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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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velocity gradient in cylindrical coordinate.

In cartesian co-ordinate ($x,y,z$), gradient of velocity($\mathbf{u}=(u_x,u_y,u_z)$)(Jacobian matrix)is defined: \begin{equation*} \nabla \mathbf{u}= \begin{pmatrix} \partial_x u_x && \...
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Find $\iiint_V z$ with $V=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$

Let $f(x,y,z)=z$ and $T=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$ Find $\iiint_T f(x,y,z) dV$ I'm having a few problems with this integral, here's ...
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Convert cylindrical coordinate displacement to cartesian

I have a set of points of a finite element mesh which when inputted into a solver (ansys) gives the displacement of each node. I can get the displacement values of each node either in r,theta(in rads),...
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Finding the volume of the intersection of a cylinder and a sphere

Let $\alpha>0$. Given the cylinder $$B_1=\{(x,y,z)\in\mathbb R^3\;|\;x^2+y^2\leq\alpha x\}$$ and the sphere $$B_2=\{(x,y,z)\in\mathbb R^3\;|\;x^2+y^2+z^2\leq\alpha^2\},$$ how can one find the ...
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33 views

Volume integral over circle not around the origin

For an assignment I am asked to find the volume of a given volume R, namely $R=\left\{(x,y,z):0\leq z\leq\sqrt{4-x^2-y^2},(x-1)^2+y^2\leq1\right\}$. I have attempted solving this using cylindrical ...
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How do I graph using cylindrical coordinates?

I need to graph the curves $x^2 + y^2 = 4$, and $x^2 + y^2 = 25$ in the cylindrical coordinate system, but I don't know how. I substituted $x$ and $y$ values for their cylindrical counterparts, but I ...
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Find the volume of the Region (Triple Integral with Cylindrical Coordinates)

Question: Find the volume of the region that is contained by the cylinder $x^2+y^2=81$, bounded above by $z=x$ and below by the $xy$-plane. I have tried the integral $\int_{0}^{2pi}\int_{0}^{9}\int_{...
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Triple integration for the volume of a given sphere

I have a problem which I've had a look on "Maths Stack Exchange" and other resources to help, but still am stuck, so any help would be most appreciated. My Problem: Set up a triple integral for the ...
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What is the volume above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$?

What is the volume above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$? I tried converting each equation to cylindrical coordinates: $z= $r, $r^2+𝑧^...
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Find the volume of solid region bounded by three cylinders.

The equations of the 3 cylinders are given by $x^2+y^2=1$, $y^2+z^2=1$ and $x^2+z^2=1$. While it is common to solve it in cylindrical coordinates via triple integral, I would like to know how to ...
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Center of mass of a planar lamina

I have to find the center of mass of a planar lamina bounded by $x=0$, $y=1/2$, and $y=x$, with the density of the lamina being $x/(1-y^2)^{1/2}$. I ended up drawing a picture, and it looks like a ...
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Triple integral conversion to cylindrical coordinates equals zero

I'm asked to convert this integral: $$\int^1_0\int_{-\sqrt{1-y^2}}^0\int^{2-x^2-y^2}_0 x\ dz\ dx\ dy $$ to cylindrical coordinates. This is what I calculated for the limits: $$\int^{2\pi}_0\int^0_{-1}...
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What's The Cylindrical coordinates for this $\int_0^6\int_{-\sqrt{6x-x^2}}^{\sqrt{6x-x^2}}\int_0^{6x-x^2-y^2}\left(x^2+y^2\right)dzdydx$

I want to convert this to cylindrical coordinates $$V=\int_0^6\int_{-\sqrt{6x-x^2}}^{\sqrt{6x-x^2}}\int_0^{6x-x^2-y^2}\left(x^2+y^2\right)dzdydx = 486π$$ I want to write it like this: $$V=\int_{\ }^{ ...
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Divergence of a vector field in cylindrical coordinates

Let $\bar{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a vector field such that $\bar{F}(x,y,z)=(x,y,z)$. Then we know that: $$\nabla\cdot\bar{F}=\frac{\partial\bar{F}_x}{\partial x}+\frac{\partial\bar{...
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1answer
71 views

Lagrangian mechanics- conservation of energy

Consider a single particle system whose Lagrangian remains the same if the position of the particle is simultaneously (i) rotated by an arbitrary angle s about the z-axis and (ii) shifted by an amount ...
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1answer
63 views

Equations of Motion in Cylindrical Co-ordinates

I've run into an interesting set of differential equations, that I'm not 100% sure where to begin- I'm not looking for a 100% complete solution, more just a push in the right direction of where I can ...
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Laplace transform of heat conduction PDE in cylindrical coordinates.

I'm trying apply the Laplace transformation to solve the non-dimensional heat conduction PDE for a hollow cylinder with convection boundary conditions and a non-homogenous initial condition. $$\frac{...
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Find radius of best fit cylinder from cloud

I want to find the radius of the cylinder that best fits a cloud of points. I used 3DReshaper to calculate the radius of the best fit cylinder for this points (format: x y z): ...
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lateral surface area of cylinder

Use cylindrical coordinates and multivariable calculus to prove that the lateral surface area of a right, circular cylinder with radius 2 and height h is 4pih. I parameterized x = rcostheta, y = ...
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deducing an area of integration in cylindrical coordinates

If I am given an area characterised by $0 \leq z \leq 1-r^{2}$ from this how can I deduce the radius r and the angle with the x-axis, $\theta$ that will span the are and I can ten integrate over i.e. ...
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49 views

Heat equation in cylindrical coordinates at origin

I'm trying to solve a heat equation in cylindrical coordinates $$\dfrac{\partial u}{\partial t} = a \left(\dfrac{\partial^2 u}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial u}{\partial r} + \dfrac{1}{...
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1answer
36 views

Integration with substitution to cylindrical coordinate

Solve the integral $$ \\A:= \int_{{x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}\leq 1}{x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}dxdydz \ $$ In my solution I have substituted to cylindrical ...
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1answer
46 views

Change of coordinates vs change of shape

It is an elementary fact that we are able to change coordinates system to new one. for example in Cartesian coordinates $x^2+y^2=1$ illustrates a circle. Changing to polar coordinates, this equation ...
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1answer
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how to find the volume of the Revolved Domain about z Axis [ volumes ] [ integrals ]

let $D$ = {$(x,0,z) | (x-1)^2 + z^2 \leq 1$} find the volume of the body obtained by revolving $D$ about the $ Z $ axis. how do i solve this with integrals ( triple / double ) . intuitive solution ( ...
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Transformation from global spherical to local cylindrical

I have the coordinates of a vortex in a global spherical coordinate on the surface of a sphere defined in terms of latitude and longitude(earth centric). Now I need to transform to a local coordinate ...
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2answers
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Find the volume of intersection between cylinders

Find the volume of intersection of the cylinder {$ x^2 + y^2 \leq 1 $} , {$ x^2 + z^2 \leq 1$}, {$ y^2 + z^2 \leq 1$}. i am having tough time finding the volume how do i solve this kind of ...
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Transformation from spherical to cylindrical coordinates

I have a coordinate on the surface of a sphere i.e latitude and longitude(two dimensions). I need to transform into a 2-D cylindrical coordinate frame i.e. r and the azimuth angle theta. Is there a ...
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With $\alpha>0$, how to calculate the volume of $E_\alpha=\{(x,y,z)\in\Bbb R^3: x^2+y^2-\alpha^2\le\alpha z\le\alpha(3\alpha-2x)\}$

With $\alpha>0$ let $E_\alpha=\left\{(x,y,z)\in\Bbb R^3:\frac1\alpha(x^2+y^2-\alpha^2)\le z\le 3\alpha-2x \right\}.$ I'm asked to find the volume of this set, which is $V_\alpha=\iiint_{E_\alpha}...
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Line element to polar coordinates

I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is: \begin{equation} \mathbf{v}=\frac{A}{r} \hat{r} + \frac{B}{r}\hat{\theta} \end{equation} So:...
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Cartesian coordinates to Cylindrical coordinates

The position vector, as a function of time and in Cartesian coordinates, of a particle is the following: $$\vec{r}(t) = (5t^2-6t)\vec{e}_1 + \cos(\sin(t))\vec{e}_2+ (8t-5)\vec{e}_3$$ I have that ...
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Find the volume enclosed between cone and rose petal

I understand that while changing an integral from cartesian to cylindrical or polar or spherical coordinates requires multiplication of a Jacobian to integration variables but what when the region is ...
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120 views

Unwrapping a cylinder onto a plane

I have an image of a label on a bottle, and I need to 'unwrap' the image as if I had taken the label off of the bottle and laid it flat on a table. Is there an easy way to transform the image, given ...
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129 views

How to evaluate the integrals in the cylindrical coordinates

Evaluate the following integral in cylindrical coordinates $$\int^{1}_{-1}\int^{\sqrt{1-x^2}}_{0}\int^{2}_{0}\dfrac{1}{1+x^2+y^2}dzdydx$$ My try: I first took the boundaries as $$-1\le x\le1\\0\le ...
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2answers
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Conversion of a Vector in a Cartesian Coordinate System to a Cylindrical Coordinate System

I'm having trouble converting a vector from the Cartesian coordinate system to the cylindrical coordinate system (second year vector calculus) Represent the vector $\mathbf A(x,y,z) = z\ \hat i - ...
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How to write basis vectors in Cartesian coordinates in terms of cylindrical coordinates?

Given a Cartesian coordinate system with basis vectors (ex, ey, ez) and a Cylindrical coordinate system with basis vectors (er, eθ, ez) Why and how does: ex = erCos(θ) - eθSin(θ) ? ey = erSin(θ) +...
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1answer
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Describing this solid in cylindrical coordinates

Let $Q$ be the solid delimitated by the paraboloid $z = x^2 + y^2$, by the cylinder $x^2 + y^2 = 4x$ and by the plane $z = 0$. In cartesian coordinates, we can write: $$Q = \{ (x, y, z) \in \mathbb{R}^...
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Converting the coordinates of a point to cylindrical coordinates with positive values.

I'm trying to convert the coordinates of point $(x,y,z) = (-2,-1,0)$ to cylindrical coordinates, with positive values for $\theta$ and $r$. I know that: $r^2 = x^2 + y^2$ so... $r = \sqrt(5)$ ...
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inward or outward normal to a surface?

I've got a conceptual problem regarding inward and outward normals. The textbook question (2nd year vector calculus) is as follows: A uniform fluid that flows vertically downward is described by ...
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Integrating $ \frac{1}{\sqrt {x^{2}+y^{2}}}-\frac{1}{\sqrt {y^{2}+z^{2}}}\ $

I am trying to solve this integral over a Cube: $$\frac{1}{\sqrt {x^{2}+y^{2}}}-\frac{1}{\sqrt {y^{2}+z^{2}}}\ $$ I can see that this that this will turn into zero because of symmetry since I am ...
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1answer
57 views

How do I revolve a general 2D coordinate system?

$\newcommand{\dd}{\partial}$ Question I wish to construct a general 3D revolved, orthogonal, curvilinear co-ordinate system, where the axis of revolution is coincident with the Cartesian $z$-axis. ...
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1answer
22 views

Volume using cylindrical coordinates

I have to find the volume of the solid which base is bounded by $$x^{2}+y^{2}+2y=0$$ and it's bounded, above, by the surface $$z=4-x^{2}-y^{2}$$ I tried to use cylindrical coordinates, where $$x=r\...
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1answer
23 views

Compute an indefinite integral through change of variables

I'm trying to compute the following integral: $\int_{-\infty}^\infty \int_{-\infty}^\infty e^{x_1^2 + x_2^2} dx_1 dx_2$ I have been thinking about a change of variables with cylindric coordinates, ...
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1answer
52 views

How to draw lines and circles on cylindrical projection map?

I am trying to draw a circle with known radius around a coordinate on a cylindrical projection map. Which is a circle around equator and egg shaped closer to the poles. And also trying to draw a line ...
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Interpreting Cylindrical equations gemotrically

I'm a bit unsure how to interpret the following equation given in cylindrical coordinates geometrically. tanθ = 1 From what I understand, this means that the angle b/w the x-axis and point in the ...
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1answer
345 views

Divergence of $1/r$ in cylindrical coordinates

In classical textbooks, like "Introduction to Electrodynamics" by J.D. Griffiths, it is given that $$\nabla\cdot\left(\frac{\widehat{r}}{r^2}\right)=4\pi\delta^3(R).$$ To prove this equality, ...
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1answer
36 views

Perpendicular vector fields in cylindrical coordinates

With two vector fields in cylindrical coordinates, I am trying to find how they may be perpendicular to each other $$ A (\rho, \phi,z) = \rho cos \phi \hat\rho + \rho sin \phi \hat\phi + \rho \hat z $...
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Derivative of vector in cylindrical coordinates with regards to $\varphi$?

We have cylindrical coordinates $(\rho,\varphi,z)$ with basis vectors $\hat e_\rho, \hat e_\varphi, \hat e_z$. Let's say we have a vector $\vec v=z\hat e_\rho + \hat e_\varphi\ + \rho \hat e_z$. How ...
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2answers
57 views

(Multivariable Calculus) Convert $\rho = \sin \phi$ to cylindrical and rectangular

Question: Consider the surface given in spherical coordinates by $\rho = \sin(\phi)$. Convert to rectangular coordinates and cylindrical coordinates. Identify the surface. By graphing the function, I'...
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2answers
112 views

How do we deal with ArcTan (or other inverse functions) of undefined values?

When converting coordinates from rectangular to cylindrical, to spherical, etc. we will eventually come across having to use $ArcTan(y/x)$, and/or $ArcCos(z/\rho)$ to derive $\theta$ and/or $\phi$. ...
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Representing displacement vectors in cylindrical coordinates and finding the distance in cylindrical coordinates?

In cartesian coordinates, we can derive the vector $\vec v_3$ by vector subtraction $\vec v_2-\vec v_1$. We then get the distance between $P$ och $Q$ by taking the absolute value of $\vec v_3$ which ...