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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Tensor identity in orthogonal coordinate systems

In Riley, Hobson & Bence's "Mathematical methods for physics and engineering" third edition, in the chapter about tensors, one of the exercises involves finding the expression for the ...
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Finding the bounds of a triple integral (spherical coordinates)

I'm currently learning how to calculate the volume of a 3D surface expressed in spherical coordinates using triple integrals. There was this exercice (from here) which asked me to find the volume of ...
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3D Sphericall Symmetric Analytical Convolution

I am trying to analytically convolve two spherically symmetric functions in 3D spherical coordinates, a 3D Gaussian and a "box" (really a radial step function). Numerical convolution yields ...
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About spherical coordinates

I will post an image which believe it is essential to understand the question: See the image: Figure book Spherical coordinates $r, \theta, \phi$ are perfectly intuitive because the angles $\theta$ ...
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Find the mass of the ice cream cone

The region looks like an ice cream cone. It is an upside-down circular cone attached to a slice of a sphere. I'm pretty sure the way you are "supposed" to solve it is with spherical ...
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Integrate a gaussian distrbution over a sphere surface

I want to integrate an off-center guassian distribtuion function over the surface of a sphere with radius of R. Here is the function: $$ f(x,y,z) = A \exp(-\frac{(x-x_{s})^2+(y-y_{s})^2}{d^2})$$ $x_s, ...
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Gradient in Spherical coordinates

I'm trying to derive the gradient vector in spherical polar coordinates: $$\nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y},\frac{\partial}{\partial z} \right)$$The method I am ...
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Showing Laplacian is rotation invariant in spherical coordinates

I am stuck at something very trivial. I want to show that the Laplacian operator $$ \frac{1}{r^2} \frac{\partial}{\partial r} \bigg(r^2 \frac{\partial f}{\partial r}\bigg)+ \frac{1}{r^2 \sin \theta} \...
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Confusion about differential forms and spherical coordinates

Let $x,y,z$ denote the standard coordinates of $\Bbb R^3$, and let $r,\phi,\theta$ denote the spherical coordinates on $\Bbb R^3-\{0\}$. It is well-known that $dx \wedge dy\wedge dz=r^2 \sin \phi ~dr\...
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Flux Through Sperical Cap

So I have the spherical Cap formed by the interception of the sphere $x^2+y^2+z^2=5$ and the plane $z=1$. Given the field $\vec{F}=(xy^2z,yz^3,y^4)$, I want to calculate the flux through the cap: $$ \...
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Transformation Matrix from Spherical coordinate system to another Spherical system with different origin

I have a vector, $\textbf{B}=(B_\rho, B_\theta, B_\varphi)$, in a Spherical coordinate system, $S$, with $\rho\in[0,\infty[$, $\theta\in[\frac{-\pi}{2},\frac{\pi}{2}]$, $\varphi\in[0,2\pi]$. The ...
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The geometric meaning of a mapping in spherical coordinates

Describe the geometric meaning of replacing $(\rho,\theta,\phi)$ by $\left(\rho,\theta,\phi+\frac\pi2\right)$ in spherical coordinates. I'm thinking a rotation of $90$ degrees about the xy plane, and ...
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Sphere's surface area element using differential forms

I am trying to use differential forms to determine the surface area element for a sphere. For a sphere of radius $r=1$. I think I am loosing something in the algebra (tried to check symbolic ...
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32 views

Integrate $\iiint_S e^{\sqrt{(x^2+y^2+x^2)^3}}$ using spherical coordinates

I need to integrate: $$\iiint_S e^{\sqrt{(x^2+y^2+x^2)^3}}$$ Where $S$ is a sphere with radius 1 and center in the origin, using spherical coordinates Using: $\iiint_s f(\rho,\phi, \theta)\rho^2 sen\...
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Finding $\iiint_R e^{\sqrt{(x^2+y^2+z^2)^3}} dA$ with spherical coordinates

How to integrate the following expression $$\iiint_R e^{\sqrt{(x^2+y^2+z^2)^3}} dA\,,$$ where $R$ is a sphere with radius $1$ centered in the origin? I did $$\iiint_R e^{\sqrt{(x^2+y^2+z^2)^3}} dA = 4\...
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Volume of a cone inside a sphere

Using spherical coordinates I have to find the volume of a cone $z=\sqrt{x^2+y^2}$ inscribed in a sphere $(x-1)^2+y^2+z^2=4.$ I can`t find $\rho$ because the center of sphere is displaced from the ...
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Integral with the indicator function using spherical coordinates

The integral of interest is: $$I = \int_{\mathbb{R}^3}\int_{\mathbb{R}^3} \boldsymbol{1}\left(\frac{1}{2}\frac{(x_2^2 - x_1^2) + (y_2^2 - y_1^2) + (z_2^2 -z_1^2)}{x_2-x_1} \in [0,1]\right) \nonumber \\...
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Convert the rectangular coordinates of the cone to cylindrical coords.

The height of the cone is $4$ and the radius is $2$ given by the inequality $0 \leq z \leq 4-2\sqrt{x^2 + y^2}$. From that information we can see we have the point $(1,0,2)$ and we can use this to ...
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Normal unit vector of sphere with spherical unit vectors $\hat r$, $\hat \theta$ and $\hat \phi$

I'm trying to find the normal unit vector at each point of the sphere $x^2+y^2+z^2 = a^2$ using cartesian and spherical unit vectors, as shown below. I was able to get it using cartesian unit vectors. ...
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Using spherical coordinates, is there an equation of a sphere not centered at the origin? If so what is it?

I am a high school teacher teaching Calculus for the first time actually, I am teaching Multivariable Calculus (Calculus 3). Its been a solid 15 years since I took Calculus 3. During a discussion of ...
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Convert function from cartesian coordinates to cylindrical and spherical

Introduction I have a point charge of magnitude $-6Q_0$. This point charge is placed in the origin of an orthogonal coordinate system. The electric field at a arbitrary point $P=(x,y,z)$ caused by ...
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Deriving the new differential wave equation when assuming cylindrical symmetry

We have the differential wave equation $$\dfrac{\partial^2}{\partial{r}^2}(r \psi) = \dfrac{1}{v^2} \dfrac{\partial^2}{\partial{t}^2}(r\psi).$$ If we assume cylindrical symmetry, then we have that $$\...
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Change of variables for coupled vectors in spherical coordinates

I am working on some calculations on coupled transition dipole moments interacting with light pulses with different polarisations. I am currently trying to find a way of writing $\theta_{\beta}$ and $\...
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Convert spherical vector field to cartesian vector field

I have a vector field defined in spherical coordinates as follows: $$\vec{F}\big\langle\rho,\theta,\phi\big\rangle = \bigg\langle\rho \sin\theta \cos\phi ,\rho^3 \cos \phi, \frac{\tan \theta}{\sqrt{r}...
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Triple or Surface Integral??

I have a difficulty to understand when I should use the surface integral over a surface or the triple integral over a space. For instance: If $S$ is the subset of the sphere with radius $1$ with $x>...
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$\int_{S}^{} z^{2} dS$ , $S$ sphere

$S$ is a sphere of radius $R$. Using, if you want to, some symmetrical argument, find $\int_{S}^{} z^{2} dS$. I used spherical coordinates: $$x=\rho \sin(\varphi )\cos(\vartheta )$$ $$y=\rho \sin(\...
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Going from cartesian to cylindrical coordinates - how to handle division with $0$

I have three point charges with the cartesian coordinates: $q_1(a,0,0) \: \: \: q_2(0,a,0) \: \: \: q_3(0,0,a) $, I want to convert these into both cylindrical and spherical coordinates. The cartesian ...
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Points on a sphere - how to get the third angle

So by the equation of a sphere $x = x_0 + r\cdot\sin\theta\cos\phi$ $y = y_0 + r\cdot\sin\theta\sin\phi$ $z = z_0 + r\cdot\cos\theta$ I can get the three coordinates from two angles and I can get the ...
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Find an equation in spherical coordinates for the surface represented by the rectangular equation [duplicate]

The rectangular equation is $$x^2+y^2-8z^2=0$$ $$x^2+y^2=8z^2$$ Know in the relationship between rectangular and spherical coords. we can manipulate our given to fit the form: $$x^2+y^2+z^2=9z^2$$ $$\...
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Find an equation in spherical coordinates for the surface represented by the rectangular equation

The rectangular equation is $$x^2+y^2-8z^2=0$$ $$x^2+y^2=8z^2$$ Know in the relationship between rectangular and spherical coords. we can manipulate our given to fit the form: $$x^2+y^2+z^2=9z^2$$ $$\...
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In spherical coordinates for the region $x^2+y^2+z^2\geq1$, $x^2+y^2=z^2\leq4$, $z\geq\sqrt{x^2+y^2}$, why does $\psi$ range from $0$ to $\pi/2$?

I was given the following task calculate: $\iiint_V{(xz^2+z)dxdydz} \space$ where $\space V = \begin{cases} x^2+y^2+z^2\geq1 & \text{inner sphere} \\ x^2+y^2+z^2\leq4 & \text{sphere} \\ z\...
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Converting integration dV in spherical coordinates for volume but not for surface?

When calculating the volume of a spherical solid, i.e. a triple integral over angles and radius, the standard $dx\,dy\,dz$ gets converted into $f(x,y,z)r^2\sin\Phi \,d\Phi \,d\Theta \,dr$. However, it ...
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How fast are you moving on the earth at a given latitude?

The radius of the earth is about $3959\,\mathrm{mi}$, so the earth is rotating at about $$\frac{1}{24\,\mathrm{hours}}\times 2\pi\times 3959 \,\mathrm{mi} \approx 1036\,\mathrm{mph}\,$$ and so someone ...
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Why is this parametrization of a hemisphere wrong?

I'm learning about surface integral, and one problem uses the surface of a hemisphere with radius $2$. $(x^2+y^2+z^2=4, z \geq 0)$ So I now want to find the parametrization $r(t, s)$. I thought I ...
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Volume of $(x^2 y^2) + (x^2 z^2) + (y^2 z^2) = 1$

I've long been intrigued by this surface which tightly hugs each axis, extending to infinity but with finite volume. But integrating this formula is beyond my powers. Any suggestions on how to do this,...
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Evaluating the Center of Mass of hollow objects using triple integrals

I have been calculating the center of masses of various objects using triple integrals, however, one thing I am struggling is calculating the center of mass of hollow objects. My first approach was to ...
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Representing a point in cartesian space as a position vector in spherical coordinates

I had a quiz in one of my physics classes the other day, and one of the questions is still bugging me. Say we have a point in 3-dimensional cartesian space with coordinates $(0, 2m, 0)$. How you would ...
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Transform the problem of finding the smallest enclosing circle on the surface of sphere to smallest circle in carthesian space

I have points on the surface of a sphere and I need to find the cutting-plane that cuts off all points and also cuts off the minimum possible volume to do so. There is the https://people.inf.ethz.ch/...
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Finding the volume of a region using spherical coordinates

I am trying to find the volume of the region bounded by $z=0, z=y^2, x=-1, x=1, y=-1, y=1$ using spherical coordinates. I realize that I don't need spherical coordinates to compute the volume since I ...
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Converting spherical electric potential to cartesian coordinates

How can I convert this spherical scalar function... $$\Phi(r,\phi,\theta)=kq\left[\frac{1}{r-d\cos(\theta)} - \frac{1}{r+d\cos(\theta)}\right]$$ to, what should be, its rectangular equivalent... $$\...
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How to integrate a pdf, $f_{XYZ}$ within a 3-D sphere?

Say that I have $f_{XYZ} = C$ that is defined within a 3-D sphere and that it's constant. If I want to obtain $f_X$, I have to integrate over $Y$ and $Z$ dimensions. What is the easiest way to do this,...
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Help With Euler Angles

I apologize in advance for the lack of meaningful formulas or calculations, I am not a mathematician and am using excel to try to compute everything. I think there is likely a simpler way to approach ...
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How to plot symmetrical bumpy sphere from spherical coordinates

I saw a previous question asking about the equation of a symmetrical bumpy sphere which yielded this paper, however the equations in the paper are beyond me. The referenced bumpy sphere equation is a ...
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1answer
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Getting geographical coordinates from pixel positions relative to known places on a map

I have a screenshot from a map of my hometown and I want to use it for a project of mine. For that I need the geographical coordinates of the point in the top left corner. I could just look it up on ...
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Volume of cylinder inside sphere, spherical coordinates

I'm asked to compute the volume of following integral in cylindrical and spherical coordinates: $$\int_0^{2R} \int_{-\sqrt{2Rx-x^2}}^\sqrt{2Rx-x^2} \int_0^\sqrt{4R^2 - x^2 - y^2}dx\ dy\ dz$$ I know ...
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Finding intersections of a spherical spiral with a geodesic segment

Long story short: Can we analytically solve for $φ$ in this equation? $\sin(\varphi) \cdot (A \cdot \cos(k \cdot \varphi) + B \cdot \sin(k \cdot \varphi)) + C \cdot \cos(\varphi)=0$ Given a point on ...
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68 views

Delta function of the Euclidean norm $\delta(|\mathbf x|)$ / in polar coordinates at origin $\delta(r)$

Several posts discuss the representation of the delta function in polar coordinates in 2D or 3D, e.g. Dirac delta in polar coordinates or Delta function at the origin in polar coordinates Does anyone ...
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Is there a faster way to find distances on a sphere than Haversine?

My task is to find the distance between the arbitrary point and many others on the sphere. Unfortunately, Haversine, even in vectorized form, does not satisfy in terms of speed, and I would like to ...
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63 views

Spherical Coordinate System and It's basis vectors

I was studying a Spherical Coordinate System. And I kinda stuck in process where it's coordinate is represent in term of vector notation . Any point $P$ in standard vector space can be represented by ...
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Converting cartesian coordinates into latitude and longitude coordinates

I'm trying to convert a position vector on a unit sphere into the latitude and longitude coordinates but I'm not sure how to do it. I know that the formula for converting the latitude and longitude ...

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