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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How to compute the following Jacobian

I need to show that the Jacobian of the n-dimensional spherical coordinates is $$\displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$$ then I have computed the Jacobian matrix, ...
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On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
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2answers
31 views

Evaluating integrals in R^m

Let $|\cdot|_m$ denote the Euclidean norm in $\mathbb{R}^m$. Then I wish to prove that $\displaystyle\int\limits_{\mathbb{R}^m}|x|_me^{-|x|_m}dx<\infty$ It's kinda embarrassing to say this, but ...
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61 views

Can anyone check if this correct?

Convert to spherical coordinates and evaluate:$$\iiint_{E}z(x^2+y^2+z^2)^{-3/2}dV$$ where E is the region satisfying the following inequalities:$$x^2+y^2+z^2\le16,z\ge 2$$ This is what i have done so ...
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2answers
42 views

How to tell if two spherical coordinates lie on the same plane

I have the rho, theta, and phi values of two points, how can one tell that two vectors are normal to the same plane by looking at their spherical coordinates?
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62 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 ...
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1answer
1k 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 ...
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1answer
69 views

Coordinates of tilted circle.

The original question is as follows: Imagine a wire located at the intersection of $x^2+y^2+z^2=1$ and $x+y+z=0$, whose density depends on position according to $\rho({\bf x})=x^2$ per unit length. ...
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2answers
64 views

Laplacian $\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r})= \frac{1}{r} \frac{\partial ^2 }{\partial r^2}(r \phi )$

Does anyone have any intuition on remembering or very quickly deriving that $$\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r}) = \frac{1}{r} \frac{\partial ^2 ...
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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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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
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} \\ ...
2
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1answer
155 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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1answer
1k views

Reverse use of Haversine formula

Alright the title is not the best. What I want to do is to change the given parameters in Haversine's formula. If we know the lat,lng of two points we can calculate their distance. I found the ...
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107 views

How do you set up the integral in spherical coordinates in the following problem?

Find the volume bounded by the surface $z = x^2 + y^2$ and $x^2+y^2 = 1$ in the first quadrant. The answer is $\pi/8$ using rectangular and cylindrical coordinates and that is the correct answer, but ...
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61 views

Finding the coordinates of the corners of an aligned pole-centered spherical square

Given a spherical square of radius $1$, with edge midpoints at $(1, x, 0)$, $(1, x, \pi/2$), $(1, x, \pi)$ and $(1, x,3 \pi/2)$ (in the spherical coordinate system of (radial distance, polar angle, ...
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0answers
110 views

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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185 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
90 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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674 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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745 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
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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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2answers
129 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
1k 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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1answer
798 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
102 views

Solving a non-linear, multivariable system of equations

I'm researching the mathematics behind GPS, and at the moment I'm trying to get my head around how to solve the following system of equations: $\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}=r_1$ ...
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2answers
63 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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1k 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
861 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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2answers
44 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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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
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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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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403 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
129 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
88 views

Manifolds, coordinate systems, books

Which books, say Lee's Introduction to Smooth Manifolds or Munkres' Analysis on Manifolds explains how the theory of a differentiable manifolds can be used to solve a problem that is expressed in a ...
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1answer
50 views

Convert $x^2+y^2+z^2=49$ to spherical coordinates

I really need your help to convert this $x^2+y^2+z^2=49$ to spherical coordinates. I tried it and I got $(R\sin\phi\cos\theta)^2+(R\sin\phi\sin\theta)^2+(R\cos\phi)^2=49$. But it says that it is ...
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1answer
36 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
110 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
35 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
134 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
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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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
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
137 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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1answer
322 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
576 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 ...