# 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.

407 questions
Filter by
Sorted by
Tagged with
8 views

### 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. ...
49 views

### What coordinate substitution should I perform to evaluate this triple integral?

I am trying to evaluate the following triple integral: $$\int_{-1}^1 \int_{-\sqrt{4-4x^2}}^{\sqrt{4-4x^2}} \int_{\sqrt{4x^2 + z^2}}^2 ye^{4x^2 + y^2 + z^2} \, dy\, dz\, dx$$...
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 ...
• 317
37 views

### 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 ...
• 317
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 ...
• 743
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 ...
58 views

• 158
26 views

### 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 ...
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, ...
• 422
66 views

### 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 ...
34 views

### 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 ...
• 1,757
23 views

### 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 ...
1 vote
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 ...
• 674
1 vote
42 views

### 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 ...
• 103
1 vote
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 ...
• 221
41 views

### 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,...
• 21
67 views

• 131
373 views

### 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.
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 ...
1 vote
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 ...
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 ...
• 141
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 ...
• 1,083
90 views

### 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 ...
• 1,100
33 views

### 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 ...
1 vote
101 views

• 33
42 views

### 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, ...
• 178
1 vote
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,\...