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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expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
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163 views

Diffusion in Spherical Coordinates with mixed BC

I have been working through the book "A Guide to First-Passage Processes" and wanted to branch out on my own doing a calculation similar to what occurs in chapter 6. My basic problem comes from the ...
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1answer
84 views

Determining new coordinates after a rotation of a sphere

Imagine that I am standing at a place on Earth, using coordinates of say N41 W74. Now the Earth's axis rolls 90 degrees, causing the N/S axis to become the equator, and rotation resumes as before. ...
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595 views

Inverse Jacobian matrix of spherical coordinates

I found inverse transformation from spherical coordinates to cartesian coordinates (on $x>0$, $y>0$ and $z>0$). I have $$ r = w_1(x,y,z) = \sqrt{x^2+y^2+z^2} $$ $$ \theta = w_2(x,y,z) = ...
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686 views

Calculate the volume between two spheres

I have to find the volume between the two spheres. Their equations are $x^2+y^2+z^2=2$ and $x^2+y^2+(z-\frac1 2)^2=\frac 1 4$ for $z>0$ The first one has center $(0,0,0)$ with $r=\sqrt 2$ and the ...
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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
2k views

Can you not rotate spherical coordinates?

I have some points that sit on the hemisphere in spherical coordinates: $\theta \in [0,\pi/2]$, $\phi \in [0, 2\pi]$ (ie so a hemisphere around the vector (1,0,0) (spherical coordinates). I should ...
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2answers
89 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 ...
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2answers
3k views

Parametric Equation for Great Circle

So I've been doing a lot of searching and haven't found exactly what I'm looking for. My math skills are a bit rusty, so I haven't had luck deriving this on my own. What I'm looking for is an ...
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2answers
2k views

Find the average value of this function

Find the average value of $e^{-z}$ over the ball $x^2+y^2+z^2 \leq 1$.
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2answers
50 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 ...
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2answers
2k 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 ...
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2answers
820 views

Probability density function for radius within part of a sphere

I would like to find the probability density function for radius within a given section of a sphere. For example, suppose I specify $\pi / 4 < \theta < \pi / 3$ and $\pi /7 < \phi < \pi ...
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4answers
773 views

Coordinates notation in Spherical polar system

There is a number of conventions for specifying coordinates in Spherical polar coordinate system: ($r$, $θ$, $φ$), ($r$, $φ$, $θ$), ($\rho$, $\theta$, $\phi$) and even ($r$, $\psi$, $θ$). The article ...
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1answer
732 views

Optimal distribution of points over the surface of a sphere

How can one generate a distribution of N points over the surface of a sphere so that the all N voronoi cells have the same area? Which is the best algorithm for this?
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2answers
43 views

Solve double integral

$$ \int_0^2 \int_0^{4-x^2} \frac{xe^{2y}}{4-y} \, dy\, dx $$ I'm stuck with this problem. I think I should change it so I integrate with respect to $dx \, dy$ but I'm not sure. Any help? Thanks
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2answers
78 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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2answers
699 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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1answer
60 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
351 views

How to verify a conversion to spherical coordinates?

Is it possible to verify if a conversion of an integral in Cartesian coordinates to spherical coordinates was done correctly other than revising it looking for mistakes? I mean, is there some kind of ...
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3answers
126 views

Doubt with bounds and integrand of $\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{2}\rho^2\sin{\phi}d\rho d\phi d\theta$

Question as follows. Find the volume of the solid enclosed between the spheres $x^2+y^2+z^2=4$ and $x^2+y^2+z^2=4z \Leftrightarrow x^2+y^2+(z-2)^2=4$. I constructed the following integral and after ...
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1answer
3k views

Converting Lat/Long coords to Cartesian X/Y, then calculating shortest distance between point & line segment

I'm having an issue with accuracy when converting Lat/Long coordinates to X,Y and then finding the shortest distance from a Point to a Line with said coordinates. The distance is off by around 40-50% ...
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1answer
28 views

Area of surface parametrized in spherical coordinates

Suppose we have a smooth, bounded, closed surface in $\mathbb{R}^3$ which can be parametrized by giving the distance from the origin as a function $r(\varphi,\theta)$ of spherical angles ...
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1answer
76 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 ...
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2answers
28 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 ...
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1answer
112 views

Laplace operator in Spherical Coordinates, a formal 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 ...
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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
66 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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1answer
126 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
1k views

Spherical harmonics expansion for a particular function

On the unit sphere, each square-integrable function can be expanded as a linear combination of spherical harmonics : $$ f(\theta,\phi) = \Sigma_{l=0}^\infty \Sigma_{m=-l}^{+l} f_{lm} Y_{lm} ...
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1answer
130 views

Volume of a 3D sphere of radius $R$ using Riemannian metric in stereographic coordinates

The question is pretty much in the title. We were also given the hint that it could be useful to use spherical coordinates when calculating the integral (the actual answer is not required, just its ...
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2answers
557 views

Discretize a circle on a sphere with a given center and radius

I would like to draw a discretized circle on the surface of a sphere (the Earth in that case). The input of the algorithm would be the center of the circle (expressed as longitude and latitude), the ...
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1answer
263 views

full hessian, spherical coordinates

The question itself is pretty simple. I am running into confusion. Seems like there is a typo in the book. I wanna check myself. Maybe I am doing something wrong. Suppose we have the function (which ...
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1answer
557 views

projection of sphere on the circumscribed cube

Suppose I have a sphere. Inside the sphere I have an inscribed cube. What I am interested in is finding out what is the latitude and longitude (or coordinates) of a point on the sphere which will be ...
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1answer
1k views

Plotting a star's position on a 2D map

First off, let me say that I'm not a Mathematical wizard. Be easy on me :) I am looking to draw a subsection of space using OpenGL on the iPhone. I have a star's RA and Dec and am needing to somehow ...
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1answer
15 views

Vector flux through a segment of a sphere

Given the vector field $\vec A(\vec r) = \vec r$, I have to calculate the vector flux through a sphere whose center is located in the origin. I want to apply Gauß-Theorem and use spherical ...
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1answer
30 views

Surface integral of $x^4+y^4+z^4$ over the sphere $x^2+y^2+z^2=a^2$

After doing regular methodology have reached upto integral shown in figure , but when i eliminate z from it it becomes very complicated to solve .Is there any other way to solve this .Thanks
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1answer
51 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 ...
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1answer
83 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 ...
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1answer
31 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 ...
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1answer
21 views

vector components and dot product with unit vector

$E_{0}\hat z=\vec E_{0}rcos\theta=E_{0}cos\theta\, \hat r$ and $E_{0}cos\theta\, \hat r\circ \hat r =E_{0}cos\theta$ This just doesn't look right to me for some reason...
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1answer
113 views

Random points in sphere with probability p(r)

How to pick random uniformly distributed points in a sphere has been asked before. The difference is that I don't want uniform distribution, rather I would like the number density to scale by ...
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1answer
300 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
53 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
41 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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3answers
696 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
702 views

Express spherical coordinates with different centers in terms of each other.

Imagine that you have two spheres with a distance $R$ from one center to the other one. Now, it is well known how one would get the cartesian position vector of each point in sphere 1 by using ...
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1answer
207 views

Convert spherical coordinates to Cartesian coordinates for a vector

So let's say I have a normalized vector $N$ given in cartesian coordinates and I have another normalized vector $V$, defined in spherical coordinates relative to the vector $N$. So $\theta_V$ is the ...
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1answer
285 views

Spherical coordinates of a unit vector around a normal $N$

So if I have a unit normal for a surface $N(x,y,z)$ and an incident unit vector $V(x,y,z)$ to that surface, how would I represent the vector V in spherical coordinates relative to the normal?
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1answer
305 views

Transformation to spherical coordinate system

If I have a sphere $T: x^{2}+y^{2}+z^{2}\leqslant 10z$ by transformation to the spherical coordinate system by the: $ x=r\cos\theta\sin\varphi\\ y=r\sin\theta\sin\varphi\\ z=r\cos\varphi $ What is ...