# 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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### Using change of coordinates to find the exact value of an integral

Use an appropriate change of coordinates to find the exact value of the integral $$\int_{-\sqrt{3}}^{\sqrt{3}}\int_{-\sqrt{3-x^2}}^{\sqrt{3-x^2}}\int_{-3+x^2+y^2}^{3-x^2-y^2}x^2dzdydx$$ My work so far:...
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### How can I tell which function of two variables is larger?

In this case, $z = 1$ and $z = \sqrt{x^2 + y^2}$. How can I tell which function is bigger to choose the upper and lower bound?
1 vote
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### Triple integral set up using cylindrical coordinates

Set up an integral in cylindrical coordinates to evaluate $\iiint_{E} x y d V$ where $E$ is the region enclosed by the cone $z=2-\sqrt{x^{2}+y^{2}}$, the cylinder $x^{2}+y^{2}=1$, and the $x y$ plane. ...
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### Steady-state heat conduction in a cylinder with discontinuity in thermal conductivity

Consider a solid cylinder of length L and radius a in which the thermal conductivity has a jump discontinuity at a point along its axis. The two bases of the cylinder are maintained at zero ...
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### Describe set in cylindrical coordinates

Describe the set in cylindrical coordinates: A = {(x,y,z) ∈ R3 : y^2 + z^2 ≤ 4, |x|≤1} My solution: We use the cylindrical coordinates r,θ,z. x,y,z expressed in cylindrical coordinates in this case: x=...
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### Value of curl at the origin when there is a singularity at the origin?

Say we have the following vector field: There is a singularity at the origin because we end up dividing by 0. I'm not sure what the value of curl is at the origin. On the one hand, we can work out ...
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### Evaluate line integral for a vector field

Given the vector field $\mathbf{E}=\mathbf{a_x}y+\mathbf{a_y}x$, evaluate $\int\mathbf{E} \cdot \text{d}l$ from $P_3(3,4,-1)$ to $P_4(4,-3,-1)$ by converting both $\mathbf{E}$ and $P_3$ and $P_4$ into ...
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### Understanding the cylindrical coordinate notation

If we have a cylindrical shell with inner radius $R_i$ and outer radius $R_a$, height $2h$ and so that it's centered in the origin of the coordinate system. And let's say that for various reasons I ...
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### Rewriting triple integrals rectangular, cylindrical, and spherical coordinates

Write three integrals, one in Cartesian/rectangular, one in cylindrical, and one in spherical coordinates, that calculate the average of the function $f(x, y, z) = x^2 + y^2$ on the region $E$ in the ...
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### Numerical solution of convection-diffusion equation in cylindrical coordinates

I want to numerically solve the 2D {convection/advection}-{diffusion/dispersion/heat} equation when it is cast in polar/cylindrical coordinates. Whereas I found many recipes of how to solve the 2D PDE ...
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### Describe curvilinear grid using coordinate functions?

A curvilinear grid around a cylinder has the following properties: The grid has $n\varphi=20$ grid points in angular direction (along a circle in the xy-plane). The grid has $nr=5$ grid points in ...
1 vote
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### Converting from cylindrical to spherical coordinates for a field

Say I have the field $$F(r,\theta,z) = 5r\hat{r}+z\hat{\theta}+\theta\hat{z}.$$ Using the conversions found in the source transformations table in the 3rd row, 2nd column of this wiki page, image here ...
1 vote
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### how to convert any vector field into another coordinate system

Say I have a field $\vec{F}(x,y,z) = A_r \hat{r} +A_\theta\hat{\theta} + A_z\hat{z}$. I'd like to change this field into both spherical and cartesian coordinates. I've seen quite a few wikipedia pages ...
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### Integrals and Area-element in Cylindrical coordinates

I was trying to solve for the moment of inertia of a solid and a hollow cylinder, and I faced a small problem. I looked through online resources and found many ways to approach the problem. One of ...
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### How can I find $\iiint\frac{xz}{1+x^2+y^2}\,dz\,dy\,dx$ where $1≤x^2+y^2≤3, 0≤z≤3$?

Compute $$\iiint\frac{xz}{1+x^2+y^2}\,dz\,dy\,dx,$$ where $1≤x^2+y^2≤3, 0≤z≤3$. I've tried it. But I'm only confused with $\theta$. I think it should be $0$ to $2\pi$, but that'll make the whole ...
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### Write down this integral as a triple integral with cylindrical coordinates

I have the integral: ${\iiint} x^2 dx dy dz$ which is bounded from above by the elliptic paraboloid $z=2-x^2 - y^2$ and from below by the upper part of the cone $z^2 = x^2 + y^2$ I want to write this ...
1 vote
A paraboloid has equation $z=a-x^2-y^2$ and a plane the equation $z=\lambda a$, where $0< \lambda < 1$. $V(A)$ is the volume of the paraboloid between its vertex and the given plane. $V(B)$ is ...