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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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Integrating over sphere

I am trying to solve integral $$I=\int \frac{dS}{\sqrt{\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}}}$$ over a sphere, where $r = \sqrt{\frac{x^2}{a}+\frac{y^2}{a}+\frac{z^2}{b}}$. I thought ...
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Why is the normal vector different in cartesian coordinates vs. spherical coordinates?

Consider the sphere $x^2+y^2+z^2=1$. Let $\mathbf x(u,v)$ be a parameterization for the sphere. Say I was trying to find specifically the normal vector given by $$ \frac{\partial \bf x}{\partial u} \...
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Subtraction of vectors in spherical space

I have 4 microphones placed in a spherical coordinate system. I know the $(r_i,\theta_i,\varphi_i)$ for each microphone $m_i$. Given a speed of sound $C$ and the direction from which the sound arrives ...
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Where will I arrive following a great circle route?

I am standing on a perfect unit sphere. I can describe any point on the surface in terms of longitude and latitude because I arbitrarily marked a "North Pole" and "Prime Meridian" on the surface. ...
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Spherical Coordinates 2

Fundementals of applied electromagnetics i am wondering where does that formula come from? (that i signed in pic.)
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How to calculate “next” point on line which is 50 meters from current point on earth? (autonomous boat)

I'm currently building an autonomous boat for which I define a path to follow. This path consists of multiple way points which are connected by straight lines. The boat doesn't need to be exactly on ...
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Reducing the Dimensionality of the Sphere in terms of the Lie Algebra

The $n$-sphere can be written as an $(n-1)$-sphere fibered over an interval $$ ds^2_{\Omega_n} = d\theta^2 + \sin^2 \theta\;d\Omega_{n-1}^2. $$ In these coordinates, when we impose that we keep $\...
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Integrals and Spherical Co-ordinates

I just do not understand how the spherical co-ordinates conversion system works. I understand the concept, but the finding the limits for p,φ,θ does not work for me (I study part-time by myself). ...
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Spherical Co-ordinates Question

I just do not understand how the spherical co-ordinates conversion system works. I understand the concept, but the finding the limits for p,φ,θ does not work for me (I study part-time by myself). ...
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When to change coordinate system when performing curl?

Consider a vector field $F = \begin{pmatrix} z\\ x^2 \\ y \\ \end{pmatrix}$. In Cartesians, $\nabla \times F = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[5pt] {\dfrac{\partial}{\...
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Triple integration for the volume of a given sphere

I have a problem which I've had a look on "Maths Stack Exchange" and other resources to help, but still am stuck, so any help would be most appreciated. My Problem: Set up a triple integral for the ...
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Find a function using cartesian coordinates of a given sphere

I'm having major difficulty with my maths problem, and any help with understanding and moving forward with the problem would be most appreciated. My Problem: Consider the sphere $$S_R=\{(x,y,z){\in}{...
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What is the volume above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$?

What is the volume above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$? I tried converting each equation to cylindrical coordinates: $z= $r, $r^2+𝑧^...
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Multidimensional integral

I am trying to understand a proof where the following equality, without any further details, appears: $$\int_{B_q}e^{-i\langle x,t\rangle}dx = c(k)\int_{-q}^q e^{-i|t|y}(q^2-y^2)^{k/2}dy,$$ where $...
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Gauss Theorem, constant flow through semisphere

So, I'm a bit rusty when it comes to multidimensional integrals, but a friend said in an exam protocol there was the question "calculate the flow(flux?) of a constant vector field through the boundary ...
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Why are we not using Dirac delta and ignoring the contribution to the surface integral from the point $r=0$?

Let $V'$ be the volume of dipole distribution and $S'$ be the boundary. The potential of a dipole distribution at a point $P$ is: $$\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{...
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How to find the volume of the part of a sphere that protrudes from a square prism?

I have a square prism with width $W$ and height $H$. So it's $W \times W \times H$, where in this case $W$ is less than $H$. The center of mass of the prism is at the origin and the center of a sphere ...
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How to prove $\displaystyle \int_V \dfrac{dx\ dy\ dz}{r^2}$ doesn't contain singularity?

Let's consider the transformation from spherical to Cartesian coordinates: $r, \theta, \phi\overset{T}{\rightarrow}x,y,z$ Let: $\vec{a}=\vec{r} (r+\Delta r,\ \theta,\ \phi)-\vec{r} (r, \theta, \phi)$...
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displacement that follow a geodesic in a spherical coordinate system

I've a spherical coordinate system $(r,\theta,\varphi)$ with: $r$ the radius, in the interval $[0, +\infty[$ $\theta$ the inclination, in the interval $[0, \pi]$ $\varphi$ the azimuth, in the ...
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Find volume in spherical coordinates? $\iiint\rho^2\sinϕ\ dρdϕd\theta$

A hemispherical bowl of radius 5-cm is filled with water to within 3-cm of the top. Find the volume of the water in the bowl? $$\int_{0}^{2π}\int_{?}^{π}\int_{?}^{5}ρ^2\sinϕ\ dρdϕd\theta$$ What value ...
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2answers
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Using spherical coordinates for triple integral

$ \int_{0}^3 \int_{0}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}} \frac{\sqrt{x^2+y^2+z^2}}{1+x^2+y^2+z^2} \ dz \ dy \ dx$ Using spherical co-ord's this becomes : $ \int_{0}^{2\pi} \int_{0}^\pi \...
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The gradient of a scalar function

I found this definition of gradient of scalar function $\Phi$: $\nabla \Phi = (g^{ij}\partial_{j}) \vec{g_{i}}$ And I know: Metric tensor of spherical coordinates $g_{11} = 1$ $g_{22} = r^2$ $g_{...
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Representing position vector

I know the position vector in spherical coordinates: $\vec r = rsin\theta cos\phi \hat r + rsin \theta sin\phi \hat \theta + rcos \theta \hat\phi$ But I do not know the derivation of it. How is the ...
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1answer
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how to graph $\theta = \phi$ in spherical coordinates

how to graph the spherical equation: $$\theta = \phi$$ in spherical coordinates $(\rho, \theta, \phi)$ where: $\rho$ = distance from origin $\theta$ = angle relative to x-axis in xy-plane $\...
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1answer
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How to sketch $\rho = \sin \phi$ and $\rho = \cos \phi$ in spherical coordinates… [closed]

What's the technique to sketch: $$\rho = \sin \phi$$ and $$\rho = \cos \phi$$ in Spherical coordinates.
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1answer
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How should I solve this triple integral?

Hi everybody I have a triple integral I can't solve: $$\iiint \sqrt {x^2+y^2+z^2} \,dx \,dy \,dz $$ Which the region is between $z=\sqrt {x^2+y^2}$ and $z=4$ . The question says after using the ...
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Finding a vector that points towards a coordinate

How to find a vector $v$ with a magnitude $m$ that starts at $(0, 0, 0)$ (or any other arbitrary coordinate) and points towards a coordinate $(x, y, z)$. For example, let $m$ be 100, and the ...
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1answer
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Studying spherical coordinates

$(1)$ Please suggest some books regarding the fundamental studies on surface and volume integrals in spherical coordinates. $(2)$ Are there any books dedicated to only elementary calculus of ...
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Expanding a function in spherical coordinates

I have a function f(theta,phi,r) in spherical coordinates. The function dies out at r->infinity (r in my case is dimensionless). Is there a natural way of expanding the function, the same way a ...
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How come $\nabla^2 f=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial f}{\partial r})$ yet $f=\int \int f'' dr dr$? [closed]

In spherical coordinate, suppose $f$ dependents only on $r$. Then $\nabla^2 f=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial f}{\partial r})$. However, to integral $f''$ back the $f$, $\...
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1answer
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Finding a great circle vector in between two places on the earths surface.

I have derived vectors from the coordinates of New York and Lisbon, but now I need to find the vector of a point between the two locations, which has a great circle distance from New York of 2790....
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partial derivative of multi-variable function

This is my original post. In the derivation, I come across to the point that I need to decompose $\frac{\partial f}{\partial v_x}$ to obtain $\frac{\partial f}{\partial v}$, $\frac{\partial f}{\...
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2answers
208 views

Solution to diffusion equation in spherical coordinates

I want to solve the equation below $$\partial_t F(r,t)= \frac{a}{r^{d-1}}\partial_r\big(r^{d-1} \partial_r F(r,t)\big)$$ where $r$ denotes the radius in spherical coordinates, and $a$ is a constant. ...
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Reference request on an equation while studying Stokes steam-functions

I came across this equation while studying Stokes steam-functions and I'm not sure how its derived. The equation is $$(r^2-\frac{a^3}{r})\sin^2(\theta)=b^2$$ I believe it is the equation of a ...
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3answers
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When is maximising a definite integral the same as maximising the integrand?

When is maximising $$\int_a^b f(x) \text{d}x$$ the same as maximising $f(x)$? Context: I was trying to find the most probable location of an electron in the ground state hydrogen atom, where the ...
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Surface area of a sphere segment. [duplicate]

I got this problem and solution in a paper but cannot find how they have solved it. Consider I have a sphere that is equally divided into two different patch(P, S). The sphere can rotate and translate(...
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1answer
36 views

Triple integrals in spherical coordinates, volume of octant

So, the question is : $S$ is the part of the sphere $ρ=a$ cut by the planes $\theta=0$ and $\theta=\frac{\pi}{6}$ in the first octant. Find the volume of $S$. I am taking the integration limits as $0≤...
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Change of variable between unit sphere and non-unit sphere

$$\int_{-\infty}^{\infty}\int_{\mathbb{S}_{r}^{n-1}}f(x)d\sigma_rdr = \int_{-\infty}^{\infty}\int_{\mathbb{S}^{n-1}}f(x)r^{n-1}d\sigma dr$$ where $\mathbb{S}^{n-1}$ is unit sphere of dimension $n-1$,...
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1answer
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Calculating the annulus of a sphere with a differential change in theta

Consider the following object: I want to calculate the area of the annulus. The annulus is within the region of $$ \theta $$ and $$ \theta + d\theta $$ The answer of the area of the annulus is ...
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Convert spherical coordinate equation to rectangular coordinates?

Find an equation in rectangular coordinates for the equation given in spherical coordinates: $\phi=\pi/6$ Equation must be such that $z \ge 0$. Here is what I did: $z = \rho\cos\phi$ $z = \cos{\pi/...
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1answer
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Incorrect radius of ellipsoid

I've seen the correct way of finding $r(\theta,\phi)$ for the purposes of integrating for area, but that left me wondering why we can't just use: $$ x = a \cdot \cos{\theta} \cdot \sin{\phi} \\ y = b ...
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1answer
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Surface area of spherical section delineated by 2 perpendicular circular planes/central angles

The problem concerns visible area based on a field of view from the center of a sphere. I was never taught spherical trigonometry so even basic terminology is hard. After trying to figure out the ...
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Need to take limit of divergence in spherical coordinates on the $\theta=0$ axis

The title pretty much says it all. I need to take the standard expression (one of the standard expressions, I guess) for the divergence of a vector and of a rank-2 tensor in spherical coordinates and ...
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1answer
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Does the Fourier transform commute with the spherical average operator?

$\newcommand{\Rcal}{\mathcal R}$For $f\colon \mathbb R^d\to \mathbb R$, write $x\in \mathbb R^d$ as $x=r\omega$, with $\omega\in\mathbb S^{d-1}$, and define $$ R f(r):=\int_{\mathbb S^{d-1}}f(r\omega)...
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Is there a way to find the angle Phi in spherical coordinates given theta, an offset secondary angle, and a distance?

Given a spherical coordinate system that gives you a point (ρ, θ, φ), its quite easy to find an (x, y, z) 3D point. My issue is that I don't really have a φ angle to use, as my secondary angle is ...
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Project position on Earth to higher altitude with local azimuth and elevation.

I have a position on Earth ($lat_1$,$lon_1$,$R_e$) of which I want to project a straight line with a known azimuth and elevation to a certain altitude, $r$. Thus obtaining new coordinates on the ...
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An example of divergence in spherical coordinates

I've found the following example in a vector calculus book: the divergence of the vector field $\vec F(x,y,z) = x\vec i + y\vec j - z \vec k$ in spherical coordinates is $$ \nabla \cdot \vec F(\rho,\...
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Defining a sphere

Let $c$ be a number, and let $a$ and $b$ be vectors in $R^3$. Let x = $(x, y, z)$. Show that the equation (x$-a)\cdot($x$-b)=c^2$ defines a sphere with centre whose position vector is $1/2(a+b)$ and ...
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spherical polar geometry change in elevation angle

how to calculate change in elevation angle if you know coordinates of two point on surface of sphere. let us say assume that a point move on the surface of sphere from [x1 y1 z1 ] = [0.1 0.1 0.9899] ...
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Show the properties of a spherically symmetric function.

Let $$K^s (\mathbf{x})=c_{k,d}k\{(\mathbf{x}^T\mathbf{x})^{1/2}\}$$ where $c_{k,d}^{-1}=\int_{\mathbb{R^d}}k\{(\mathbf{x}^T\mathbf{x})^{1/2}\}d\mathbf{x}$, $\mathbf{x}=(x_1,\cdots,x_d)^T\in \mathbb{R^...