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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19 views

Conversion to Spherical Coordinates

http://www.physics.usu.edu/Wheeler/QuantumMechanics/QMOrbitalAngularMomentum.pdf Unfortunately, this is the first time I've come across a conversion to spherical coordinates and I'm pretty lost. The ...
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218 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
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1answer
112 views

Finding the limits of a triple integral.

Evaluate: $$\iiint_V (x^2+y^2)\:\mathrm{d}x\:\mathrm{d}y\:\mathrm{d}z,$$ where $V$ is the region in the positive octant bounded by the sphere $\|\vec{r}\|=a$. I am unsure how to get the limits of ...
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0answers
236 views

Volume enclosed by two spheres (triple integral, cylindrical coordinates)

The question: Find the volume of the solid enclosed by the sphere $x^2 + y^2 + z^2 - 6z = 0$ , and the hemisphere $x^2 + y^2 + z^2 = 49 , z ≥ 0$ I set up the triple integral ...
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3answers
228 views

Find volume of the cap of a sphere of radius R with thickness h

I have to determine the volume aka the formula for the volume for this spherical cap of height h and the radius of the sphere is R. Two methods: *I just need help setting up the triple integrals ...
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1answer
125 views

Finding the volume using spherical coordinates

I am stuck on the limits of integration for rho. I tried 0 but that didn't work. Is this because I'd be finding the area under the $z=2$ plane? Question: Write a ...
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1answer
29 views

3D Fourier transforms of $e^{-\beta r} $ and $re^{-\beta r} $

I am trying to find the integrals $$\large\int\limits_{\mathbb{R}^3} e^{-\beta \left|\vec{r}\,\right|}e^{i \vec{q} \cdot\,\vec{r}} \mathop{d^3r}$$ $$\large\int\limits_{\mathbb{R}^3} ...
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1answer
200 views

How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian ...
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109 views

$\operatorname{div}$ and $\operatorname{grad}$ in spherical coordinates. Formula from general relativity goes crazy

I try to calculate the gradient of a function and the divergence of a vector field in spherical coordinates. Nothing special so far, but a formula that I learned in a general relativity lecture ...
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2answers
380 views

Equation for calculating azimuth between two points

Does anybody know an equation or approximation for calculating the azimuth as a function of latitudes and longitudes of both the points. For example I have Princeton, NJ is at 40.3571° N, 74.6702° ...
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98 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
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19 views

Projecting spherical components of a variable point to the unit vector of a fixed point

Consider the following vector function in spherical coordinates: $\mathbf{v} (r, \theta, \phi) = V_{\phi} (r, \theta, \phi) \mathbf{a}_{\phi} = A \delta(r - k) \displaystyle \delta \left( \theta - ...
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1answer
38 views

Problem with Laplacian while treating polar coordinates as special case of spherical coordinates.

I thought that polar coordinates ($r, \phi$) can be viewed as a special case of cylindrical coordinates ($\rho, \phi, z$) with $z=0$, or as spherical coordinates ($r, \theta, \phi$) with ...
2
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1answer
164 views

Find the volume of the solid bounded by the surface given in spherical coordinates by $R = 4-3\cos(\phi)$.

It is worth noting that $R$ in this case denotes the distance from origin to a point $P$ in space. You may be more familiar with $\rho$ instead of $R$. Here is my attempted solution: I am assuming ...
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2answers
130 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 ...
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1answer
8 views

Uniform sampling from part of sphere surface

I'd like to pose a question about uniform sampling on the surface of a sphere. I searched this site, and uniform sampling on a sphere surface seems to be quite a common problem. The common solution ...
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2answers
283 views

How to move a camera in “Flight Simulator” style (Roll, Pitch, Yaw) using a joystick

I am working on a video game where the camera movement (what the viewer sees) is controlled by a joystick. I want the camera movement to act like a flight simulator meaning the following: When the ...
2
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3answers
217 views

Find volume between two spheres using cylindrical & spherical coordinates

I've got two spheres, one of which is the other sphere just shifted, and I'm trying to find the volume of the shared region. The spheres are $x^2 + y^2 +z^2 = 1$ and $x^2 + y^2 +(z-1)^2 = 1$ I know ...
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0answers
58 views

Finding the coefficients of a partial differential equation after a change of coordinates.

I'm stuck in one of the mathematical steps of my physical problem. I've been following the derivation of my equations (starting at section 4) from this article Symmetric Euler-Angle Decomposition of ...
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98 views

Calculate Phi & Theta in Sphere

My target is to calculate phi and theta in a sphere. I built a panoramic viewer. In this viewer the user can load a picture. This picture will be load onto an sphere with a radius of 50. The user ...
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1answer
34 views

Trouble with understanding a spherical coordinate system.

We have a sphere with $r=1$, and we want the coordinates of $C$. $A$ is the north pole, and $AB$ is our prime meridian. See picture: I'm familiar with an $(x,y,z)$ coordinate system, but not so ...
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1answer
100 views

Transforming a matrix from cartesian to spherical coordinates

Consider a variable matrix $$\left[\begin{array}{ccc}a_{11}(x,y,z) \quad a_{12}(x,y,z) \quad a_{13}(x,y,z)\\ a_{21}(x,y,z) \quad a_{22}(x,y,z) \quad a_{23}(x,y,z)\\ ...
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2answers
58 views

Triple integral over a sphere with parameter $2n$?

I need to intergrate $x^{2n}+y^{2n}+z^{2n}$ over a sphere of equation $x²+y²+z²=1$. I have thought of changing the coordinates from cartesian to spherical but I don't know how to deal with the ...
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2answers
72 views

Multivariable Calculus Volume of Integration Question

I have an integral $$ \iiint\sqrt{x^{2} + y^{2} + z^{2}\,}\,{\rm e}^{-\left(x^{2} + y^{2} + z^{2}\right)}\, {\rm d}x\,{\rm d}y\,{\rm d}z $$ The integration region is bounded by the sphere ...
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1answer
31 views

Sphere to Caresian Coordinates and vis versa

I want to convert from Cartesian to sphere coordinates and back Here is my code ...
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1answer
33 views

Curving points to a sphere

I think math.stackexchange is the right place to post this, but if not, feel free to tell me. I have a series of points to be plotted on a sphere (Each one has a latitude and longitude value). These ...
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1answer
39 views

Spherical symmetry math

For spherical symmetry how the last four equations calculations is done? ccan you explain please? For reference see the equations 44
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1answer
44 views

Spherical Coordinates Representation

I just wanted to know what the set of all points in which spherical coordinates can be shown in more than one way is? I think it is only the origin but I am not sure
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1answer
62 views

Numerical solution needed for the quadratic equation (spheres)

There are three point coordinates $A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3)$, where $z_1=z_2=z_3$ Now, $(x-x_1)^2+(y-y_1)^2+(z-z_1)^2=d_1^2$ this equation can be written for my system as ...
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2answers
77 views

Computing distance from line to point in geodetic environment

Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ...
4
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2answers
111 views

Integral in 3 dimensions

I am trying to integrate $$ \iiint \delta(|\mathbf r| -R)\:\mathrm{d}^{3}\mathbf{r} $$ I know that $ \int f(r) \delta(r-R) d^3 \mathbf r =f(R) $, but when I try to apply this here I end up ...
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2answers
878 views

Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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1answer
154 views

How to calculate this integral in 3 dimensions involving the Dirac delta function?

How would I go about calculating the integral $ \int d^3 \mathbf r {1\over 1+ \mathbf r \cdot \mathbf r} \delta(\mathbf r - \mathbf r_0) $ where $\mathbf r_0 = (2,-1,3)$ My attempt so far: I have ...
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0answers
61 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
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1answer
58 views

Spherical coordinates + Laplacian

If $f(x, y,z)=f(r\sin(\phi)\cos(\theta), r\sin(\phi)\sin(\theta),r\cos(\phi))$, what is the value of $\Delta f$, in terms of $r$, $\phi$ and $\theta$?
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2answers
160 views

How to find all 3 orthogonal vectors to a 4D vector

For a program I'm writing, I need to find the vectors orthogonal to a given vector rotated at an arbitrary angle, and in 4D. It is a unit vector. For 3D, I found the two orthogonal vectors like ...
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66 views

Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
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2answers
281 views

How to calculate volume of a cylinder using triple integration in “spherical” co-ordinate system?

Lets have a cylinder given by $x^2+y^2=1$ which is cut from the top by plane $z=2$ and bottom by $z=-2$.I am having problem regarding the limits of ρ for the equation ∭ ρ sin^2ϕ dρ dϕ dθ where ϕ is ...
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0answers
52 views

Integrating Over Truncated Sphere

Consider the volume bounded by the sphere of radius R (with center $r_0$ = R/2 z) and the plane z = 0. Evaluate $\int dS$ over the surface of the truncated sphere. I'm not sure how to integrate over ...
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2answers
649 views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
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0answers
122 views

Conversion from Spherical Coordinates to Cartesian Coordinates aligned along arbitrary polar axis

I have spherical coordinates $w = (\theta, \phi)$ such that $\theta$ is the angle between $w$ and the polar axis (let's assume $z$ is up). Assuming $w$ is a unit vector, the conversion to cartesian ...
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2answers
407 views

Interpolating GPS coordinates

I can't profess to being a hardcore mathematician, I'm a computer scientist by nature, so please take it easy on me! There are a couple of similar questions on this, however, none seem to discuss the ...
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1answer
62 views

How to derive curl in spherical coordinates

This is one of those questions where I know I am making a dumb mistake someplace and I am trying to check where it is. $$ \begin{vmatrix} \frac{\bf\hat r}{r^2\sin\theta} & ...
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3answers
102 views

Writing triple integrals in spherical coordinates over nonspherical/nonconical regions

Defining upper and lower limits of integration for $\rho$, $\theta$, and $\phi$ is relatively easy when writing a triple integral in spherical coordinates if the region of integration is defined by ...
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1answer
109 views

Angled Spherical Sector - formulas?

I am glad to be here. First off, please excuse my almost saddening lack of knowledge. Math (in general) isn't my strong point. I might ask some really basic questions, you have been warned :) The ...
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51 views

How to rotate point on a spherical coordinate

I am looking for formula that shows hot to find the spherical coordinates of a point after the axis rotated. Assume that I am on earth and I have the coordinate of a point in lan/lot (which I believe ...
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1answer
54 views

Energy of particles on sphere and uniform rotation

I have a computer program containing some particles on a unit sphere, characterized by their positions $\{(\theta_i,\phi_i)\}$. They have a total energy given by the arc distance between particle ...
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1answer
100 views

Project point onto line in Latitude/Longitude

Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D. Diagram:
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71 views

A problem on vector calculus on spherical coordinate systems

I'm wondering whether I could get some idea on a problem I've been working on, here it is: h and g are two functions defined on the surface of a unit sphere, and their values depend on the ...
2
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1answer
182 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} \\ ...