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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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 ...
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0answers
18 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: ...
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1answer
31 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 ...
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1answer
49 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 ...
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1answer
26 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 ...
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2answers
37 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 ...
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0answers
10 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 ...
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1answer
67 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 ...
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21 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 ...
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1answer
19 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 ...
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1answer
54 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 $$ ...
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3answers
49 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 ...
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1answer
23 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system (r, θ, φ), θ can range from 0 to 2pi, but φ only varies from 0 to pi. Why is that?
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0answers
15 views

How shall I find the angle between two vectors inside a sphere using spherical coordinates.

How shall I find the angle between two vectors inside a sphere using spherical coordinates. I want to compute the angle between two vectors by spherical coordinates only and not by any ...
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0answers
27 views

Finite difference expressions for spherical coordinates

I have information in a spherical grid, that is, I have a value for each combination of $\rho$, $\psi$, $\theta$ ($\rho$ - distance from origin to center of grid cell, $\psi$ - elevation angle to ...
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2answers
48 views

How are arc components of a spherical system derived?

I am studying a flight dynamics book (see Flight Dynamics by Stengel) and am rusty on spherical coordinates. Commonly, aerospace coordinates use a North/East/Down right-hand system. So $z=-h$, ...
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2answers
67 views

Rotate a unit sphere such as to align it two orthogonal unit vectors

I have two orthogonal vectors $a$, $b$, which lie on a unit sphere (i.e. unit vectors). I want to apply one or more rotations to the sphere such that $a$ is transformed to $c$, and $b$ is transformed ...
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3answers
78 views

Volume of the region outside of a cylinder and inside a sphere

The cylinder is $x^2 +y^2 = 1$ and the sphere is $x^2 + y^2 + z^2 = 4$. I have to find the volume of the region outside the cylinder and inside the sphere. The triple spherical integral for this ...
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0answers
12 views

Representation of a cone in 3D

I need to find the representation of a cone in the 3D space with the following criteria: It's tip is located at the origin. It opens in the positive direction of the axis (it’s one-sided). The ...
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1answer
49 views

analytic solution poisson equation spherical coordinates

I'm trying to analitically solve a poisson equation $\nabla^{2}p(r,\theta,\phi)=f(r,\theta)$ in spherical coordinates; the boundary condition is $p(\infty,\theta,\phi)$=0. I'm quite used to the ...
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0answers
13 views

Integrating over particular grids to obtain Spherical Harmonic coefficients

Theoretically the spherical harmonic expansion coefficients of a function $f$ should be calculated via a continuous integration: $$F_{lm} = \int_{0}^{2\pi}\int_{0}^{\pi} ...
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0answers
34 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...
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3answers
43 views

2-Sphere surface coordinate dimension

Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...
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2answers
61 views

How can I evaluate this integral over a sphere, with surface area element instead of volume element?

$$\int_S (x^2 + y^2)d\sigma,$$ where S is the sphere of radius 1 centered at (0,0,0) and $\sigma$ is surface area. I would like some hints on how to proceed. This is tricky, since I am not being ...
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1answer
58 views

Help with limits of integration in spherical coordinates

$$\int_\Omega F(\theta,\phi) \sin(\phi) \, d\Omega$$ where $\Omega$ represents the (outer) surface of a spherical cap on a unit sphere. Specifically, let the center of the spherical cap be in some ...
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1answer
33 views

Problem in deducing gradient in spherical coordinates.

I know the differential displacement in spherical coordinate as $$dr \cdot \widehat{r}+ r d\theta\cdot\widehat{\theta} + r\sin\theta d\phi\cdot \widehat{\phi}$$. But I can't figure out how the ...
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1answer
32 views

Spherical Parametrization of a Cholesky Decomposition

[Background: Trying to build up my math base knowledge (self-taught) to follow an explanation on page 291 of this document, dealing with spherical parametrization to estimate unknown ...
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0answers
39 views

Calculating two rotation angles from xyz coordinates for dummies

This post is a bit verbose so that others who come later may benefit from my thick headedness. I am attempting to construct a primitives composition and constructed solids geometry parser/processor ...
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0answers
17 views

Particular Solutions to Time Independent Nonhomogeneous PDE's in Spherical Coordinates

I find myself working with PDE's like: $$\nabla^2f = G$$ or $$\nabla^2 f + k^2f = G$$ in spherical coordinates. I know that the homogeneous form of these equations (Laplace's and Helmholtz) have ...
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2answers
36 views

How do I calculate the angles between a point on a sphere and each unit vector in $\Bbb R ^3$?

Given the Cartesian coordinates of any point $p$ on the surface of a sphere in $\Bbb R ^3$, how do I calculate the angles between each axis $(x, y, z)$ and the vector $n$ defined by origin $o$ and ...
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2answers
108 views

Geometry: What is being calculated here?

Context: I am a computer graphics programmer looking at a code-implementation. I need help understanding a function that has neither been documented properly nor commented. Given a circle with ...
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0answers
24 views

periodic radial basis function

A have a point cloud ,described in spherical coordinates, which I need to fit with a smooth surface. I'm trying to do this with a bivariate radial basis function network, which operates on a spherical ...
4
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1answer
47 views

Coordinates on the sphere not global?

I'm reading a book on differential geometry and some part of the introduction I do not understand but I'm curious to understand it. Maybe someone can try to explain those parts to me. "Each point on ...
2
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1answer
39 views

Why must power series count up by integers 0,1,2.. in 3D harmonic oscillator in spherical coordinates?

http://www.physicspages.com/2013/01/17/harmonic-oscillator-in-3-d-spherical-coordinates/ http://quantummechanics.ucsd.edu/ph130a/130_notes/node244.html These are two links that have roughly the same ...
3
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2answers
38 views

Using spherical coordinates to find volume of a region

Use spherical coordinates to find the volume of the region lying above $z = \sqrt{3x^2+3y^2}$ and within the $x^2+y^2+z^2=2az$, $a>0$. So far I know that the first graph is a cone and the second ...
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1answer
37 views

Finding volume of a cone using triple integral

The cone has the formula: $x^2 + y^2 = z^2 , 0≤z≤2$ So I used the cylindrical coordinates to get the following answer: $$\int_0^{2\pi}\int_0^2\int_0^2 dz\,rdr\,d\theta = 8\pi$$ In the solution of ...
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1answer
20 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
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1answer
12 views

How do I get vectors orthogonal to the one generated by the spherical coordinate formula?

Given a formula: F : ℝ → ℝ → ℝ3 F(θ,φ) = (cos(φ)*sin(θ), sin(φ)*sin(θ), cos(θ)) what are the formulas: ...
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21 views

Proving that the Jacobian for spherical coordinates and the Jacobian for integrating over a sphere coincide

I want to use the formula for switching an integration from spherical to cartesian coordinates : $\int_{RS^{n-1}} \! f(x) \, \mathrm{d}\sigma (x)$=$\int\int...\int ...
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1answer
22 views

volume using spherical coordinates

Let $$V = \{(x, y, z): x^2 + y^2 ≤ 4 , 0 ≤ z ≤ 4\}$$ be a cylinder and let $P$ be the plane through $(4, 0, 2), (0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I'm having ...
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27 views

Solving LES containg spherical coordinates

i have a three-dimensional parametric equation of a line, where the directional vector is normalized and converted to spherical coordinates to calculate a angle offset. It looks like this: ...
3
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1answer
95 views

Circle from three points on surface of sphere

I need to compute the circle on the surface of a sphere given three points on that very surface. It is very easy to do that in Euclidean geometry, but the sphere has no x and y, but just two angles ...
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0answers
27 views

Comparing paired objects in 3d space using direction angles.

I have data about paired objects in 3d space, where each object is defined by three components. What I would like to do is compare the orientations of the paired objects and see if one group of those ...
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0answers
6 views

Midpoint of two n-vectors

Currently I am using n-vectors (more information can be found here) for an accurate and singularity-free representation of coordinates on the earth's surface. For various reasons, I need to compute ...
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0answers
43 views

Intersection of two spherical caps in $(n+1)$-dimensional Euclidian space

I want to compute the intersection area of two spherical caps whose tops are placed on two orthogonal axes in $\mathbb{R}^{n+1}$. In spherical coordinates, the surface element is given by $$ \,dS = ...
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0answers
21 views

Normal form of plane

I propose a simple equation of plane generalized from 2D Normal form of straight line as follows, starting from Straight line Normal form: $$ x\, \cos \alpha + y \sin \alpha = p. \tag{1}$$ ...
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28 views

Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
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1answer
117 views

PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + ...
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1answer
29 views

Integrating a sphere by discs vs shells (spherical coordinates)

I am getting very confused about the following. Let's say I want to find the volume of a sphere. I can start with a circle having circumference $2\pi R\cos\theta$. I can multiply by $R d\theta$ and ...
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15 views

Cover the entire sphere with a moving cone

Assume you have a cone with half-angle theta in a 3D cartesian space, with the vertex of the cone in the origin, and that you want to rotate the cone along a curve so that it covers the entire sphere. ...