# 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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### 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 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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### 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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### 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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### (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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### 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 ...