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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How to express unit vectors as partial derivatives in spherical coordinates?

In some General Relativity text, I found that the unit vectors are expressed as partial derivatives. In particular, an axisymmetric problem was dealt with using spherical coordinates, where the unit ...
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Integral on the sphere of $\cos^n(\theta)$

I'm struggling with an integral. I have reduced it to the form $$ \int \cos^n(\theta)d\Omega d\rho$$ over the unit sphere, where $\theta$ is the angle associated to the $z$-axis. The radial integral ...
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Derive curl of vector field in spherical coordinates

I need to calculate curl of $F$, and show that it is conservative on this region. A vector field $F$, defined on a simply-connected region $r > 0,\; \frac{\pi}{4}< \theta < \frac{3 \pi}{4},\; ...
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Conversion from NED to 'flat earth' coordinates.

I have a flat earth problem of a missile that needs to return to launch pad. The solution to this problem (using convex optimization in case you are interested) is then meant to be fed to a simulator ...
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Do $\frac{\partial{z}}{\partial{\phi}}=0$ implies $\frac{\partial{\phi}}{\partial{z}}=0$? [duplicate]

If $\frac{dy}{dx}=0$, then $\frac{dx}{dy}$ is unbounded. I'v tried to derive $\nabla$ in sc and there is: $$ z=rcos(\theta)\\ \frac{\partial}{\partial{z}}=\frac{\partial{r}}{\partial{z}}\frac{\partial}...
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Spherical Mean of Schwartz Function Decreases Rapidly - Stein, Shakarchi, Fourier Analysis.

I'm studying Stein and Shakarchi's Fourier Analysis book, and I'm stumped on page. 189, where it talks about the spherical mean of a function. The book defines the spherical mean of a complex-valued ...
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Spherical harmonics orthogonality

I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin(\theta) e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'} (\theta,\phi) \mathrm d\theta \mathrm d\phi $$ I've tried to use the ...
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find the point two lines intersect in 3d with a mixture of cartesian and spherical coordinate system knowns

I have line 1: originating from origin (0,0,0) and magnitude (length) of 7 I have line 2: originating from a shifted position (0, 0, 0.4) and the polar and azimuth are also known (see picture) I ...
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Deriving the partial chain rule using first principles

Can someone derive the chain rule for ∂/∂x(T) using first principle where T is a function of x(r,θ,φ),y(r,θ,φ),z(r,θ,φ) ? The equation goes like: ∂T/∂x=∂T/∂r(∂r/∂x)+∂T/∂θ(∂θ/∂x)+∂T/∂φ(∂φ/∂x) ...
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Length of a line to the surface of a sphere when line does not originate from center

Here is what I know: I have a 3D sphere of radius 7 I have a line (call it line1) with a known polar and azimuthal angle leaving a point that is shifted from the center/origin (x0, y0, z+0.4). I am ...
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Integral of spherically symmetric scalar function over non centered cube?

I have a radially (spherical) symmetric scalar function: $$ W(r,h)= \frac{1}{\pi h^3} \begin{cases} 1-\frac{3 r^2 \left(1-\frac{r}{2 h}\right)}{2 h^2} & 0\leq \frac{r}{h}<1 \\ \frac{1}{4} \...
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Rectangle and ellipse earth coordinats from minor, major axis and orientation

I am trying to calculate area geographic coordinates on earth of different forms from given parameters: Latitude and Longitude in decimal degree of the center point Major and Minor axis in feet Major ...
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Evaluate the following multiple integral: [closed]

$\displaystyle \iiint_D \cos(𝑥+2𝑦+3𝑧)dV, D=\{(x, y, z): x^2+y^2+z^2≤ 1\}$
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How does Terrain Compensation work?

Does anybody know the math behind terrain compensation? All I know is that to make these calculations, sensors such as accelerometers and gyroscopes, are mounted in different axes to compensate for ...
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How can I calculate the sun's angle relative to a window's normal?

The following is given: The sun's azimuth $0 \le a \lt 360$ in degrees, that is the horizontal angle. North would be $a=0$, south would be $a=180$. The sun's elevation $-90 \le e \le 90$ relative to ...
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How to use spherical polar integration on the Poisson equation?

I want to use spherical polar integration on the Poisson Equation, $\nabla^2\phi=4\pi G\rho$ in order to equate it to Euler's equation when the velocity derivative is zero so we are left with, $\nabla\...
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Adapt the derivatives to spherical coordinates

Let's consider the usual spherical coordinates: $$ \begin{cases} x = r \sin \theta \cos \phi\\ y = r \sin \theta \sin \phi\\ z = r \cos \theta \end{cases} $$ with $r \in \mathbb{R}^+$, $\theta \in [0, ...
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Integration of a coordinate function over a hemisphere

Let $\mathbb{S}^n = \{ (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} : x_0^2 + \cdots + x_n^2 = 1 \}$ be the unit sphere in $\mathbb{R}^{n+1}$ and let $\mathbb{S}^{n}_+ = \mathbb{S}^n \cap \{x_0 \geq 0 \...
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Convex hull and bounding circle of a set of points on a sphere?

Given a finite set of random points on the unit sphere (defined in spherical coordinates), are there formulas giving the center and radius of the smallest circle (on the sphere) that contains all of ...
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How can I calculate my position, if I have 3 points coordinates and distance? [closed]

How can I calculate my position, if I have 3 points coordinates and distance from every coordinate to my position. All coordinates by longitude and latitude.This is example
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Equation to convert longitude and latitude of any point if The Great Pyramid is the new north pole [closed]

In GCS coordinate system the North Pole is +90, +0; Great Pyramid of Giza is +29.9, +31.1; and ...
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Using spherical coordinates to evaluate $\iiint_{E}z dV$ where $E$ lies above paraboloid $z = x^2 + y^2$ and below the plane $z=2y$

I was initially stuck trying to solve this question using spherical coordinates. I eventually got it by using cylindrical coordinates: $$\int_0^\pi \int_0^{2\sin\theta}\int_{r^2}^{2r\sin\theta}rz dz ...
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Change of Variables involving Norms

Here is a change of variable that we did in the class and I wonder what is the justification behind the change: $$ \int_{\mathbf{R}^N} \int_{\mathbf{R}^N} \frac{|f(x + h) - f(x)|^p}{\| h \|^{N + sp}} \...
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2 votes
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Range of $\phi, \theta$ in $\int_0^{\pi/4} \int_0^{\pi/2} \int_0^{2\sin\phi \sin\theta} \rho^3\sin\phi \sin\theta d\rho d\theta d\phi$

The question: A solid bounded by the (y,z)-plane, the (x,y)-plane, the cone $x^2 + y^2 = z^2$, and the surface $x^2 + y^2 + z^2 - 2y = 0$. Suppose a density of a chunk of metal of the shape of this ...
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If $(r,\theta,\phi)$ are spherical coordinates representing $(c_1, c_2, c_3)$, what is the difference between $(c_1, c_2, c_3)$ or $(r,\theta,\phi)$?

In the Cartesian coordinate system we identify a point in space by the three coordinates $x$, $y$, and $z$. In the spherical coordinate system, we identify a point in space by the three coordinates $r$...
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Find angle between vectors

Let $v_1\cdots v_6$ be six unit vectors in $3$D and $\theta_{ij}$ denote the angle between the vectors $v_i$ and $v_j$. These vectors will form, in total, $^6C_2=15$ angles between them. Suppose we ...
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How to calculate the middlle coordinate/point on earth between two coordinates? [closed]

Good Afternoon. I need help for a component of my math IA. I need to calculate the middle point between two coordinates on earth to make calculations based on this. I (stupidly) attempted to use the ...
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How do I change a 3D equation into a Spherical coordinates

I know how to change 2D Cartesian equations into polar equations, however I'm having some difficulty with a 3D equation and converting that into a Spherical coordinate system. I am trying to take the ...
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Flux of a vector field over a sphere

I have been given $\bar{F}=x\hat{x}+xy\hat{y}+xyz\hat{z}$, and I need to compute the flux over a sphere of radius $2$ which I assume is centered at the origin. I have already computed this using the ...
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Calculating the spherical harmonic of θ=π/2

This is a very simple question, yet I'm not sure how to approach it. I want to calculate the spherical harmonic: $$ Y_{l m}^{*}(\theta = \pi/2, \phi) $$ I know the general formula: $$ Y_{l m}^{*}(\...
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equidistant mapping from icosahedron to sphere

I am struggling to find an equidistant mapping that maps points from an icosahedron to a sphere, such that the points are spaced evenly (as much as possible) on the sphere. A similar mapping from cube ...
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About transforming angles from one coordinate system to another

Say I have a gimble's wanted direction (in elevation phi and azimuth theta) in coordinate system xyz. Now I'm rotating the gimbal it by r about the x axis, p about the y axis and q about the z axis to ...
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Spherical decomposition for a radial Dirac delta?

Is this a spherical coordinate decomposition for a radial Dirac delta? $\delta(\overrightarrow r-\overrightarrow r_0)=\frac{\delta(r-r_0)\delta(\theta -\theta_0)\delta(\phi-\phi_0)}{r^2sin(\theta)}$
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Intersection of cone and sphere

I have the following problem: There is a sphere (Earth) and a cone (the FOV of a satellite orbiting Earth). So the tip of the cone is at the satellite's center orbiting Earth, and the wide part of the ...
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Solve multiple integral $\int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz$ [closed]

I found this integral in a book. I thought it was easy to solve it, but I've not been able to do it. I tried to use spherical coordinates, however, I'm not sure what the limits are in such coordinates....
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What is the step by step integration of ds for a great circle in terms of spherical coordinates? [closed]

Wikipedia gives the metric as: I found the equation below on line, but have not been able to find it again. K = Gaussian curvature. Because the circumference equation involves, K, it could be ...
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How do I map a Sphere to a Sphube (a special Superellipsoid)?

UPDATE: I realized the formula actually works fine (once I started using $2/r$ instead of $r$), but the mesh I was working with was not a perfect sphere to begin with, which led to skewed results, as ...
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Metric tensor and dot product in spherical coordinates

I saw there are many questions on this subject, but they don't really solve my doubt. Reading "Tensor Calculus for Physics" there is a point (chapter $3.5$) where it says that we can define $...
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Expected value of $\cos^2\theta$ between two 3D vectors?

Consider two randomly oriented and independent vectors in 3D space. The angle between these two vectors is $\theta$. What is the expected value of $\cos^2\theta$ ? I have been reading some lecture ...
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In Spherical Mercator Tile coordinates, how to go one zoom-level deeper but span the same area?

For an arbitrary tile region defined by (z1, x1, y1) in Standard Web Mercator Tile format (Spherical Mercator) , how can we increase ...
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Spherical coordinates of two Sphere Equations (bounds for integral)

I'm given two spheres; $x^2+y^2+z^2=1$ and $x^2+y^2+z^2=2$. I'm trying to convert from rectangular to spherical coordinates to integrate the region between the two spheres, and my bounds so far are $1\...
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Exponential of scalar product (with i) integrated over the sphere

Let $ \vec{x}=\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} \in \mathbb{R}^3, |\vec{x}|=x $ and $ \vec{Y}=\begin{bmatrix} Y_1\\ Y_2\\ Y_3 \end{bmatrix} $ Consider an expression $ f(\vec{x})=e^{i \vec{...
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How can we show that $\int_S \frac{dS\cos\alpha}{r^2}=4\pi$ in spherical polar coordinates $(r,\theta,\phi)$?

To find the solid angle subtended at a point O by an arbitrary surface element $d{\vec S}=dS\hat{{n}}$, one joins the peripheral points of $d{\vec S}$ to O by straight lines which generates a cone at ...
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Convergence of $\int_{\epsilon<|x-y|<1}\frac{x_j-y_j}{|x-y|^{n+1}}dy$ a.e. (Riesz transforms)

I am trying to show that the following limit exists a.e. for $x\in \mathbb{R}^n$: \begin{equation}\lim_{\epsilon\to 0}I_\epsilon(x):=\lim_{\epsilon\to 0}\int_{\epsilon<|x-y|<1}\frac{x_j-y_j}{|x-...
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Can spherical coordinates be used to trace a line between 2 points? [duplicate]

I am trying to figure out how to trace a path between two points with my goal being to enter this info into a computer program in order to iterate down the line. My first thought was to use spherical ...
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Calculating an increasing angle in Spherical Coordinates

I'm making a program that generates lines in 3D space. One feature that I need is to have an incrementally increasing angle on a line (a bending line / curve). The problem is simple if the line exists ...
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A Question on Spherical Method for volume.

Im currently studying, calculus in spherical method. I encounter some confusion with the function $f(ρ,θ,φ)$ in the following formula $$ \displaystyle\iiint_{E} f(ρ,θ,φ) ρ^2\sinφdφdρdθ $$ Taking the ...
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High-order Sobolev function on sphere

For a bounded $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, there are various definitions of fractional Sobolev spaces (a.k.a. Sobolev-Slobodeckij spaces) on $\partial \Omega$, either by using ...
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$\sum_{cyc} \frac{x^2+x}{x^2 +yz} \ge 4$ Inequality on a unit sphere's first octant

Given the portion of the sphere which is in the first octant: $x^2+y^2+z^2=1$ ; $ x,y,z>0$ , show the following cyclic inequality: $$ \sum_{cyc} \frac{x^2+x}{x^2 +yz} = \frac{x^2+x}{x^2 +yz}+ \frac{...
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Triple integral $\iiint 2x$ $dxdydz$ on the region of space defined by $S={x^2 +y^2 + z^2 \le 1 }$

I have done the following triple integral $ \displaystyle \iiint 2x ~dx~dy~dz~$ on the region defined by $S={x^2 +y^2 + z^2 \le 1 }$. Turning to the spherical coordinates I get $$ \left\{ \begin{...
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