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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1answer
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Evaluating a Gaussian integral on $\mathbb{R}^{n}$.

For $t>0$ I want to show that $$\frac{1}{(4\pi t)^{n/2}}\int_{\mathbb{R}^{n}}e^{\frac{-\|x\|^{2}}{4t}}dx=1$$ So far, I have $$\begin{aligned} \frac{1}{(4\pi t)^{n/2}}\int_{\mathbb{R}^{n}}e^{\...
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17 views

Determining the spherical coordinate parametrization of an area in $\mathbb{R}^3$.

Let $\epsilon_0,\epsilon_1,\epsilon_2$ be three iid random variables with a symmetric distribution and let $\lambda>1$. I want to calculate $$ P(\epsilon_0>0 \quad;\quad\epsilon_1>-\lambda\...
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1answer
25 views

Converting coordinates

I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just ...
2
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1answer
35 views

Triple integral in spherical / cylindrical coordinates - where's the error? Exercise check

I have done an exercise in two different ways but I have obtained two different results and I can't understand what's wrong. Please, help me. Given: $V=\{(x,y,z)\in R^3: x^2+y^2+z^2\leq 1, \frac{...
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1answer
24 views

Finding the volume of Torus, Jacobian of spherical substitution.

I thought to find the volume of a Torus, like I would a sphere, where the spherical substitution was: $$x=r\cos\varphi\sin \theta , y= r\sin\varphi \sin \theta, z=r\cos \theta \\ g(r,\varphi,\theta)\...
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0answers
22 views

Reversible function

I need help. For which $(r, θ, φ) ∈ \mathbb{R}^3$ is the function $$f(r,\theta,\varphi)=\begin{pmatrix}x(r,\theta,\varphi)\\ y(r,\theta,\varphi)\\z(r,\theta,\varphi)\end{pmatrix}=\begin{pmatrix}r\sin ...
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0answers
12 views

Looking for a particular parameterization of $S^n$

Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$. Is there any ...
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8 views

Three-dimensional integral using spherical coordinates

In the past I have calculated integrals like $$ I_1(q,p,\alpha) = \int \frac{d^3{\bf k}}{(2\pi)^3} \frac{1}{|{\bf k - q}|^2 + \alpha^2}\bigg(\frac{{\bf p}\cdot{\bf k}}{k^3} \bigg), $$ where $\alpha$ ...
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3answers
74 views

Triple Integrals in Spherical Coordinates where (z-2)^2

$$\iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}}$$ for $x^2+y^2+z^2 = 1$ I've used spherical coordinates, like this: $x=\rho sin\phi cos\theta$; $y=\rho sin\phi sin\theta$; $z=\rho cos\phi$ and $J=\rho^2 ...
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1answer
3k views

Rotating a point in spherical coordinates around Cartesian axis

If I have a point in spherical coordinates, and I rotate it around one of the Cartesian axes, what will be the new spherical coordinates for the point? Both spherical and Cartesian coordinate systems ...
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1answer
435 views

Finding the volume between a cone and a sphere

I have to find the volume between the sphere $x^2+y^2+z^2=1$ and below the cone $z=\sqrt{x^2+y^2}$ using Spherical Coordinates. Here is what I have so far: Transforming the cone part gives: $\begin{...
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0answers
12 views

How to sketch functions in polar and spherical coordinates by hand on paper?

I've been practicing drawing surfaces in different coordinates. I can do the easier ones but no I am completely stuck on the following two: Say we define spherical coordinates as follows: so that $...
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0answers
19 views

Adding Pitch/Yaw to Spherical Coordinates

Im running a program on a simulator where I have a stationary camera at global position $(0, 0, 0.48)$ with a pitch $=-28$ and yaw $=0$ rotation. The reference for this pitch and yaw are the global ...
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1answer
13 views

spherical parametrization

spherical parametrization for different equations A=$x^2+y^2+z^2=5, z\geq0.$ B=$(x-1)^2+(y-1)^2+z^2=1.$ C=$2x^2+3y^2+4z^2=1.$ for A i know that the answer would be $r\left(\theta,\phi\right)=\...
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0answers
18 views

sphere centered at 1,1,0 prametrization

spherical equation is usually as $\,x^2+y^2+z^2=1$ centered at (0,0,0) and we know this $\,\rho=\sqrt{1} , \,x=\rho\sin\phi\cos\theta , \,y=\rho\sin\phi\sin\theta , \,z=\rho\cos\phi$ the ...
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0answers
19 views

shperical coordinates limits

how to write the limits for the integration in spherical coordinates to get the volume of 1- the solid inside the cylinder $$r= \sin(\theta)$$ bounded from above by $$z=\sqrt{1-x^2-y^2}$$ and ...
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0answers
35 views

Calculate next point on sphere a certain distance away

I have a sphere where I know the radius, origin and a single point $p^1$ on the surface by its cartesian coordinates ('world' x/y/z). I'm struggling trying to calculate the position of another point $...
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18 views

Evaluate the integral $f(x,y,z) = x$ within $x^2+4y^2+9z^2 \leq 1$ and $x \geq 0$ and also $y \geq 0$

I am asked to evaluate the integral $f(x,y,z) = x$ within $x^2+4y^2+9z^2 \leq 1$ and $x \geq 0$ and also $y \geq 0$ using a change of variables. Should I proceed with spherical coordinates? If so, is ...
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0answers
26 views

Using cylindrical coordinates evaluate $\int_{0}^{2} dx \int_{0}^{\sqrt{2x-x^2}} dy \int_{0}^{a} z \sqrt{x^2+y^2} dz$

I am asked to solve the following problem: Using cylindrical coordinates evaluate $\int_{0}^{2} dx \int_{0}^{\sqrt{2x-x^2}} dy \int_{0}^{a} z \sqrt{x^2+y^2} dz$ Before doing that long ...
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1answer
54 views

Find limits of integration for the interior region of sphere with center $(a,0,0)$ and radius $a$ using spherical coordinates

I am asked to find limits of integration for the interior region of sphere with center $(a,0,0)$ and radius $a$ using spherical coordinates. How can one do that? I know that one may use $$ x = r \...
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1answer
21 views

How to determine $\varphi$ in spherical coordinates

Assume that I would like to integrate some continuous a.e. function $f(x,y,z)$ over the following set: $ a^2_1 \le x^2 + y^2 +z^2 \le a^2_2$, and $z\ge c^2(x^2+y^2)^{1/2}$. So, in a case I would like ...
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2answers
18 views

Find limits of integration for region under sphere $x^2+y^2+z^2=a^2$ inside cone $x^2+y^2=z^2$ and above $0xy$

I am asked to find the limits of integration for region under sphere $x^2+y^2+z^2=a^2$ inside cone $x^2+y^2=z^2$ and above $0xy$. Should I use spherical coordinates or cylindrical coordinates? Is it ...
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2answers
16 views

Limits of integration spherical coordinates

I have seen a lot of exercises where they solve a triple integral using spherical coordinates. But I'm confused about the limits that one should use. For example when they integrate over a sphere like ...
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1answer
22 views

Jacobian matrix for change of variables from Cartesian coordinate system to Spherical (Geographic) coordinate system.

I am trying to obtain the Jacobian matrix for a change of variables from Cartesian coordinate to spherical coordinates. My spherical coordinate system is a conventional right-handed Geographic ...
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1answer
47 views

Equation of a great circle passing through two points

I've searched everywhere for something to help me with this problem, but I can't find anything. What I want to calculate is the midpoint between two locations (latitude and longitude) on a sphere. The ...
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2answers
26 views

Set up the triple integral for region between cylinders $x^2 + y^2 = 9 \quad x^2 + y^2 = 16 \quad z = 4+x^2$ and $0xy$ plane

I ran into a problem that I am not sure about the correct answer. The question is: Set up the triple integral for region between $x^2 + y^2 = 9 \quad x^2 + y^2 = 16 \quad z = 4+x^2$ and $0xy$ ...
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Volume enclosed between two spheres using spherical coordinates [duplicate]

The question reads use spherical coordinates to find the volume of the solid enclosed between the spheres $x^2+y^2+z^2=4$ and $x^2+y^2+z^2=4z$. The first sphere is a sphere of radius 2 centered at ...
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1answer
22 views

Derivative of spherical coordinates [closed]

Why are the r dot terms (eg -r^dotsincos in the z derivative) in the derivatives of the spherical coordinates? Differentiation, as I've understood it, is differentiating a function with respect to a ...
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0answers
10 views

Finding Surface area of a shape given its spherical coordinate equation

Is the surface area of the shape defined by $\rho = 4\cos(\theta)\sin(\theta) $ given by the following? $$\int_0^{2\pi}\int_0^\pi\sqrt{1 + 0 + 16\cos^2(2\theta)}\ \rho^2\sin(\phi)\ \ d\phi\ d\theta$$...
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1answer
38 views

Cube in Spherical Coordinates not centred at the origin

I`ve seen that there are already a couple of questions about how to describe a cube in Spherical Coordinates. However they are all centred at the origin. I would like to describe a cube in Spherical ...
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1answer
26 views

Triple integrals using spherical coordinates

I'm trying to integrate this using spherical coordinates (this is the only information given by the way). My issue is understanding how to find the range of $φ$ and $θ$. I know that $0≤ρ≤3$. But for $...
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1answer
23 views

How to Find Equation of Line Given Latitude, Longitude, Heading

I need to find the equation of a line given X and Y coordinates (latitude and longitude) and a heading in degrees. I can assume that 0 degrees is North. So for example, I might have that the point ...
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1answer
67 views

How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ $$\partial\Omega:=\{x\in\...
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35 views

Question the convergence of $\iiint_{x^2+y^2+z^2\geq1}\frac{e^{\sin(x+y+z)}}{(x^2+y^2+z^2)^p}$ in dependence of $p$

Question the convergence of $$\iiint_{x^2+y^2+z^2\geq1}\frac{e^{\sin(x+y+z)}}{(x^2+y^2+z^2)^p}$$ in dependence of $p$. In class we did $$\iiint_{x^2+y^2+z^2\geq1}\frac{e^{-x^2-y^2-z^2}}{\sqrt{x^2+y^...
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1answer
21 views

Find the volume of the region outside cone and inside sphere.

Find the volume of region outside the cone $\varphi = \frac{\pi}{4}$ and inside the sphere $\rho =4cos(\varphi)$. Solution Attempt: I can visualize the surfaces and see that the volume is two ...
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1answer
31 views

Plotting Spherical Coordinates

I'm trying to plot the Poisson Kernel, where a = 1, so the resulting equation would be $$P(r,\theta) = \frac{1-r^2}{1-2r\cos(\theta)+r^2}$$ $0\leq r <a = 1 ,\textrm{ } -\pi \leq \theta \leq \pi$ ...
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1answer
29 views

Sperical coordinates and the divergence theorem

Use the divergence theorem to calculate $\int \int F\cdot dS$ where $F=<x^3,y^3,4z^3>$ and $S$ is the sphere $x^2+y^2+z^2=25$ oriented by the outward normal. I have found that $div(F)=<3x^2,...
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0answers
34 views

Laplacian in spherical coordinates - brownian motion

Consider the Laplacian equation on the unit sphere, for a vector $f$. $\theta$ is polar angle, and $\phi$ is azimuthal angle. The Laplacian in spherical coordinate is : $$ \Delta f = {1 \over r^2} {\...
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1answer
23 views

Point along great circle line (aka arc) closest to a target point on the ground

Given: an arc (aka a great circle line, not a straight-line) defined by two arbitrary end points (which I can express in lat/lon/altitude or earth-centered fixed (ECF) 3D 'cartesian' space). Think ...
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0answers
67 views

Computing the vector Laplacian in spherical coordinates using metric tensor

I want to compute the Laplacian of a vector in spherical coordinates using metric tensor. I started only with the first component but I have obtained a different formula! Let be $\vec{v} $ a vector ...
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0answers
24 views

Getting topological objects from the “cube” of $T^3$

One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$. To get $S^...
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1answer
96 views

Equation of a circle in spherical coordinates

What would be the equation of an arbitrary circle rotated along some angle theta around the X-axis in spherical coordinates? For simplicity we may assume that it is a circle with constant radius r. ...
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0answers
27 views

Find the volume of a shape using spherical coordinates.

Find the volume of the shape formed inside $x^2+y^2=z$ and $x^2+y^2=2$ using spherical coordinates. I know I need to do a triple integral but I can't quite get the integral limits.
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Minimizing arc length on unit sphere (geodesics)

I just completed a Calculus IV course and taught myself basic Calculus of Variations, and wanted to extend some of the basic principles of optimization from planes to surfaces. The arc length ...
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29 views

Let $r(s)$ be a regular closed curve that is in the sphere $S^2$. Prove that $\int_{\gamma}\tau(s)ds=0$ $\gamma$-map of this curve, $\tau$ torsion

Let $r(s)$ be a regular closed curve that is in the sphere $S^2$. Prove that $$\int_{\gamma}\tau(s)ds=0$$ $\gamma$-map of this curve, $\tau$ the function of torsion of this curve. Every year this ...
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0answers
61 views

Trajectory on a sphere

I've asked a question before concerning a parallel problem, and I read a wikipedia page on spherical caps (Nominal Animal), which gave me an idea to do the following: I have the Cartesian coordinates ...
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1answer
41 views

Is there a solution to the equation $tan({\phi})=\frac{0}{0}$

I've been reading about conversion from Cartesian ($x,y,z$) to Spherical (r, $\theta$, $\phi$) coordinates. The formula to find the value of ${\phi}$ is given as: $\tan({\phi})=\frac{y}{x}$ My ...
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1answer
299 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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0answers
50 views

Estimating the distance between two coordinates but without using Euclidean distance

Bill opens up "Café Finder" on his phone, and it tells him that it will take him 10 minutes to get to his nearest Starbucks to grab a triple-shot frapa-crapa-flat-white, so he decides to walk. 20 ...
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0answers
15 views

Find the Radius of Sphere using TDOA

My goal is to calculate Position of impact using Trilateration. I followed this guide on wikipedia : Trilateration Wikipedia I don't know how to find the Radius R1,R2,R3.(Normally it is PA,PB,PC,...