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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Volume of solid inside surface in spherical coordinates.

Find the volume of the solid inside the surface defined by the equation $\rho=8\sin \phi$ in spherical coordinates So far I've set up an integral in spherical coordinates with $\rho$ from $0$ to $\...
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Surface of a torus in terms of Legendre polynomials

The equation of a spheroid is $$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2}$$ Its surface can be expressed as $$ r = a \left( 1 - \frac{2}{3} \epsilon P_2(\cos \theta) \right) $$ where $r$ is the ...
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2answers
60 views

Using Divergence Theorem to evaluate the flux over a sphere

Above is the question. I've try to find the divergence of F and parameterize the sphere using spherical coordinates. Below is my work. Then I use online integral calculator(just to avoid human error) ...
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0answers
25 views

Spherical Triangle — Angle from one Point without using North

I'm designing a tool for students and right now im working with coordinates. I have the plan to shift the north pole in my code to Munich and I would like to do this by working with a spherical ...
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0answers
78 views

Triple integrals - spherical coordinates

Integrate the function: $$f(x,y,z)=1/\sqrt{(x^2+y^2)(x^2+y^2+z^2)}$$ over the region $R$ which is the set of all points outside the sphere $x^2+y^2+(z-1)^2=1$ but inside the sphere $x^2+y^2+z^2=4$. ...
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1answer
79 views

Applied Mathematics: Spherical Polar Coordinates and Newton's Second Law

I've been attempting this question but can't seem to find a solution. Question: A particle of mass $m$ moves under the influence of a force which, in spherical polar coordinates, only acts in the ...
3
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1answer
74 views

Inclusion, pullback of differential form

Let $\omega=x\,dy\wedge dz +y\,dz\wedge dx+z\,dx\wedge dy$ or in spherical coordinates (unless I had made some mistake) $\omega=r^3\cos \theta\, d\phi\wedge d\theta$. Now I want to find $i^*\omega$ ...
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1answer
62 views

Using spherical coordinates, find the volume.

Find the volume of the solid that lies in the first octant above the cone $z=\sqrt{3(x^2+y^2)}$ and inside the sphere $$x^{2}+y^{2}+z^{2}=4z $$ using spherical coordinates: So here is what I have ...
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0answers
27 views

formula for getting the cross product in spherical coordinates, given the two vectors.

I am using this coordinate system: consider these two vectors in spherical coordinates: $$\vec A=A_r \hat r +A_\phi \hat \phi + A_\theta \hat \theta= A_r\sin{\theta_A} \cos{\phi_A} \hat i +A_r\...
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60 views

Parameterization of Ellipsoid

I have a question asking me to evaluate $\iint_\Sigma \mathbf{F} \cdot \mathbf{n}~dS$, where $\Sigma$ is the lower half of the ellipsoid $z = -2 \sqrt{1 - x^2 - y^2}$ with $\mathbf{n}$ directed ...
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2answers
54 views

substituting spherical coordinates to evaluate an integral.

I have to evaluate $$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{-\sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}} \, dz \, dy \, dx$$ using spherical coordinates. This is what I have come up with \begin{align} &...
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0answers
50 views

Plot in spherical coordinates in matlab

I have this vector function in sperical coordinate system $$\vec{E} =\frac{p}{4\pi E_{0}r^3}(2\cos(u) \hat{r} + \sin(u)\hat{u} ),$$ where $p$ and $E_{0}$ are known. I want to plot this function in ...
3
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1answer
63 views

Conversion from Cartesian to spherical coordinate for vectors - ray tracing application

I'm implementing a ray tracer that support physically based rendering, so is based on various BRDF models. At the moment I'm focused on Oren-Nayar and Torrance-Sparrow model. Each one of these is ...
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1answer
23 views

$\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$ Where $W$ is the solid delimited by $y=4$ and $y=\sqrt{x^2+z^2}$

$\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$, $W$ is limited by the plane $y=4$ and the paraboloid $y=\sqrt{x^2+z^2}$. I'm trying to solve with spherical coordinates, however I got stuck in the following ...
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1answer
19 views

What is the easiest way to find the radius and center of the circle of intersection between two spheres?

If given two spheres $S_1$ and $S_2$, of radius $r_1$ and $r_2$, centered at 3-space points $P_1$ and $P_2$, respectively. What is the easiest way to find the radius and center of the circle of ...
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1answer
28 views

Determine coordinate of a point on unit sphere

Let $S$ be unit sphere in $\mathbb R^3$ center at $O(0,0,0)$. Let $A=(x_1,y_1,z_1),B = (x_2,y_2,z_2)$ be two points lying on the sphere $S$. Let $M$ be center of $AB$ which lies on the geodesics $AB$. ...
3
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1answer
58 views

Determine a point lie in bisect area between 2 circles on sphere

Given 2 points A,B,O on sphere of radius $R$. Point O is in middle of AB. E and F are deviation from O by geodesic distance $d$ (angle between EF and AB is $90^o$). Consider 2 circles $C_1,C_2$ on ...
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1answer
33 views

Proving $v$ is harmonic

Let $u$ be a harmonic function in $\mathbb{R}^3$ and let $a > 0$. Show that the function $v$ defined in spherical coordinates by $v(r,\theta,\psi )=\frac{a}{r}u(\frac{a^2}{r},\theta,\psi)$ is ...
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1answer
27 views

Find critical points using spherical coordinate

again! Let $E:=\left\{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 \le 4 \right\}$ and $f:E \to \mathbb{R}$ the function define by $$f(x,y,z):= \frac{z}{1+x^2+y^2+z^2}.$$ How can I determine the ...
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0answers
16 views

How to integrate a distribution in spherical coordinates over a circle?

I have an angular distribution $\frac{s \sigma}{d\Omega} = \frac{d\sigma}{d(cos\theta)d\phi}$. How can I calculate it over a circle which lies on the plane $X = dist$, has radius $r$ and its centre is ...
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2answers
267 views

Find the center of a circle on a sphere given 2 points and its radius [closed]

How can I find the set of the centers of the circles on sphere that pass through 2 given points and have pretedermined radius, using spherical coordinates? Assume that the radius of sphere is $R$. (...
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0answers
39 views

Locate a point on sphere with equal distance

Given 3 points A (lat1, lon1), B(lat2, lon2), O(lat3,lon3) on earth with geometric location longitude and latitude and a distance d, where O is middle point of A and B. Let GCD denote the great circle ...
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26 views

Find a point on extended great circle with given distance

Given 2 points on earth with longitude and latitude coordinate A(lat1, lon1), B(lat2, lon2), and a distance d. Find coordinate (in longitude and latitude) of 2 points C, and D on extended of great ...
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1answer
15 views

Particularly Problematic Spherical Polar Problem

The question is as follows: Using spherical polar coordinates, find the volume of the solid specified by R $\leq$ 3 and $0 \leq \theta \leq \frac{\pi}{3} $. I have two big questions about this ...
3
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1answer
296 views

why is the laplacian of 1/r equal zero outside the origin?

In spherical coordinates, the laplacian can be written as: $\nabla^2 = \frac{1}{r}\frac{\partial^2}{\partial r^2}r + \frac{1}{r^2 \sin \theta}\frac{\partial}{\partial \theta}(\sin \theta \frac{\...
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1answer
25 views

Volume of Region using Spherical Polar Coordinates

Find the volume of the region D bounded by the hemisphere $y=\sqrt{4-x^2-z^2}$ and the planes $y=x\ $,$\ y= \sqrt3x$ by using polar coordinates. Working: I have calculated $x^2+y^2+z^2=4$ so am I ...
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1answer
33 views

Triple Integral with Spherical Polar Coordinates Problem

I am having trouble understanding the whole concept of spherical polar coordinates, so any help is appreciated! The full question is: Let D be the region bounded by the surface of the hemisphere $z = ...
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0answers
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Is the $r$-axis in spherical coordinates the same as the $z$-axis?

If Cartesian coordinates have an $x$-axis, $y$-axis, and $z$-axis, do spherical coordinates have an $r$-axis, a $\theta$-axis, and a $\phi$-axis? Since the Cartesian $z$-axis is just the set: $$\{(0,...
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0answers
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How to intergate the cross section over the surface of a detector?

My beam moves along the $X$ axis. I know the cross section $\frac{d \sigma}{d \Omega}$. My rectangular detector is perpendicular to the $XY$ plane and its surface is perpendicular to the line ...
3
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1answer
86 views

Surface integral of a partially constant Dirac delta

I am trying to integrate the product of a function and a partially constant delta function over a sphere of constant radius $r$. The integral is of the form $\int^{2\pi}_0 \int^{\pi}_0 f(\mathbf{r})...
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1answer
36 views

Dual Spaces vs Dual Bases

I'm trying to wrap my head around differentiable manifolds and tensors. I partially worked through a question which asked me to use the metric tensor and the line element in spherical polar ...
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0answers
35 views

Marsden Triple Integration, Spherical Coordinates

This comes from Marsden Vector Calculus book. I think I've done it correctly, but I'm not very confident in my work. The integral is $$\int_0^3\int_0^\sqrt{9-x^2}\int_0^\sqrt{9-x^2-y^2} \frac{\sqrt{x^...
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1answer
63 views

How to integrate a part of a sphere limited by a rectangle?

I have a function which depends on the solid angle $\frac{df}{d \Omega} = \frac{df}{d\phi d\theta} $. I want to integrate it over a part of the sphere limited by a rectangle. How should I set the ...
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0answers
25 views

Coordinate system for evenly-distributed points on a sphere

Suppose I have a sphere with a set of evenly-distributed points on its surface: What is the coordinate system to use to represent these points, where each adjacent point is a unit step? Spherical ...
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0answers
15 views

Relationships between positions on a sphere in a global coordinate system

Imagine I have a camera which is observing an object. The camera can move around the abject, by traversing a virtual sphere, where the radius of the sphere is the distance between the object centroid ...
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0answers
60 views

First person camera rotation (conversion between spherical to cartesian coordinates)

A similar question has been asked before on Stack Overflow but no satisfying answer. Basically I'm following this tutorial, and I can't exactly figure out how the guy deduced these two equations to ...
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0answers
94 views

Triple integral in spherical coordinates: Finding $\phi$ limits?

The question asks to convert to spherical coordinates then evaluate. So for this question, I manage to get the bounds of theta and row right, but I got the bounds of phi wrong. According to the ...
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0answers
32 views

Is it possible to calculate the lat/lon of a point knowing the extent coordinates?

I would like to know where it's possible to determine the latitude and longitude corresponding to a point inside a cartesian (planar?) representation of a geographical map, knowing the x/y coordinates ...
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29 views

Finding moment of intertia using spherical coordinates

I have taken a picture of the problem with my work. The answer I am getting is incorrect. I think I am doing something wrong after the following. Any help is appreciated `
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1answer
39 views

Parameterising of scalarproduct of surface gradient with vector field

Given is this operation $\left((\mathbb{1}-\hat{r}\hat{r})\cdot \nabla\right) \cdot v(r) = \left(\nabla-\hat{r} (\hat{r} \cdot \nabla)\right) \cdot v(r)$, where $r=(x,y,z) \in \mathbb{R}^3$, $\hat{...
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1answer
64 views

Find volume of region lying above $z=0$, below $z=4-x^2-y^2$ and inside extruded disc $x^2+y^2=2^2$

I am working on the following homework problem: Find the volume of the region that lies above the plane $z=0$, below the surface $z=4-x^2-y^2$ and inside the extruded disc $x^2+y^2=2^2$. I think ...
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2answers
189 views

Evaluate the integral by changing to spherical coordinates.

$$\int_{0}^{6} \int_0^{\sqrt{36-x^2}} \int_{\sqrt{x^2+y^2}}^\sqrt{72-x^2-y^2} xy~ dzdydx $$ I tried converting it and I ended up with $$\int_0^{2\pi}\int_0^{\pi}\int_0^{\sqrt{72}}\left[p\sin(\phi)\...
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0answers
19 views

Spherical to Cartesian coordinate ellipsoid overlap

I have two geographic coordinates; latitude and longitude , separated by few meters. I need to draw an ellipsoid of same major and minor axes ,centered around the geographic co-ordinates I used ...
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0answers
49 views

How to get parallels of tilted Equator?

I have a Great Circle on Earth, which is not an Equator nor Meridian, and it's not parallel to these. I have four geographical coordinate pairs for it, separated by 90 degrees, so I can use these in ...
3
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2answers
42 views

$\iiint_Ez(x^2+y^2+z^2)^{-3/2}dV$ where $E=\{(x,y,z):x^2+y^2+z^2\leq16, z\geq2\}$

Convert to spherical coordinates and evaluate: $$\iiint_Ez(x^2+y^2+z^2)^{-3/2}dV$$ where $E$ is the region satisfying the following inequalities: $x^2+y^2+z^2\leq16$, $z\geq2$. When I drew out the ...
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0answers
10 views

Scale factors and metric in cylindrical and spherical coordinates - isotropy of space [duplicate]

In cylindrical (polar) coordinates, the scale factors are $$h_r=1$$ $$h_{\theta}=r$$ $$h_z=1$$ Would it be correct to say that $h_i$ do not depend on $\theta$ because space is isotropic (has the same ...
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0answers
14 views

Shortest distance (in km) between a lat/long and a line between two lat/longs?

I know that Haversine can be used to calculate the absolute distance between two lat/long, but is there any implementation of the shortest distance (in km) between a lat/long and the line that is ...
1
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1answer
33 views

dot-product spherical

I want to calculate dotproduct of $ e_r *e_\phi $ they are unit vectors in spherical. where my spherical coordinates is $(r,\phi,\theta)$ My attempt is first to convert them to cartesian: which ...
2
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2answers
52 views

cylindrical coordinates - base vector integral

Hello I have a problem where $\hat{p}$ is a basevector in a cylindrical system $$\int_{0}^{\pi/4}\hat{p} d\theta$$ I know that; $\hat{p} = \hat{x}\cos\theta + \hat{y}\sin \theta$ $$ \hat{x}\int_{0}^...
2
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1answer
38 views

Integration of function $\sqrt{1 - \sum_{i}x_{i}^{a_{i}}}$ by using spherical coordinates in D dimensions

Let's have integral $$ I = \int \limits_{0}^{\text{T.P.}}dx_{1}...dx_{D}\sqrt{1 - \sum_{i}x_{i}^{a_{i}}}, \quad \text{T.P.}: \quad 1 - \sum_{i}x_{i}^{a_{i}} = 0 $$ By rewriting it in coordinates $x_{i}...