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?
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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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Transformation of derivatives from cartesian to cylindrical coordinates

It is well known that for some function $\phi$ its derivatives have the following relations $$\left[\begin{array}{l} \frac{\partial \phi}{\partial r} \\ \frac{\partial \phi}{\partial \theta} \\ \frac{\...
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An example of integral with cross product

Suppose I have a uniform vector field ${\mathbf{f}}({\mathbf{x}}):\mathbb{R}^3 \to \mathbb{R}^3$. Let us fix a cylindrical coordinate system such that ${\mathbf f}({\mathbf x})=f\,\hat z$, where $f$ ...
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Cylindrical Coordinate Dirac delta Expression for Sphere Without Function Composition Inside Dirac delta?

Given the cylindrical coordinate $(\rho,\phi,z)$ (ISO) Dirac delta expression for a sphere: $$\delta(sphere(\rho,z))$$ WHERE $r_0$ is the sphere's radius and: $$sphere(\rho,z) = \sqrt{\rho^2+z^2}-r_0$$...
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Cylindrical Coordinate Convolution Between Dirac Deltas for a Plane and Sphere

Assuming this the correct way to express the cylindrical coordinate convolution between Dirac deltas for a $\phi,\rho$-plane and a sphere of radius $r_0$ centered at the origin: $$\delta(\sqrt{\rho^2+...
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Using spherical coordinates to evaluate $\iiint_{E}z dV$ where $E$ lies above paraboloid $z = x^2 + y^2$ and below the plane $z=2y$

I was initially stuck trying to solve this question using spherical coordinates. I eventually got it by using cylindrical coordinates: $$\int_0^\pi \int_0^{2\sin\theta}\int_{r^2}^{2r\sin\theta}rz dz ...
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Is the following Fourier Transform in cylindrical coordinates correct?

I am trying to solve the integral $$\int_ {Cylinder}e^{-i\vec{k}\vec{r}}dV=\int_0^Rrdr\int_0^{2\pi}d\phi\int_0^Le^{-ik_zz}e^{-i(k_xx+k_yy)}dz$$ I tried to rewrite it using polar coordinates and solved ...
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Numerical Solution of transient heat equation with source in cylindrical co-ordinates

I am attempting to use the finite difference scheme on a uniform grid in $r$. I have a no flux boundary condition at $r=0$ which is proving to be a pain to implement. I'm having difficulty in doing ...
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2 answers
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Computing : $\displaystyle \int \frac {m + n\cos (x-x_0) }{ [a+b\cos (x-x_0)]^{3/2} } \mathrm{d}x $ with integration of $E$ and $F$

Yesterday I tried to find an expression to predict the behaviour of Magnetic Field due a circular loop at any point in the space, in cylindrical co-ordinates. At the end, I am stuck with some ...
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Ambiguity on a vector representation in a cylindrical coordinate system

I am reading a book about relative motion. While defining the problem statement, the book assumes a figure like the one below: There is an inertial center at point $O$, a chief object at point $A$, ...
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Equation of an Oblique Cylinder in 3d Plane and checking if a point lies within it or not

I am doing a research project and trying to figure out a problem to find whether a point in 3d space ( x,y,z) lies within an imaginary cylinder or not. Going by the basic principles of coordinate ...
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Surface area of a cone using cylindrical coordinates

I know this question was asked already, but I would really like to resolve some problems I still have with it. An area element in cylindrical coordinate is $\quad rd \theta dz \quad $ and $ \quad r ...
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Finding the volume with triple integrals

I want to find the volume of a function described by: $$ G= \{(x,y,z)|\sqrt{x^2+y^2} \le z \le 1, (x-1)^2+y^2 \le 1\}$$ This question can be best solved in cylindrical coordinates. So if I follow that ...
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How can I do asymtotic expansion for the parabolic cylinder function of negative function?

Obtain an asymptotic expansion for the parabolic cylinder function of negative argument: $$ U(a,-z)=\frac{\mathrm{e}^{-\frac{1}{4} z^{2}}}{\Gamma\left(\frac{1}{2}+a\right)} \int_{0}^{\infty} t^{a-\...
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1 vote
1 answer
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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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Change of basis: Cartesian to tilted and shifted cylindrical in 3D

As a part of a little project I'm doing there is a part where I calculate some density at a point in 3D space. The trick part is that I have the map of densities based on cylindrical coordinates $(r,z)...
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triple integral on cone

Hello everyone I have to calculate $\int\int\int (x^2+y^2+z^2)^\alpha dxdydz$ on the cone $z=\sqrt{x^2 + y^2}$ which has a height of 1 and base circumference $x^2+y^2=1$. $\alpha >0$. I considered ...
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A higher dimensional generalization of cylindrical coordinate system?

Does a higher dimensional generalization of a cylindrical coordinate system exist such that there is $N$ scalar dimensions and $M$ polar dimensions, and consequently does there exist a higher-...
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$\int_Z z^2 x^2 = \int_{-1}^1 \int_0^{2\pi} \int_0^1 z^2 r^2 \sin(\theta)^2 \,\,dr \,d\theta \, dz$

$Z = \{(x,y,z)\in \mathbb{R}^3 | -1 < z < 1, \,\ x^2+y^2 \leq 1\}$ Calculate $\int_Z z^2x^2$. My approach: Polar coordinates: $x = r \sin(\theta),\,\,y = r \cos(\theta)$. Then: $\int_Z z^2x^2 = ...
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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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2 votes
1 answer
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Does order of integration matter, while integrating over a cone?

Suppose I want to find the center of mass of a cone, from its vertex. It is a right circular solid cone of radius $a$. About the $z$ axis, the center of mass is given by : $$z_{com}=\frac{\int dm\,z}{\...
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Area of intersection between a sphere and a cylinder.

I was trying to solve this question and I am pretty sure my result is correct, yet I was given wrong marks for it and the teacher refuses to elaborate. So I come to you: Calculate the volume of an ...
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Transformation of tensor depending on another tensor

I have a tensor $\mathbb{T} = T_{xx} \mathbf{e}_x \otimes \mathbf{e}_x + T_{xy} \mathbf{e}_x \otimes \mathbf{e}_y + T_{xz} \mathbf{e}_x \otimes \mathbf{e}_z + \dots$ in cartesian coordinates that I ...
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Normal unit vector in cylindrical coordinates

Suppose I have a surface in cylindrical coordinates given by $z=f(r,\theta)$. How can I proceed to find the normal unit vector of this surface? My initial guess was to evaluate it's gradient, which ...
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Volume and surface area of a solid in cylindrical coordinates

Suppose that initially I have a right triangle in the $xy$ plane given by the points $a=(0,0)$; $b=(0,B)$ and $c=(B,B)$, such as shown below: Since from now on, I will work with polar coordinates, I ...
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How to compute the cross product using cyclindrical coordinates? [duplicate]

can anyone help explain what formula could be used to compute the cross product in cylindrical coordinates? Is it the same for Cartesian coordinates?
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Which integral has the right limits or order?

If I have the integral $$ \iiint_S f dV $$ with $S$ as $$x^2+y^2+z^2\leq 18, z\leq \sqrt{x^2+y^2}, z\geq 0$$ and then I make a change to $$ x=r\cos\theta, y=r\sin\theta, z=z$$ is not difficult to see ...
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How do you convert this to cylindrical coordinates

So I'm practicing on how to convert cartesian to cylindrical and I'm not sure how to go about the $z$-coordinate. This is what I'm trying to convert: $U = \{(x,y,z):0\leq x^2+y^2\leq 2, 0\leq z\leq 6-...
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Compute the volume of the domain in $R^3$ given by the inequalities

: $x^2+y^2+z^2\leq100$. $x^2+y^2\leq99$ $x\geq0$ $y\geq0$ $z\geq0$ I tried to use cylindrical coordinates but could not identify my limits for $z$.
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4 votes
1 answer
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Volume inside an ellipsoid and offset cylinder

I want to find the volume of the region inside the ellipsoid $$\frac{x^2}{4}+\frac{y^2}{4}+z^2=1$$ and the cylinder $$x^2+(y-1)^2=1$$ I tried shifting the axes so that the cylinder was centered at ...
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How to derive a Jacobian to transform from a Rotated Cartesian C.S. to Cylindrical where the y axis is orthogonal?

Usually, cylindrical coordinates are defined with the z-axis normal to the curvilinear plane. See this image from Wikipedia. However, I want to derive a Jacobian matrix to convert from a "shifted&...
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Curl of a vector field on all parts of space

Find $ \nabla \times \vec{F} $ of a given vector field $\vec{F}=\frac{{\hat{\phi}}}{r}$ in cylindrical coordinates on all points of space . I calculated $ \nabla \times \vec{F} $ in all points except ...
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Gradient, Laplace operator

Could someone help me with the following question? The scalar field f and the vector field F in cylindricalcoordinates are given by $\displaystyle f=\frac{cos(\theta)}{r}$ respective $\displaystyle\...
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How do I parameterize the intersection of $x^2-y^2=11$ and $z=xy$ [closed]

The minus sign is totally throwing me off. If it were $x^2 + y^2$ I know they would be functions of the sin and cos of the parameter, but I can find no trig identity to make this work.
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Using cylindrical coordinates compute $\int_B z\, dx\, dy \,dz$

Using cylindrical coordinates compute $\int_B z \,dx \,dy \,dz$ where $B$ satisfies $1 \le z \le 2, x^2+y^2 \le 1$ My workings out: I get the following for the integral $$\int_1^2z\,dz\int_0^{2\pi}d\...
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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 ...
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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 ...
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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 ...
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Volume of the intersection of two planes and a paraboloid

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 ...
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Best coordinate system for integrating through lens-like geometry?

I am looking for the best coordinate system to help simplify this problem I am looking at. Suppose I have a beam of light rays entering normally into a focusing device (blue rectangle in figure). Upon ...
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