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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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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31 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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24 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
20 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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1answer
37 views
+50

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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2answers
15 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 ...
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1answer
17 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
36 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
24 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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0answers
8 views

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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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\ ...
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1answer
34 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
18 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
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 ...
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1answer
20 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
65 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 $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ ...
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1answer
29 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 ...
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1answer
21 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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23 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 ...
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64 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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1answer
71 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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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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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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25 views

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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58 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
40 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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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 ...
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9 views

The smallest bounding sphere of a prolate spheroid domain

Let $\Omega\subset \mathbb{R}^3$ be a prolate spheroid domain. Denote by $d$ its interfocal distance and by $b$ the surface of the region occupied by $\Omega$. The question is how to prove that the ...
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1answer
26 views

Angle between arc of two points on a unit sphere and $xy$-plane [closed]

Suppose I have two points on a unit sphere whose spherical coordinates are $A(\theta_1,\phi_1)$ and $B(\theta_2,\phi_2)$, what is the angle between $xy$-plane and arc $AB$? Maybe I can draw a ...
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20 views

Conversion of laplacian form cartesian to spherical coordinates 2nd part

Using the same method of conversion of laplacian from cartesian to spherical coordinates and changing $\psi$ to $u$, I am trying to finish the demostration of the spherical laplacian. However, I have ...
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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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2answers
29 views

Can someone please help me convert this triple integral to spherical coordinates?

I'm not sure how to approach the problem, though I know it needs to be broken up into two integrals before it can be evaluated based on the way the answer input is set up. I do not know how to start ...
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1answer
36 views

Write the equation $4x^{2}+4z^{2}=5$ in spherical coordinates.

Write the equation $4x^{2}+4z^{2}=5$ in spherical coordinates. I used the facts that $$ \begin{align} x&=ρ\sin\theta\cos\phi\;,\\ z&=ρ\cos\phi\;, \end{align} $$ And ended up with: $ 4 (ρ^2 ...
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1answer
82 views

The centre of the earth

I'm a real beginner here (first post and first foray into math since high school, trying to catch up), so I'm going to try my best to explain my problem in mathematical terms then follow up with an ...
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1answer
31 views

sphere arc intersection

Given: an arc defined by two end points (which I can express in lat/lon/altitude or earth-centered fixed (ECF) 3D 'cartesian' space) a sphere defined by a center (lat/lon/alt or ECF) and a radius ...
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0answers
41 views

Integration in spherical coordinates

I needed to solve the following integral on one of my exercise sheets, which seemed not too difficult: $ \phi(\vec{r}) = \dfrac{1}{4\pi\epsilon_0} \int\limits_0^{\infty} dr' \int\limits_0^{\pi} ...
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1answer
29 views

Find a circle on sphere using spherical distance

I have a sphere with radius $R$. On this sphere I also have a point $P_1$ written in spherical coordinates, so I know $\theta_1$, $\phi_1$ and $R$ for this point (same as on this picture). I also ...
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0answers
12 views

spherical radial transformation

several papers mentioned a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty \int_{\mathbf{z}'\mathbf{z} = 1} f(r\mathbf{z}) ~r^{n-1} ...
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1answer
21 views

Rewrite equation using cylindrical and spherical coordinates.

I want to rewrite the equation $z=x^2-y^2$ using cylindrical and spherical coordinates. The cartesian coordinates are of the form $(x,y,z)$. The spherical coordinates are of the form $(\rho, \theta, ...
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16 views

How to setup/evaluate a triple integral to show an interesting result in physics?

I know this isn't the physics forum, but the task i'm struggling with is purely mathematical. My task is as follows; Let $A$ be a sphere centered at origin with radius $R$ and assume $a \geq R$. ...
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3answers
159 views

Show that $\nabla\cdot\left(\dfrac{\mathbf{e}_r}{r^2}\right)=4\pi\delta(\mathbf{r})$ using the divergence theorem.

The book answer goes as follows: By the divergence theorem, in spherical coordinates we find ...
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0answers
31 views

Marginals/conditionals of a normalized Gaussian vector

It is well - known that if $x=(x_1,...,x_n)^T\sim{N(0, \sigma^2I)}$, then its normalized version is uniformly distributed on the unit $n-1$ - sphere: $$ ...
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1answer
214 views

Triple integral to find the mass of the intersection between two spheres

I've got two unit spheres, one is centered at $(0,0,0)$ and the other at $(0,0,1)$, the intersection of these two spheres is my region $R$. I would like to find the integral: $$\iiint\limits_R ...
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0answers
20 views

Vectors and dot product in spherical coordinate system [duplicate]

Let $\vec{v_1}$ and $\vec{v_2}$ given: $\overrightarrow{V_1} = r_1\hat{u_r} + \theta_1\hat{u_\theta} + \phi_1\hat{u_\phi} \\ \overrightarrow{V_2} = r_2\hat{u_r} + \theta_2\hat{u_\theta} + ...