration in spherical coordinates

This problem comes from computation of magnetic field. $\vec r'$ and $\vec r$ are vectors, and represent different variable respectively. The integration is for $r'$ in Volune $V$'. Thank you!
2
votes
1answer
73 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 ...
0
votes
1answer
36 views

Project a line onto a sphere to calculate parameterized spherical coordinates

I have a line segment and I want to find the arc that it projects to on a sphere. I know there are two arcs; I'm interested in the one that's closest to the line (or intersects it). The easy way to ...
0
votes
2answers
24 views

Volume of solid inside surface in spherical coordinates.

Find the volume of the solid inside the surface defined by the equation $\rho=8\sin \phi$ in spherical coordinates So far I've set up an integral in spherical coordinates with $\rho$ from $0$ to $\...
0
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0answers
19 views

Surface of a torus in terms of Legendre polynomials

The equation of a spheroid is $$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2}$$ Its surface can be expressed as $$ r = a \left( 1 - \frac{2}{3} \epsilon P_2(\cos \theta) \right) $$ where $r$ is the ...
0
votes
2answers
60 views

Using Divergence Theorem to evaluate the flux over a sphere

Above is the question. I've try to find the divergence of F and parameterize the sphere using spherical coordinates. Below is my work. Then I use online integral calculator(just to avoid human error) ...