Questions tagged [cylindrical-coordinates]

Questions on cylindrical coordinates, a coordinate system where points in space are represented by their distance to the $z$-axis ($r$), the angle the line joining the orthogonal projection of the point on the $xy$-plane and the origin makes with the positive $x$-axis ($\theta$) and the $z$ coordinate of the point.

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Given a vector field in spherical coordinates, compute the flux through a disk at z = -d

I want to compute the flux of the magnetic field $B = \frac{\mu_0m}{4\pi r^3}(2cos(\theta)\vec{e_r}+sin(\theta)\vec{e_\theta})$ through the disk at $z=-d$ with radius b centered around the z-axis. ...
Merkel_Bot's user avatar
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What coordinate substitution should I perform to evaluate this triple integral?

I am trying to evaluate the following triple integral: \begin{equation} \int_{-1}^1 \int_{-\sqrt{4-4x^2}}^{\sqrt{4-4x^2}} \int_{\sqrt{4x^2 + z^2}}^2 ye^{4x^2 + y^2 + z^2} \, dy\, dz\, dx \end{equation}...
Christopher Miller's user avatar
4 votes
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51 views

evaluate the volume of solid

Consider the paraboloid $(\mathcal{P}): z=x^2+y^2$ and the plane $(\mathcal{Q}): 2x+2y+z=2$. Let $\mathcal{S}$ be the solid region bounded above by $(\mathcal{Q})$ and below by $(\mathcal{P})$. Find ...
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set the limits of integration of the spherical coordinates between two paraboloids and a plane

Find the volume of the solid $\mathcal{S}$ enclosed laterally by the paraboloids $\mathcal{P}_1$ of equation $z = x^2 + y^2$ and $\mathcal{P}_2$ of equation $z = 3(x^2 + y^2)$ and from above by the ...
Student's user avatar
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7 votes
2 answers
210 views

Volume of cylindrical wedge of intersecting cylindrical shells

Two cylindical shells of equal radius are inserted one into the other at various angle between the axes (I tried to give an example with the pic attached). What is the maximum volume for the ...
user967210's user avatar
2 votes
1 answer
87 views

Solution of the transient heat equation for an infinite domain with a circular hole and angular symmetry

I mean to solve the heat equation in 2D $$ \frac{\partial T}{\partial t} = \alpha \frac 1r \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right) \tag{1} $$ in an infinite domain ...
sancho.s ReinstateMonicaCellio's user avatar
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General Solution To Helmholtz Equation in Cylindrical Coordinates Seems To Be Wrong

On the web, it is given that the general solution to the Helmholtz equation ($\nabla^2u+ku=0$) in cylindrical coordinates is $$\sum_{m=0}^\infty\sum_{n=0}^\infty{[A_{mn}\cdot J_m(\sqrt{k^2 - n^2}\cdot ...
Remeraze's user avatar
1 vote
1 answer
56 views

Area of a plane enclosed by a cylinder

I have the cylinder $x^2+y^2=R^2$ and the plane $z=ax+b$ ($R,a,b$ are constants). I need to find the area of the ellipse that is the region of the plane enclosed by the cylinder. I've done it with a ...
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Integral over unit vectors in different coordinates systems

In one of my problems, I have to integrate: $$ \int_0^{2\pi} \vec{e}_\phi \: d\phi $$ What the solution to my problem seems to say is that because $$ \vec{e}_\phi = cos\phi \, \vec{e}_x + sin\phi \, \...
tonetillo 4's user avatar
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How to approach multivariable integration problems?

I have a problem that seems to be a cylindrical conversion problem, but I could not find bounds for r. The problem asked me to find the volume bounded by $z = (2x)^2 + y^2$ and $z+y^2 = 2$. I first ...
Knight of the darkmoon's user avatar
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65 views

A paradox on curl equations in cylindrical and spherical coordinates

Let $\mathbf{A}=\sin(\theta)\hat{\phi}$ be an azimuthal vector field in either cylindrical (cylindrical radial, azimuthal, vertical)=$(\rho,\phi,z)$ or spherical (spherical radial, colatitude, ...
Aria's user avatar
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3 votes
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Struggling with a Calculus III problem :$\iint x+y\ dS,$ where $S(u,v)=2\cos(u)\vec i+2\sin(u)\vec j+v \vec k$ and $0\le u\le \pi/2;\;0 \le v \le9.$

So I have had Calc III many moons ago but I cannot seem to solve this problem for my son who is taking it now. Worked it four ways and got four different answers. Hoping someone here can set me ...
Transonic's user avatar
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Using cylindrical coordinates to solve volume of revolution question

Is there a way to solve a volume of revolution question using cylindrical coordinates by using three iterated integrals $dr,dz,d\theta$ ? This is the question: I can solve the volume when this is ...
Aurora Borealis's user avatar
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The center of mass of a semiellipsoid

I am trying to find the center of mass of a semiellipsoid using cylindrical coordinates. $$ \frac{r^2}{a^2}+\frac{z^2}{b^2} \leq 1 $$ with $z < 0$ and density = 1. I know that the center of mass is ...
Mike Gotier's user avatar
1 vote
1 answer
54 views

Understanding the Meaning of $y \geq x$ in Cylindrical Coordinates during a Variable Transformation.

I am seeking clarity / intuition on the meaning of the condition $y \geq x$ when performing a change of variables from Cartesian to Cylindrical Coordinates for volume integration. In the context of ...
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Find the volume of a body formed by a cylinder and a hyperboloid

Find the volume of a body formed by a cylinder $x^2 + y^2 = 9$ and a hyperboloid $x^2 + y^2 + 9 = z^2$ My solution Let's try to build the body data When viewed from above, it will look like a circle ...
Nick Schemov's user avatar
1 vote
0 answers
42 views

Integral representation of Dirac Delta in cylindrical coordinates

Dirac delta have the representation $$ (2\pi)^4\delta^4(x) := \int e^{ik.x} d^4 k $$ I would like to know how such integral representation realized in cylindrical coordinates. I tried the following ...
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When integrating by an area defined on the xy plane, dA=dx*dy. In cylindrical space, dA=rdrdθ. This doesn't line up with what I calculated. Why?

I understand that we're calculating the area of an infinitesimal polar rectangle, and summing up many of them. My teacher kind of glossed over why this produced rdrd$\theta$. I tried verifying by hand,...
Max Son's user avatar
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Combining the two polar Navier-Stokes equations into a single PDE using vorticity and streamfunction

I am starting with the following reduced form of the Navier-Stokes equations in polar coordinates. $$u_r\frac{\partial u_r}{\partial r}+\frac{u_\theta}{r}\frac{\partial u_r}{\partial\theta}-\frac{u_\...
WnGatRC456's user avatar
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Rotation in cylindrical coordinates and exterior derivative

Suppose a 1-form $A$ of $\mathbb{R}^3$ is represented as $A= A_r (r,\theta,z)dr + A_\theta (r,\theta,z)d\theta + A_z(r,\theta,z)dz$ using cylindrical coordinate system $(r,\theta, z)$. The external ...
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How to convert Cartesian coordinate to polar coordinate in $-180°$ to $+180°$

I wanted to plot temperature at circumference of a circle. I extracted the data by converting cartesian coordinates into polar coordinates by using ATAN2 function, but the plot is having negative and ...
Liril Silvi's user avatar
1 vote
0 answers
83 views

surface integral with vector field, using cylindrical coordinates

I want to compute a surface-integral, but using cylindrical coordinates. I'll denote the Cartesian coordinate system on $\mathbb{R^3}$ by $\mathbb{R}^3_{xyz}$, and the cylindrical by $\mathbb{R}^3_{\...
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Calculating the volume of $(x-z)^2+(y-z)^2 \le \sin^2(z) $

I am trying to calculate the volume of $(x-z)^2+(y-z)^2 \le \sin^2(z) $ and $0 \le z \le \pi $. I am having trouble parameterising the equation and setting the bounds for the parameters. I think the $-...
Buhgro's user avatar
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calculation rules of unit vector

I cannot figure out how to derive a formula in a scientific paper (LINDBORG, 2007, DOI: 10.1175/JAS3864.1). I will list all the information needed below: The starting point of the derivation is this ...
Gaelthorn's user avatar
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4 votes
2 answers
277 views

Calculating Triple Integral using Cylindrical Coordinates

I'm given $ E $ is located in $ x^2 + y^2 = (z-1)^2 $ and between $z = 0$ and $z=2$. I used level curves to graph this out, and as I see it is a circular cone. First, I set up my region, $$ E = \Big\{(...
CodedRoses's user avatar
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1 answer
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Converting a rectangular equation to cylindrical coordinates

If I have an equation like $x^2+y^2+4z^2=10$, would the cylindrical equation then just be $r^2=10-4z^2$? I found this answer, but it just seems like it was too easy to find.
Kaeden George's user avatar
0 votes
2 answers
135 views

Expressing $ \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}=0$ in cylindrical coordinates

The problem says Show that when Laplace's equation $$ \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}=0 $$ is written in cylindrical ...
iwjueph94rgytbhr's user avatar
1 vote
0 answers
34 views

Every term in 3D Laplacian is separated by two spatial dimensions?

From Richard Haberman's Applied Partial Differential Equations with Fourier Series and Boundary Problems, 4th Edition, page 28, chapter 1.5, in the context of mnemonics to remember the Laplacian in ...
JongJu Park's user avatar
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0 answers
35 views

How to calculate solid angles of segments of hemisphere from abstract points

I am imagining a half-sphere which has been cut, pizza-style, into many slices, which may vary in size. I want to specify a point inside the semi-sphere at random and be able to identify what solid ...
Post169's user avatar
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2 votes
1 answer
75 views

How to change from cartesian coordinate to cylindrical coordinates

Consider the triple integral $$\iiint_{K} \frac{z}{2+ x^2 + y^2} dV$$, where K is the region defined by $z \geq \sqrt{x^2 + y^2}$ and $x^2 + y^2 + z^2 \leq 9$. The question then asks me to rewrite the ...
Need_MathHelp's user avatar
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1 answer
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Converting a cross product typically found in electrodynamics between coordinate systems

Context There are numerous posts on mathstackexchange and physicsstack exchange that seek clarity regarding conversion from a Cartesian coordinate system to curvilinear coordinate system, or viceversa ...
Michael Levy's user avatar
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How to transform unit vectors from one cylindrical coordinate system to another one displaced and tilted with respect to it?

I have one cylindrical coordinate system attached with a laboratory device and another cylindrical coordinate system that is attached with an object in it. The object is symmetric about the z-axis in ...
sumant ola's user avatar
1 vote
0 answers
101 views

Laplace's equation in cylinder — does a solution exist?

7.9.2. (c) Solve Laplace's equation inside a semicircular cylinder, subject to the boundary conditions $$ \dfrac{\partial}{\partial z} u (r, \theta, 0) = 0, \quad \dfrac{\partial}{\partial z} u (r, \...
Melanie's user avatar
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1 vote
1 answer
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Integrating over a Truncated Elliptic Cone

So let's say that we have I can only describe as a "truncated elliptic cone" (TEC) as on the picture: Let's further say that this TEC is of height $h$, and let's say that it is centered at ...
StormyTeacup's user avatar
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1 vote
1 answer
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Definition of a particular precession angle

I am trying to deduce an expression in terms of the $x$,$y$,$z$ coordinates of three points $P_1$, $P_2$ and $P_3$. Given the relatively small motion of points $P_1$ and $P_2$, I want to find the ...
Giano's user avatar
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0 votes
2 answers
141 views

Doubt in locating a point in cylindrical coordinate system

Suppose a vector $\overrightarrow{A} = A_r\overrightarrow{a_r} + A_\phi\overrightarrow{a_\phi} +A_z\overrightarrow{a_z}$ represents a point $P$ in cylindrical coordinate system and $\overrightarrow{A} ...
Shreyansh Kuntal's user avatar
-1 votes
1 answer
99 views

Computing the integral $\iiint_D\frac{z}{(x^2+y^2+z^2)^{\frac{3}{2}}}~\mathrm{d}x~\mathrm{d}y~\mathrm{d}z$

Computing the integral $$\iiint_D\frac{z}{(x^2+y^2+z^2)^{\frac{3}{2}}}~\mathrm{d}x~\mathrm{d}y~\mathrm{d}z$$ where $D$ is the portion of the ball $x^2 + y^2 + z^2 \leq 16$ that satisfies $z \geq 2$. (...
Adrián's user avatar
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What is the unity matrix in cylindrical coordinates?

If $e_{\rho},e_{\varphi},e_z$ are the unit vectors in cylindrical coordinates, one can check \begin{align} (e_{\rho}\otimes e_{\rho})\cdot (e_{\rho}\otimes e_{\rho})= e_{\rho}\otimes e_{\rho}, (e_{\...
homersimpson's user avatar
1 vote
1 answer
90 views

Divergence on the hyperbolic plane vs $3D$ divergence in cylindrical coordinates

I wonder if there is a connection between the divergence on the hyperbolic plane and the divergence in 3D expressed in cylindrical coordinates. According to Divergence operator on the hyperbolic plane ...
user99432's user avatar
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2 answers
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Distance from a point on a torus to the origin

This question may be slightly subtle, but I'd like to better understand why my first attempts to solve the following problem failed. I am defining a point on a torus with minor and major radii $0<r&...
Greg von Winckel's user avatar
3 votes
1 answer
1k views

Divergence of a tensor in cylindrical coordinates

I'm having a hard time finding a source with an actual derivation on this. I understand the divergence of a vector field in cylindrical coordinates. However, for a tensor, how do I go from this $$\...
Felix's user avatar
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1 answer
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Integral curve equations conversion to cylindrical coordinates

Consider an electric field (or whatever 3-component vector field you want) $\mathbf{E}=\left(E_x, E_y, E_z\right)$. Let $\mathbf{r}(s) = (x(s), y(s), z(s))$ be the parametric equation of a field line, ...
Trisztan's user avatar
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1 vote
0 answers
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Compute divergence of a matrix

Related to the question Compute $a\otimes a$ in cylindrical coordinates consider \begin{align} A(\rho,\varphi,z)=\begin{pmatrix} a_{11}(\rho,\varphi,z) & a_{12}(\rho,\varphi,z) & a_{13}(\rho,\...
homersimpson's user avatar
1 vote
0 answers
40 views

Compute $a\otimes a$ in cylindrical coordinates

I want to compute $a\otimes a$ if $a\in \mathbb{R}^3$ is given in cylindrical coordinates, i.e. $a=a_{\rho}e_{\rho}+a_{\varphi}e_{\varphi}+a_z e_z$, where $e_{\rho}=(\cos(\varphi),\sin(\varphi),0),e_{\...
homersimpson's user avatar
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63 views

Question about the definition of axisymmetric vector fields

i just started to study axisymmetric vector fields. In the definition it says a vector field $V$ in cylindrical coordinates $V(\rho,\varphi,z)$ is axisymmetric if it does not depend on the angular, i....
homersimpson's user avatar
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0 answers
47 views

Doing the riemann sum in cylindrical shells using left endpoints?

The picture shows how the method of cylindrical shells is explained using a riemann sum of midpoints, but can we do this sum using left end points or right end points?
Hfdssdjns's user avatar
1 vote
1 answer
132 views

Finding the volume between sphere and hyperboloid

I am trying to find the volume between the sphere: $x^2+y^2+z^2=9$ and the hyperboloid $x^2+y^2-z^2=1$. I set the integral as: $$\int_{0}^{2\pi}\int_{-3}^{3}\int_{\sqrt{1+z^2}}^{\sqrt{9-z^2}}rdrdzd\...
Buhgro's user avatar
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1 vote
0 answers
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Calculating Distance for Circ Cross Section in Cylindrical Coordinates Analogously to Distance for Rectangular Cross Section in Cartesian Coordinates

I have two cross sections: one is a rectangular cross section in cartesian coordinates and the other is a semicircle of radius R (the bottom half of a circle) in cylindrical coordinates. I have an ...
k12345's user avatar
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1 answer
285 views

Calculating the surface area of a spherical cap using cylindrical coordinates

I am trying to calculate the surface area of the spherical cap : $x^2+y^2+z^2=R^2$ and $z\ge h$. I parameterized the surface as $$x=\sqrt{R^2-z^2}\cos(\phi)$$ $$y=\sqrt{R^2-z^2}\sin(\phi)$$ $$z=z$$ I ...
Buhgro's user avatar
  • 49
2 votes
1 answer
42 views

When I reverse the order in the triple integral with cylindrical coordinates, I get different volumes

I want to compute the mass of the region between the sphere \begin{equation} x^2 + y^2 + z^2 = 9 \end{equation} and the hyperboloid \begin{equation} x^2 + y^2 - z^2 = 1 \end{equation} for $z \geq 0$ ...
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