Questions tagged [spherical-coordinates]

Question 1

Evaluate the integral $$\iiint_{D} e^{(x^{2}+y^{2}+z^{2})^{3/2}} \, dV$$ where $D$ is the region above the cone $z=\sqrt{x^{2}+y^{2}}$ and below the hemisphere $z=\sqrt{1-x^{2}-y^{2}}$ using spherical ...

Question 2

According to this wiki page here, given the vector spherical harmonic $$ \boldsymbol{\Phi}_l^m = \mathbf{r} \times \nabla Y_l^m $$ its Laplacian is, $$ \nabla^2 \boldsymbol{\Phi}_l^m = -\frac{l(l+1)}{...

Question 3

I derived the following product formula for the volume of an $N$-ball: $$V_{N}(R)=\frac{2^{N-1}\pi^{\frac{N}{2}}R^{N}}{N}\prod_{k=1}^{N-2}\frac{\Gamma(\frac{k+1}{2})}{k\Gamma(\frac{k}{2})}$$ The ...

Question 4

I recently had a homework problem for my physics class that required me to evaluate the following spherical integral $$ \int\int\int \alpha e^{- \beta r^3} {|\sin(\theta)|}^2 \sin(\phi) \ dr d\theta d\...

Question 5

Is there a 3D coordinate transform which turns rotation in cartesian coordinates into translation in the transformed coordinate system? It would be sufficient if the transformation has the desired ...

Question 6

I have a specific example, computing the volume within part of a hemisphere of radius $2$ centered at the origin, and bounded above by the plane $z = 1$ (if it were a cone or pyramid, this volume ...

Question 7

Is the only reason the polar angle is measured from the north pole to the south pole (0$^\circ \leq \phi \leq 180^\circ$) by convention so that it has no negative measurement values ?$\;$ I find ...

Question 8

I have a plane in 3D space defined in terms of two variables, $u$ and $v$: $$ \begin{aligned} & x × \frac{\sqrt{\frac{2}{3}} × \left(\sin\left(\frac{π}{8}\right)^2 × \left(u^2 + v^2 + 2\right)...

Question 9

I have a unit sphere divided into 14 patches: 6 identical square-ish sections and 8 identical triangle-ish sections, with the arrangement and all of the symmetries of a cuboctahedron. The boundaries ...

Question 10

Fix $a > 0$. Let $\textbf{F} = \left\langle xz, x, y \right\rangle$, and let $S$ be the surface given by $$x^2 + y^2 + z^2 = a^2 \qquad y \geq 0$$ Compute $$\iint_{S} \textbf{F} \cdot \,d\textbf{r}$...

Question 11

In this paper, Montina gives a reformulation of the Kochen-Specker model for spin measurements of an electron (section B). To prove that it really does reproduce the probabilities given by the Born ...

Question 12

Evaluate the integral over $\mathbb{R}^n$ $$\int_{\mathbb{R}^n} \frac{|x|^{2}}{|x|^{8} + 2|x|^{4}\cos \left( \theta \right) + 1} \, dx \quad \textrm{where } \theta \in \left( 0, \frac{\pi}{2} \right)$$...

Question 13

I am interested in solving the heat equation in spherical coordinates for a situation where there is a “pulse” at $t=0$ given at $r=0$. I think this means that a Dirac delta pulse is the initial ...

Question 14

I have a closed region on the unit sphere expressed as a series of azimuth, elevation points $(\theta,\phi)_i$. I need to calculate the solid angle subtended by the enclosed region. I know that the ...

Question 15

I have two concentric spheres: Inner sphere with radius r₁ Outer sphere with radius r₂, where r₁ < r₂ On the surface of the outer sphere, there is a spherical triangle (defined by three geodesic ...

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Questions tagged [spherical-coordinates]

Answer Key Mistake? Triple Integral in Spherical Problem

Laplacian of vector spherical harmonic

Showing $\frac{2^{N-1}\pi^{N/2}R^N}N\prod_{k=1}^{N-2}\frac{\Gamma(\frac{k+1}2)}{k\Gamma(\frac k2)}=\frac{\pi^{N/2}}{\Gamma(\frac N2+1)}R^N$

Evaluating $\int \alpha e^{-2 \beta r^3}dr$

3D coordinate transform that turns rotation into translation

Bounds of integration in spherical polar coordinates: Volume of base of hemisphere bounded above by a plane

Is it legitimate to alter the spherical coordinate system?

Is it possible to solve this system of equations to convert the equation of a plane to spherical coordinates?

Is it possible to write a set of 2D functions that are each positive for only one section of this divided sphere?

How can $\iint_{S} \textbf{F} \cdot \,d\textbf{r}$ be determined more efficiently?

Evaluating an integral over spherical lunes

Evaluating $\int_{\mathbb{R}^n} \frac{|x|^{2}}{|x|^{8} + 2|x|^{4}\cos \left( \theta \right) + 1} \, dx $

Heat equation in spherical coordinates with Dirac delta function as initial term

Calculating the solid angle of a 'polygon' in spherical coordinates

3D Volume Enclosed by a Spherical Cap and a Point on an Inner Sphere

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