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

3
votes
1answer
78 views

Surface integral of a partially constant Dirac delta

I am trying to integrate the product of a function and a partially constant delta function over a sphere of constant radius $r$. The integral is of the form $\int^{2\pi}_0 \int^{\pi}_0 ...
1
vote
1answer
36 views

Dual Spaces vs Dual Bases

I'm trying to wrap my head around differentiable manifolds and tensors. I partially worked through a question which asked me to use the metric tensor and the line element in spherical polar ...
0
votes
0answers
32 views

Marsden Triple Integration, Spherical Coordinates

This comes from Marsden Vector Calculus book. I think I've done it correctly, but I'm not very confident in my work. The integral is $$\int_0^3\int_0^\sqrt{9-x^2}\int_0^\sqrt{9-x^2-y^2} ...
2
votes
1answer
57 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 ...
0
votes
0answers
21 views

Coordinate system for evenly-distributed points on a sphere

Suppose I have a sphere with a set of evenly-distributed points on its surface: What is the coordinate system to use to represent these points, where each adjacent point is a unit step? Spherical ...
0
votes
0answers
13 views

Relationships between positions on a sphere in a global coordinate system

Imagine I have a camera which is observing an object. The camera can move around the abject, by traversing a virtual sphere, where the radius of the sphere is the distance between the object centroid ...
0
votes
0answers
46 views

First person camera rotation (conversion between spherical to cartesian coordinates)

A similar question has been asked before on Stack Overflow but no satisfying answer. Basically I'm following this tutorial, and I can't exactly figure out how the guy deduced these two equations to ...
0
votes
0answers
88 views

Triple integral in spherical coordinates: Finding $\phi$ limits?

The question asks to convert to spherical coordinates then evaluate. So for this question, I manage to get the bounds of theta and row right, but I got the bounds of phi wrong. According to the ...
0
votes
0answers
32 views

Is it possible to calculate the lat/lon of a point knowing the extent coordinates?

I would like to know where it's possible to determine the latitude and longitude corresponding to a point inside a cartesian (planar?) representation of a geographical map, knowing the x/y coordinates ...
0
votes
0answers
29 views

Finding moment of intertia using spherical coordinates

I have taken a picture of the problem with my work. The answer I am getting is incorrect. I think I am doing something wrong after the following. Any help is appreciated `
0
votes
1answer
32 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
votes
1answer
63 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 ...
0
votes
2answers
174 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 ...
0
votes
0answers
14 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 ...
1
vote
0answers
48 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 ...
3
votes
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 ...
0
votes
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 ...
0
votes
0answers
12 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 ...
1
vote
1answer
28 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 ...
2
votes
2answers
50 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; $\hat{p} = \hat{x}\cos\theta + \hat{y}\sin \theta$ $$ ...
2
votes
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 ...
1
vote
1answer
38 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 ...
0
votes
2answers
64 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 ...
1
vote
2answers
29 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 ...
0
votes
1answer
29 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
votes
0answers
61 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 ...
1
vote
0answers
57 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 ...
1
vote
2answers
74 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
vote
1answer
57 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 ...
1
vote
0answers
30 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 ...
0
votes
0answers
24 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 ...
0
votes
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 ...
1
vote
2answers
75 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
votes
1answer
412 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
votes
1answer
40 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 ...
0
votes
0answers
32 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 ...
1
vote
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 ...
0
votes
2answers
86 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 ...
0
votes
2answers
93 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 ...
0
votes
0answers
30 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: ...
2
votes
1answer
47 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 ...
1
vote
1answer
276 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 ...
0
votes
1answer
30 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 ...
3
votes
2answers
156 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 ...
0
votes
0answers
30 views

lat/lon spherical coordinates to equidistant spherical coordinates

How to transform spherical data expressed in latitude/longitude pairs (parallels/meridians) in a new set of pair expressed just in parallels pairs? In other words, I need to transform data expressed ...
1
vote
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 ...
0
votes
0answers
49 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 ...
0
votes
1answer
33 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 ...
1
vote
1answer
79 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 $$ ...
1
vote
3answers
560 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 ...