Questions tagged [spherical-coordinates]
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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Proof of one of the equations from Schwartz's QFT book
I am attempting to prove an integral that appears in the appendices of the book "Quantum Field Theory and Standard Model," specifically equation (B.25):
$$ \dfrac{1}{(2\pi)^{d}}\int\limits_{\...
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Conversion from machine-local polar coordinates (yaw, pitch, roll) to global (azimuth, elevation)
I'm writing a 3D LOGO module in Python.
I recently realized that my approach is flawed. What I implemented is: the turtle besides right/left commands which change its azimuth in the xy plane, received ...
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Calculating Cosine Similarity Between Two Points in Hyperspherical Coordinate
I'm working with points in an n-dimensional hyperspherical coordinate system, in other words, my points are in the shape $(r, \theta_1, \theta_2, ..., \theta_{n-1})$. I want to calculate the angle ...
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Seeking Cartesian to Hyperspherical Coordinates Conversion Examples in Higher Dimensions
I've recently coded a function that transforms Cartesian points into hyperspherical points. While I've verified its accuracy for 2D and 3D points, I'm finding it challenging to test for higher ...
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Understanding the Formulas for Cartesian to Hyperspherical Coordinate Transformation
I am in the process of coding a function to convert Cartesian coordinates to hyperspherical coordinates. However, I've encountered some confusion regarding the transformation formulas.
On the ...
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Jacobian of azimuth and elevation angles with respect to unit vector
We know that azimuth ($\theta$) and elevation ($\phi$) angles can represent a unit vector as $\mathbf{e}=\begin{bmatrix}\cos\theta\cos\phi \\ \sin\theta\cos\phi \\ \sin\phi\end{bmatrix}$. It is easy ...
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Poisson equation spherical-radial only coordinates with second type boundary condition at the origin
I have an issue with a basic Poisson's equation:
$$ \frac{1}{r^2} \partial_r r^2 \partial_r V = - f(r)$$
with $r > 0$, $f(r) > 0$ has the following property:
$$ \int_0^{\infty} r^2 f(r) d r = 1 $...
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products of real spherical harmonics
Based on the description in wikipedia and the book: Modern Quantum Mechanics (Sakurai & Napolitano), any product of two complex spherical harmonics follows the contraction rule:
$$Y_{\ell_1}^{m_1}...
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Volume of a sphere cut out by a cylinder
The goal is to find a formula for the volume of a sphere $ x^2 + y^2 + z^2 \le R^2 $ cut by a cylinder $ (x - \frac{R}{2})^2 + y^2 \le \frac{R^2}{4} $
I was able to solve it using double integrals as ...
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Divergence in cartesian coordinates conflicts with spherical divergence.
Consider the vector
$$
\vec{a}=f(r)\hat{r}+r\hat{\theta}+r\hat{\phi}
$$
Where $\vec{a}$ is a vector in spherical coordinates and $f(r)$ is a function of $r$.
Let's calculate $\nabla\cdot(\vec{a}\times\...
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Constructing transformation of an extended spherical movement
I have a frame of reference positioned on a surface of a sphere with the $z$-axis always pointing towards the center of this sphere, which is at distance $d$.
Now given:
spherical angle $\phi$
...
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Parametric Surfaces Appearing completely different in graphing software than in original paper
I tried to plot the parametric surfaces from this paper to try to play around with the variables and potentially adapt it for other purposes. However I attempted to plot the surfaces in both Geogebra ...
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Finding common tangent lines of two circle in spherical surface [closed]
I am able to find common tangent lines of two circles in 2D space ( point is x,y ).
but the problem is can I use the equation to be used in spherical surfaces ( ...
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Show that $K(z,t) \equiv \int_{\mathbb{R}^3}{e^{i(z,\xi)}\dfrac{\cos {|\xi|t}}{|\xi|^2}d\xi} = K(0,0,|z|,t)$, $z \in \mathbb{R}^3$ in spherical system
Show that $K(z,t) \equiv \int_{\mathbb{R}^3}{e^{i(z,\xi)}\dfrac{\cos {|\xi|t}}{|\xi|^2}d\xi} = K(0,0,|z|,t)$, $z \in \mathbb{R}^3$ when switched to spherical coordinates. According to my textbook this ...
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Sphere Coordinates
How do you convert the set $$ K = \{((x, y, z) \in \mathbb{R^3}
| x^2 + y^2 + z^2 ≤ 25, -4 ≤ x ≤ 4, -4 ≤ y ≤ 4, -4 ≤ z ≤ 4)\} $$ into spherical coordinates?
It's evident that the radius ranges from 0 ...
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Integral over angular spherical coordinates vol. 3
This question is related to two older posts of mine: Integral over spherical angular coordinates vol. 2 and Integral over angular spherical coordinates. None of the above contain the answer to what I ...
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What is the parametric path of an equalateral spherical triangle in the first octant on the unit sphere with the edges equadistance from each plane?
I want to clarify a bit. The vertices need to be the same distance a path that follows from (1,0,0), (0,1,0), and (0,0,1) and converges at ($\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}...
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How do I find edge points on an arc given radius, sweep and start angle
I'm trying to draw an arc with rounded edges in one of my project, but i'm having hard time figuring out Math to draw the rounded edged on an arc.
this is what I ...
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Finding the center of a series of points on the surface of a sphere
I have a series of points on a unit sphere that are given in azimuth, elevation coordinates, where azimuth has a domain of -180 < azimuth <= 180 degrees, and elevation has a domain of -90 <= ...
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Explanation of expanded spherical harmonics?
I’m reading through A Physical Introduction to Suspension Dynamics and I am having trouble understanding the jump between equations $(2.5)$ and $(2.6)$ in the photo. How do the partial derivatives in $...
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Volume of intersection of two partial spheres having origins at different coordinate frames using spherical coordinates
Assume I have a world coordinate frame $\mathbf{w}$.
Assume I have a second coordinate frame that can be parameterized as a $4\times4$ homogenous transformation matrix with respect to the world ...
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Euler angles to ENU coordinates
Given the LLA coordinates and Euler angles (orientation) of a phone, where alpha = beta = gamma = 0 when the top of the phone is pointing north and facing up, I would like to find the unit ENU ...
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Spherical Helmholtz-wave equation with variable wave number
I am looking for a function of variable $r > 0$ (spherical radius), $f(r) \geq 1$, that obeys to:
$$ (\Delta_r + \alpha \delta(\boldsymbol{r}) ) f(r) = 0$$
with $\alpha > 0$ the only condition $...
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When are spherical harmonic expansions valid?
It is known that a square integrable function on the sphere can be expanded in a basis of spherical harmonics,
$$
f(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l c_l^m Y_l^m(\theta,\phi)
$$
where $\...
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Deriving the gradient operator in spherical coordinates
This is a sort of problem where I know what to do but do not completely understand what I am doing.
I have been taught how to derive the gradient operator in spherical coordinate using this theorem
$$\...
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Change of variable under surface integrals
I'm seeking clarification on the following identity involving surface integrals and partial derivatives:
$B_{\rho} = B(y, \rho) \subset \mathbb{R}^n$ represents the ball centered at $y$ with radius $\...
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How are the tangential divergence and tangential Laplace operators defined in spherical coordinates?
Suppose that the 2D vector field
$$\boldsymbol{u} (x,y,z) = u_x (x,y,z) \hat{\boldsymbol{e}}_x + u_y (x,y,z) \hat{\boldsymbol{e}}_y$$
satisfies the property
$$
\boldsymbol{\nabla}_\parallel \cdot \...
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Evaluating the triple integral of $ f(x,y,z) = e^{(x^2+y^2+z^2)^\frac{3}{2}} $
as part of a math tutorial I am supposed to evaluate the following integral:
$$ \int_{-1}^1
\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}
\int_{-\sqrt{1-x^2-y^2}}^\sqrt{1-x^2-y^2}
e^{(x^2+y^2+z^2)^\frac{3}{2}}
...
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How can calculate the area of the region as follows?
Thank you for reading!
I want to calculate the area as follows.
$$
z > 0,
x^2 + y^2 + z^2 = r^2 ,
\left|\frac{x}{\sqrt{x^2+y^2+z^2}}\right| < \frac{\sqrt{2}}{2} ,
\left|\frac{y}{\sqrt{x^2+y^2+z^...
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Parametrize 4-d Torus to standard Torus equations
Based on a comment here, the following 4-d parametrization represents a Torus (it also happens to lie completely on $S^3$):
$$
x=3/5 \cos(\theta)\\
y=3/5 \sin(\theta)\\
z=4/5 \cos(\phi)\\
w=4/5 \sin(\...
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Converting between two spherical coordinate systems with an application to astronomy
I am currently working on a sub-problem of a larger problem which involves being able to efficiently align an equatorial telescope mount with greater precision than the conventional amateur method, ...
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evaluate $\iiint\limits_{B}^{} \cosh \left ( x+y+z \right ) dxdydz$, where $B=\left \{ \left ( x,y,z \right ) \in R^3|x^2+y^2+z^2\le 1 \right \} $
Evaluate $\iiint\limits_{B}^{} \cosh \left ( x+y+z \right ) dxdydz$, where $B=\left \{ \left ( x,y,z \right ) \in R^3|x^2+y^2+z^2\le 1 \right \} $
Since B is a unit ball which centers at the origin,
...
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Spherical → Spherical Coordinate System Matrix Transformation, Assuming Common Origin. Geographic vs Geomagnetic North Pole
Background: I would like to model the Earth's magnetic field with the magnetic dipole approximation. This is a low-level approximation with the following magnetic vector field in spherical coordinates ...
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Extract Spherical coordinates of panorama from camera pose information and panorama (+pixel) size for correct position in a 3D space.
I am working with spherical image representation. I have the following camera pose information:
[pose.rotation.x, pose.rotation.y, pose.rotation.z, pose.rotation.w],
[pose.translation.x, pose....
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How to change from cartesian coordinate to cylindrical coordinates
Consider the triple integral
$$\iiint_{K} \frac{z}{2+ x^2 + y^2} dV$$, where K is the region defined by $z \geq \sqrt{x^2 + y^2}$ and $x^2 + y^2 + z^2 \leq 9$.
The question then asks me to rewrite the ...
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Converting a cross product typically found in electrodynamics between coordinate systems
Context
There are numerous posts on mathstackexchange and physicsstack exchange that seek clarity regarding conversion from a Cartesian coordinate system to curvilinear coordinate system, or viceversa ...
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Magnitude of Vector Field at a point in Spherical Coordinates
I am fuzzy on the process for the following question. If I could get a walk through on the solution process I would be very appreciative. The soln manual only gives the answers.
Field and Wave ...
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Dirac delta function in spherical cordinates
I am studying Dirac Delta Function in Spherical Polar Cordinates. I found this expression
$$\delta\ ^3 (\vec r -\vec r_{o}) = \frac{ \delta\ (r -r_{o})\delta\ (θ -θ_{o})\delta\ (φ -φ_{o} )}{r^2 \sin ...
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Laplace–Beltrami of a function in spherical coordinates
In a proof I want to compute the laplacian of a function in spherical coordinates. We know that : $$|x|^2\Delta=(r\partial_r)^2+(n-2)r\partial_r+\Delta_{\mathcal{S}^{n-1}}.$$
For a function $f$, I ...
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What is the equation of warped disk in Cartesian or Spherical coordinate system?
My teacher gave us the assignment to find the moment of inertia of any shape you want. So I decided to find the moment of inertia of our milky way galaxy.
I found out that our galaxy is shaped like a ...
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How to find the mean distance between a point inside a ball and the surface of this ball?
Say I have a sphere of radius $R$, I would like to know the average distance between a point $M$ inside the ball and the sphere of radius $R$.
Because of the spherical symmetry, we can say without ...
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How to find center of an arc given a geographical start point, end point and arc angle?
I am working in a way of drawing transitions in some kind of "Flight Plan" between two "waypoints" using the heading at the beginning of the arc and at the end of an arc. Given an ...
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How to apply the definition of line integral when we have a vector field in spherical coordinates?
Here is the definition of line integral as it appears in Apostol's Calculus, Volume II
Let $\pmb{\alpha}$ be a piecewise smooth path in n-space defined on an
interval $[a,b]$, and let $\pmb{f}$ be a ...
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Why does $\int_{-\infty}^\infty e^{ar}\nabla^2\nabla^2 e^{br} = \int_{-\infty}^\infty e^{br}\nabla^2\nabla^2 e^{ar}$ for a self-adjoint operator?
If the Laplacian operator in spherical coordinates is:
$$ \nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)$$
then for $a<0$, $\int_{r=0}^\infty e^{ar}\...
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How to calculate the hemisphere using triple integrals?
How to solve this $\displaystyle\iiint _K (y + x^2) \mathrm{d}x\mathrm{d}y\mathrm{d}z$
where $K$ is the hemisphere $x^2 + y^2 + z^2 \leq 4$, $z \geq 0$.
This is what I tried so far:
I used variable ...
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If $T(x,y,z)$ and we know we know $x=x(r,\theta,\phi)$, $y=y(r,\theta,\phi)$, $z=z(r,\theta,\phi)$, can we write $T(r,\theta,\phi)$?
This question is about notation when writing a function composition.
Consider a scalar field $T(x,y,z)$. It's gradient, using Cartesian coordinates and the standard basis $(\hat{i},\hat{j},\hat{k})$ ...
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How do I find the surface integral over a paraboloid in spherical polar coordinates
The question I'm trying to solve asks us to find $\iint_{S}\vec{r}\cdot \vec{dS}$ over the surface described by the paraboloid $z = a^2 - x^2 - y^2$. They offer the parameterisation that $x=a\sin(\...
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What is the distribution of Cartesian coordinate x, y, z if the spherical coordinate r, $\theta$ and $\phi$ are all Gaussian distribution?
Assuming that I know the spherical coordinate r, $\theta$, $\phi$ are all Gaussian distributions, what will be the distribution of the x,y,z in Cartesian coordinate if I transform r, $\theta$, $\phi$ ...
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How do I evalute the value of an given only the radii of 2 spheres??
The question basically. I was able to find rho and theta $rho$ and $theta$ as $\int_1^5\int_0^2pi$,(2pi is the top limit), respectivley. I can't find $phi$ though and I'm assuming thats wrong. My ...
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Integral over spherical angular coordinates vol. 2
This question is heavily related to this old post of mine Integral over angular spherical coordinates. However, now I have a different integral, which is, I believe, convergent. I write down the ...
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