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).

learn more… | top users | synonyms

0
votes
1answer
37 views

Spherical coordinates on cartesian straight lines

I'm trying to solve this problem: Compute the volume of the solid bounded by: the surface $(z+1)^2=x^2+y^2,$ the surface $4z=x^2+y^2,$ above the $xy$ plane. I want to do it with spherical ...
2
votes
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 ...
1
vote
0answers
65 views

integrate over quadrant of sphere

It is known that $\int_{x\in S}\exp(\kappa\mu^Tx)dx$ where S is the surface of the unit sphere is $\frac{(2\pi)^{p/2} I_{p/2-1}(\kappa) }{\kappa^{p/2-1}}$ where $p$ is the number of dimensions and $I$ ...
0
votes
1answer
27 views

Envisioning Spherical Coordinates

Is there a way to envision these two equations in spherical coordinates without plotting a bunch of points? I'm interested in what the surfaces look like. $$\rho=\sin\theta\sin\phi$$ ...
1
vote
2answers
362 views

Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
0
votes
2answers
391 views

Find volume above cone within sphere

My objective: Using spherical coordinates, set up and compute an integral to find the volume of the ice-cream-cone shaped solid lying above the cone $z = \sqrt{x^2 + y^2}$ and below the sphere ...
1
vote
0answers
147 views

Convert geodetic coordinates to cartesian coordinates

I am working on some simulation software that will represent a number of entities in a defined geographic area in the world. The part of the software that I am currently working on is to implement ...
1
vote
0answers
47 views

Why did my teacher integrate \varphi in the opposite direction while solving a triple integral for volume in spherical coordinates?

In Calculus III class today we learned how to evaluate triple integrals in spherical coordinates. One of the example questions we worked on was to use a triple integral to find the volume of a shape. ...
1
vote
1answer
176 views

Changing to spherical coordinates to evaluate the integral

$$\iiint_D \,dz\,dy\,dx$$ where the region $D$ is defined as followed: $$0<z<\sqrt{9-x^2-y^2}$$ $$0<y<\sqrt{9-x^2}$$ $$0<x<3$$ I got the corresponding spherical coordinates for ...
1
vote
1answer
155 views

Line-of-Sight Angle on Sphere

I'm trying to calculate the angle (in degrees) between two latitude/longitude pairs, but with a twist. Most calculations I see use the Great Circle / bearing method, but this does not seem correct ...
1
vote
1answer
334 views

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces?

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces (explicitly)? A derivation is here but its conclusions seems not right? The expected ...
4
votes
1answer
223 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
vote
0answers
31 views

angle difference in hyper-spherical coordinate

The question is kind of intuitive: Consider a points $p$ in $\mathbb{R}^d$ ($d$-dimensional Euclidean space). $$p=\{x_1, x_2, \ldots, x_d\}$$ We can always transform it into spherical coordinate. ...
0
votes
2answers
42 views

Derivation for the integrating term in line integrals and volume integrals in spherical coordinates

Can anyone refer me to, or respond with, the derivation for the integrating term in line integrals $dl=dr\hat{r}+rd\theta\hat{\theta}+r\sin\theta\ d\phi\hat{\phi}$ and volume integrals ...
0
votes
2answers
78 views

Divergence of vector in spherical coordinates

How should I calculate the divergence for $$\vec{V}=\frac {\vec{r}}{r^2}$$ Is it possible to convert it from spherical coordinates to cartesian?
0
votes
1answer
47 views

Converting to Spherical Coordinates that have a Large Azimuth?

I've run into a problem converting Cartesian coordinates to spherical coordinates. Say I've got a vector/point $p=(-1,5,7\frac{2}{3})$. Obviously, finding the polar angle/inclination isn't going to ...
2
votes
0answers
137 views

Fourier transform of an exponentially singular radial function

I am trying to compute the 3D Fourier transform of a spherically symmetric function of the form $$f(r) = e^{\frac{1}{r} e^{-r}} - 1\, ,$$ which entails the integral $$\begin{aligned}F(k) =& \int ...
3
votes
0answers
89 views

Computing volume element in spherical coordinates

Suppose $y = (r, \theta^1, \theta^2)$ are spherical coordinates in $(\mathbb{R}^3,g)$. What is the $d\text{vol}$ in these coordinates? I solved it but I don't know if it's right. My solution: We ...
0
votes
2answers
50 views

How can I solve these two tough integrals?

\begin{equation*} J_{1} = \int_{0}^{\sqrt{{\pi}/{6}}} \int_{y}^{\sqrt{{\pi}/{6}}} \cos{(x^2)}\,dx\,dy \end{equation*} \begin{equation*} J_{2} = \int\int_{E}\int z e^{(x^2+y^2)} + xe^{x^8}\,dV, ...
0
votes
1answer
207 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
1
vote
1answer
242 views

How to describe the region inside a sphere and below a cone in cylindrical and spherical coordinates?

If E is the region of space located inside the sphere $x^2 + y^2 + z^2 = 4$ and below the cone $z = \sqrt{3x^2 + 3y^2}$ How may I describe E in cylindrical and spherical coordinates? And how may I ...
2
votes
1answer
198 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 ...
1
vote
1answer
208 views

Writing the squared sine as a Legendre polynomial of cosine

I'm just getting learning about how Legendre polynomials come about when considering product solutions in spherical coordinates with azimuthal symmetry. I'm trying a problem on my own, and I'm a bit ...
1
vote
2answers
61 views

Under what conditions can we change the order of integration in any coordinate system?

I thought of this question while mindlessly setting up another integral in spherical coordinates - I usually swap the orders of integration in spherical coordinates without thinking, and I'm not ...
4
votes
1answer
310 views

Electrostatic Potential Energy integral in spherical coordinates

I'm having trouble with evaluating an integral that arises from attempting to find the total energy of an electrostatic system consisting of two point charges, which involves an integral over all ...
0
votes
1answer
35 views

Integration in d-dimensional spherical coordinates

Can someone please tell me why $$\left(\int_0^\infty dr\, e^{-r^2}\right)^d=\int_0^\infty dr\,r^{d-1}S_d e^{-r^2}?$$ Why doesn't $d$ end up joining the exponential?
0
votes
3answers
108 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\,{\rm d} V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
1
vote
2answers
100 views

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 ...
2
votes
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 ...
0
votes
0answers
25 views

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?
0
votes
0answers
49 views

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 ...
1
vote
0answers
156 views

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.. ...
3
votes
1answer
73 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 ...
1
vote
1answer
764 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 ...
9
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 ...
0
votes
1answer
328 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 ...
2
votes
3answers
1k 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 ...
0
votes
1answer
73 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 ...
1
vote
0answers
41 views

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, ...
3
votes
0answers
25 views

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 \, ...
0
votes
1answer
97 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 ...
1
vote
0answers
147 views

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 ...
2
votes
2answers
129 views

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: ...
0
votes
1answer
111 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. ...
2
votes
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 = ...
1
vote
1answer
44 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 ...
3
votes
1answer
106 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} ...
7
votes
0answers
137 views

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, cylindrical or spherical coordinates ...
2
votes
1answer
72 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 ...
6
votes
2answers
12k 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 ...