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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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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237 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
143 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
419 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
3k 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
371 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
49 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
962 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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129 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
1k 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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1answer
60 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 ...
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1answer
286 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
384 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
17 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
501 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 ...
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3answers
666 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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1answer
68 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
289 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
74 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
79 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
43 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
39 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
47 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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60 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
80 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
111 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 ...
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2answers
119 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
1k 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
280 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
122 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
69 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
421 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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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
1k 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
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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
1k 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
243 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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3answers
993 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
85 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
131 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
149 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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1answer
62 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
149 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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1answer
305 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} \\ ...
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1answer
62 views

Consider the region shared by ρ=8cos(φ) and ρ=4. Find the volume of the region.

Consider the region shared by ρ=8cos(φ) and ρ=4. Find the volume of the region. I know it involves a triple integral, but do not understand how to set up the integral.
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2answers
697 views

How to calculate the angle between two vectors, defined by 3 points on the earth?

I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values. ...
3
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1answer
178 views

Integration in n-spherical coordinates

I'd like to compute the following integral: $$I = \int_{\mathbb{R}^n} {\rm d}^n x \; \frac{e^{i \vec x \cdot \vec k}}{\vec x^2}$$ My first step is to use generalized spherical coordinates and then I ...
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3answers
1k views

Solve the triple integral $\iiint_D (x^2 + y^2 + z^2)\, dxdydz$

How does one go about solving the integral: $$ \iiint_D (x^2 + y^2 + z^2)\, dxdydz, $$ where $$ D=\{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + z^2 \le 9\}. $$ I believe I am supposed to convert to ...
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1answer
154 views

Divergence in spherical coordinates

On the one hand there is an explicit formula for divergence in spherical coordinates, namely: $$ \nabla \cdot \vec{F} = \frac{1}{r^2} \partial_r (r^2 F^r) + \frac{1}{r \sin \theta} \partial_\theta ...
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2answers
108 views

can a great circle route be predicted from initial condition?

Assuming the world is a sphere with no wind, can the great circle route of a vessel be predicted from the current position $\{\phi_i,\lambda_i\}$ and the current true course $\theta_i$? Presently, ...