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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0
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3answers
701 views

Center of mass of one octant of a non-homogenous sphere

Find the center of mass of that part of the sphere $x^2+y^2+z^2 \le a^2$ having $x,y,z \ge 0$ (that is, the part in the first octant) With density given by $\rho(x,y,z)=(x^2+y^2+z^2)^{3/2}$ It ...
2
votes
3answers
34 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 ...
1
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3answers
138 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
20 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 ...
3
votes
2answers
4k views

building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $ x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta $ $$ \begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin ...
6
votes
1answer
3k 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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0answers
23 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 ...
0
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1answer
23 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 ...
1
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0answers
5 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 ...
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0answers
24 views

Computing the spherical coordinates in n-dimensions [duplicate]

This time I want to compute the Jacobian of the spherical coordinates in n dimensions, so it needs to give me the following result: $$\displaystyle ...
3
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1answer
48 views

How to compute the following Jacobian

I need to show that the Jacobian of the n-dimensional spherical coordinates is $$\displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$$ then I have computed the Jacobian matrix, ...
1
vote
1answer
196 views

Laplace operator in spherical coordinates, abstract approach

I'd like to show the well-known formula of the Laplacian operator for euclidean $\mathbb{R}^3$ in spherical coordinates: $$ \Delta U = \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial ...
4
votes
3answers
47 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
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1answer
18 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
votes
2answers
87 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 ...
2
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0answers
25 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 ...
0
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1answer
12 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 $0$ ...
2
votes
2answers
30 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
23 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 ...
6
votes
2answers
425 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
0
votes
1answer
28 views

Parameterising of scalarproduct of surface gradient with vector field

Given is this operation $\left((\mathbb{1}-\hat{r}\hat{r})\cdot \nabla\right) \cdot v(r) = \left(\nabla-\hat{r} (\hat{r} \cdot \nabla)\right) \cdot v(r)$, where $r=(x,y,z) \in \mathbb{R}^3$, ...
0
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1answer
12 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. ...
1
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3answers
49 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
23 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
vote
1answer
31 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 ...
0
votes
2answers
645 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 ...
1
vote
0answers
41 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 ...
-3
votes
1answer
39 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
vote
1answer
39 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 ...
2
votes
1answer
52 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
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0answers
15 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!
0
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1answer
18 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
17 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 ...
7
votes
5answers
22k views

What is the general formula for calculating dot and cross products in spherical coordinates?

I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
0
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0answers
14 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
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2answers
41 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 ...
0
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0answers
17 views

Find The Volume Enclosed By p=2cos(φ)sin(φ)^2

So I've been trying to find the volume enclosed by this area for hours and when I do the triple integral with φ from 0 - π, θ from 0 - 2π and p from 0 to 2cos(φ)sin(φ)^2 I get 0!!! QQ Please help me ...
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0answers
23 views

Spherical Triangle — Angle from one Point without using North

I'm designing a tool for students and right now im working with coordinates. I have the plan to shift the north pole in my code to Munich and I would like to do this by working with a spherical ...
0
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0answers
63 views

Triple integrals - spherical coordinates

Integrate the function: $$f(x,y,z)=1/\sqrt{(x^2+y^2)(x^2+y^2+z^2)}$$ over the region $R$ which is the set of all points outside the sphere $x^2+y^2+(z-1)^2=1$ but inside the sphere $x^2+y^2+z^2=4$. ...
4
votes
1answer
186 views

Transformations from n-sphere coordinates to cartesian coordinates.

I was wondering how one would proceed to convert between coordinate systems in $ \mathbb R^n $. For $ \mathbb R^2 $ the conversion is easy and just basic trigonometry. Given $(r, \theta)$ we can ...
1
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0answers
56 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 ...
2
votes
1answer
41 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 ...
3
votes
1answer
54 views

Conversion from Cartesian to spherical coordinate for vectors - ray tracing application

I'm implementing a ray tracer that support physically based rendering, so is based on various BRDF models. At the moment I'm focused on Oren-Nayar and Torrance-Sparrow model. Each one of these is ...
2
votes
1answer
32 views

Inclusion, pullback of differential form

Let $\omega=x\,dy\wedge dz +y\,dz\wedge dx+z\,dx\wedge dy$ or in spherical coordinates (unless I had made some mistake) $\omega=r^3\cos \theta\, d\phi\wedge d\theta$. Now I want to find $i^*\omega$ ...
0
votes
1answer
49 views

Using spherical coordinates, find the volume.

Find the volume of the solid that lies in the first octant above the cone $z=\sqrt{3(x^2+y^2)}$ and inside the sphere $$x^{2}+y^{2}+z^{2}=4z $$ using spherical coordinates: So here is what I have ...
3
votes
2answers
51 views

substituting spherical coordinates to evaluate an integral.

I have to evaluate $$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{-\sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}} \, dz \, dy \, dx$$ using spherical coordinates. This is what I have come up with \begin{align} ...
0
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0answers
22 views

formula for getting the cross product in spherical coordinates, given the two vectors.

I am using this coordinate system: consider these two vectors in spherical coordinates: $$\vec A=A_r \hat r +A_\phi \hat \phi + A_\theta \hat \theta= A_r\sin{\theta_A} \cos{\phi_A} \hat i ...
0
votes
0answers
37 views

Parameterization of Ellipsoid

I have a question asking me to evaluate $\iint_\Sigma \mathbf{F} \cdot \mathbf{n}~dS$, where $\Sigma$ is the lower half of the ellipsoid $z = -2 \sqrt{1 - x^2 - y^2}$ with $\mathbf{n}$ directed ...
0
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0answers
28 views

Plot 8 points on the Unit sphere satisfy some conditions

I want to plot $8$ points on the unit sphere such that: •for $N = 8$: a “twisted cuboidal” shape, consisting of two parallel rings of four equally-spaced spots, with the rings symmetrically placed ...
0
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1answer
23 views

$\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$ Where $W$ is the solid delimited by $y=4$ and $y=\sqrt{x^2+z^2}$

$\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$, $W$ is limited by the plane $y=4$ and the paraboloid $y=\sqrt{x^2+z^2}$. I'm trying to solve with spherical coordinates, however I got stuck in the following ...