What is Cylindrical coordinates: Definition and 234 Discussions

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.
The origin of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis.
The axis is variously called the cylindrical or longitudinal axis, to differentiate it from the polar axis, which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction.
Other directions perpendicular to the longitudinal axis are called radial lines.
The distance from the axis may be called the radial distance or radius, while the angular coordinate is sometimes referred to as the angular position or as the azimuth. The radius and the azimuth are together called the polar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position, or axial position.Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, accretion disks in astronomy, and so on.
They are sometimes called "cylindrical polar coordinates" and "polar cylindrical coordinates", and are sometimes used to specify the position of stars in a galaxy ("galactocentric cylindrical polar coordinates").

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  1. S

    A Heat conduction equation in cylindrical coordinates

    I've been studying a few books on PDE's, specifically the heat equation. I have one book that covers this topic in cylindrical coordinates. All the examples are applied to a solid cylinder and result in a general Fourier Bessel series for 3 common cases that can be found easily with an online...
  2. H

    I Generating Divergence Equation In Cylindrical Coordinates

    This is from an old E&M exam question where we were asked to derive the formula for the divergence of a vector field in cylindrical coordinates using Taylor's Approximation and the fundamental definition of the divergence: ∇⋅A = Lim V→0 { ( ∫S A⋅da ) / V } The vector field, A, is defined in...
  3. C

    Line element in cylindrical coordinates

    First I took the total derivative of these and arrived at $$ dr=\frac{\partial r}{\partial x}dx+\frac{\partial r}{\partial y}dy \quad\rightarrow \quad r²dr=xdx+ydy $$ $$ d\phi=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy \quad\rightarrow \quad r²dr \phi=-ydx+xdy $$...
  4. chwala

    I Cylindrical coordinates -Curvilinear

    Why are the coordinates seemingly used when the symmetry is around ##z## axis? Any particular reason why not ##x## or ##y##. In transforming from Cartesian to cylindrical form; I can see that ##z## is not considered when determining ##r##. Can we also use ##x## and ##z## assuming that the...
  5. T

    A General solution to Laplacian in cylindrical coordinates

    I am trying to model the voltage function for a very long cylinder with an assigned surface charge density or voltage. Then the solution inside the cylinder is: $$\sum_{n=0}^{\infty}A_n r^n cos(nθ)$$ And$$\sum_{n=0}^{\infty}A_n r^-n cos(nθ)$$ outside. Is that correct
  6. Onyx

    B Creating Metric Describing Large Disk

    How can I create a metric describing the space outside a large disk, like an elliptical galaxy? In cylindrical coordinates, ##\phi## would be the angle restricted the the plane, as ##\rho## would be the radius restricted to the plane. I think that if ##z## is suppressed to create an embedding...
  7. chiyu

    I Vector calculus: line element dr in cylindrical coordinates

    We were taught that in cylindrical coodrinates, the position vector can be expressed as And then we can write the line element by differentiating to get . We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
  8. E

    I PDE - Heat Equation - Cylindrical Coordinates.

    Would method of separation of variables lead to a solution to the following PDE? $$ \frac{1}{r} \frac{ \partial}{\partial r} \left( kr \frac{ \partial T}{ \partial r}\right) = \rho c_p \frac{\partial T }{ \partial t }$$ This would be for the transient conduction of a hollow cylinder, of wall...
  9. S

    Angle between normal force and radial line for cylindrical coordinates

    so I was wondering. there is this normal force on the can from the path. And there's this formula to find the angle between the radial line and the tangent or also between the normal force and either the radial or theta axis. the formula is ##\psi = r/dr/d\theta##. The thing is that here they...
  10. Vladimir_Kitanov

    Motion in Cylindrical Coordinates

    7:03 what is second component of a(theta)? this -> 2 * r' * (theta)' I understand everything except that.
  11. M

    Mathematica Plot a vector valued function in cylindrical coordinates

    Hi PF! I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?
  12. L

    A Curl in cylindrical coordinates -- seeking a deeper understanding

    I calculate that \mbox{curl}(\vec{e}_{\varphi})=\frac{1}{\rho}\vec{e}_z, where ##\vec{e}_{\rho}##, ##\vec{e}_{\varphi}##, ##\vec{e}_z## are unit vectors of cylindrical coordinate system. Is there any method to spot immediately that ##\mbox{curl}(\vec{e}_{\varphi}) \neq 0 ## without employing...
  13. nuclearsneke

    I Convert cylindrical coordinates to Cartesian

    Good day! I am currently struggling with a very trivial question. During my studies, I operated with a parameter called "geometrical buckling" for neutrons and determined it in cylindrical coordinates. But thing is that we usually do not consider buckling's dependence on angle so its angular...
  14. J

    3D Laplace solution in Cylindrical Coordinates For a Hollow Cylindrical Tube

    Here is the initial problem and my attempt at getting Laplace solution. I get lost near the end and after some research, ended up with the Bessel equation and function. I don't completely understand what this is or even if this i the direction I go in. This is a supplemental thing that I want to...
  15. D

    I Question about the vector cross product in spherical or cylindrical coordinates

    Hi If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b. Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...
  16. A

    Problem with a triple integral in cylindrical coordinates

    Good day here is the solution J just don't understand why the solution r=√2 has been omitted?? many thanks in advance best regards!
  17. SebastianRM

    I How to obtain the determinant of the Curl in cylindrical coordinates?

    I have a vector in cylindrical Coordinates: $$\vec{V} = \left < 0 ,V_{\theta},0 \right> $$ where ##V_\theta = V(r,t)##. The Del operator in ##\{r,\theta,z\}$ is: $\vec{\nabla} = \left< \frac{\partial}{\partial r}, \frac{1}{r}\frac{\partial}{\partial \theta}, \frac{\partial}{\partial z}...
  18. cwill53

    How to set bounds in cylindrical coordinates analytically?

    I'm trying to evaluate the following integral in cylindrical coordinates. $$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_x^{\sqrt{1-x^2}}e^{-x^2-y^2} \, dy \, dx \, dz$$ After attempting to set the bounds in cylindrical coordinates, I got $$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_{\rho \cos\varphi...
  19. M

    Determining an object's velocity in cylindrical coordinates

    I got the answer for velocity and acceleration. But I don't know how to draw the shape of the particle's motion over time. How to draw it? should we change a,b,c,e into a numbers or not? or we may not to change a,b,c,e? Please help me how to draw the shape of particle's motion over time?
  20. K

    Calculating crossproduct integral, Parametrization

    i) I approximate the solenoid as a cylinder with height L and radius R. I am not sure how I am supposed to place the solenoid in the coordinate system but I think it must be like this, right? The surface occupied by the cylinder can be described by all vectors ##\vec x =(x,y,z)## so that...
  21. G

    Line integral where a vector field is given in cylindrical coordinates

    What I've done so far: From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1). We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z. We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...
  22. M

    Convert cylindrical coordinate displacement to Cartesian

    Summary: I can't figure out how the solver carries out the conversions from cartesian to cylindrical coordinates and vice-versa. 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...
  23. Luke Tan

    I Transforming Vector Fields between Cylindrical Coordinates

    In dealing with rotating objects, I have found the need to be able to transform a vector field from cylindrical coordinate systems with one set of coordinate axes to another set. For eg i'd like to transform a vector field from being measured in a set of cylindrical coordinates with origin at...
  24. O

    I Radius vector in cylindrical coordinates

    I am starting to learn classical physics for my own. One exercise was, to calculate the vector r (see picture: 1.47 b). The vector r is r=z*z+p*p. I don’t understand this solution. My problem is: in a vector space with n dimensions there are n basis vectors. In the case of cylindrical...
  25. Z

    Kinematics in Cylindrical Coordinates

    Homework Statement A small bead of mass m slides on a frictionless cylinder of radius R which lies with its cylindrical axis horizontal. At t = 0 , when the bead is at (R,0), vz = 0 and the bead has an initial angular momentum Lo < mR sqrt(Rg) about the axis of the cylinder where g is the...
  26. Mason Smith

    Cylindrical coordinates: unit vectors and time derivatives

    Homework Statement Homework EquationsThe Attempt at a Solution I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates. I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am...
  27. Q

    Cylindrical Coordinates: Line Integral Of Electrostatic Field

    Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...
  28. M

    I Converting from spherical to cylindrical coordinates

    I have the coordinates of a hurricane at a particular point defined on the surface of a sphere i.e. longitude and latitude. Now I want to transform these coordinates into a axisymmetric representation cylindrical coordinate i.e. radial and azimuth angle. Is there a way to do the mathematical...
  29. S

    Cartesian to Cylindrical coordinates?

    Homework Statement I want to convert R = xi + yj + zk into cylindrical coordinates and get the acceleration in cylindrical coordinates. Homework Equations z The Attempt at a Solution I input the equations listed into R giving me: R = i + j + z k Apply chain rule twice: The...
  30. W

    I Curl in cylindrical coordinates

    So, let me derive the curl in the cylindrical coordinate system so I can showcase what I get. Let ##x=p\cos\phi##, ##y=p\sin\phi## and ##z=z##. This gives us a line element of ##ds^2 = {dp}^2+p^2{d\phi}^2+{dz}^2## Given that this is an orthogonal coordinate system, our gradient is then ##\nabla...
  31. M

    Convert a spherical vector into cylindrical coordinates

    Homework Statement Convert the vector given in spherical coordinates to cylindrical coordinates: \vec{F}(r,\theta,\varphi) = \frac{F_{0}}{arsin\theta}\bigg{[}(a^2 + arsin\theta cos\varphi)(sin\theta \hat{r} + cos\theta \hat{\theta}) - (a^2 + arsin\theta sin\varphi - r^2 sin^2\theta)...
  32. BrandonUSC

    Radial Acceleration in Polar/Cylindrical Coordinates

    My question is why isn't the radial component e→r of acceleration in cylindrical coords simply r'' ? If r'' is the rate at which the rate of change of position is changing in the radial direction, wouldn't that make it the radial acceleration? I.e, the acceleration of the radius is the...
  33. sams

    I A Question about Unit Vectors of Cylindrical Coordinates

    I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}##...
  34. ys98

    I Integrating unit vector ρ

    When doing integration such as \int_{0}^{2\pi} \hat{\rho} d\phi which would give us 2\pi \hat{\rho} , must we decompose \hat{ρ} into sin(\phi) \hat{i} + cos(\phi) \hat{j} , then \int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi , which would give us 0 instead? Thanks
  35. Remixex

    Stress and Strain tensors in cylindrical coordinates

    Homework Statement I am following a textbook "Seismic Wave Propagation in Stratified Media" by Kennet, I was greeted by the fact that he decided to use cylindrical coordinates to compute the Stress and Strain tensor, so given these two relations, that I believed to be constitutive given an...
  36. F

    Cylindrical Coordinates Triple Integral -- stuck in one place

    Homework Statement Use cylindrical coordinates to evaluate triple integral E (sqrt(x^2+y^2)dv where E is the solid that lies within the cylinder x^2+y^2 = 9, above the plane z=0, and below the plane z=5-y Homework EquationsThe Attempt at a Solution So i just need to know how to get the bounds...
  37. E

    Showing E.dl is 0 - Why cylindrical coordinates?

    Homework Statement A point charge +Q exists at the origin. Find \oint \vec{E} \cdot \vec{dl} around a circle of radius a centered around the origin. Homework EquationsThe Attempt at a Solution The solution provided is: \vec{E} = \hat{\rho}\frac{Q}{4\pi E_0a^2} \vec{dl}=\hat{\phi}\rho d\phi...
  38. M

    Change integration limits for cylindrical to cartesian coord

    Homework Statement I want to change the integration limits of an integral in cylindrical to cartesian coordinates. For example the integral of function f(r) evaluated between b and R: ∫ f(r)dr for r=b and r=R (there is no angular dependence). For write de function in cartesian coordinates...
  39. B

    Schrodinger equation in cylindrical coordinates.

    Hi guys! For nuclear case, I need to write an Schrodinger equation in cylindrical coordinates with an total potential formed by Woods-Saxon potential, spin-orbit potential and the Coulomb potential. Schrodinger equation can be written in this form: $$[-\frac{\hbar^2}{2m}(\frac{\partial...
  40. I

    How to express velocity gradient in cylindrical coordinates?

    Homework Statement The vlasov equation is (from !Introduction to Plasma Physics and Controlled Fusion! by Francis Chen): $$\frac{d}{dt}f + \vec{v} \cdot \nabla f + \vec{a} \cdot \nabla_v f = 0$$ Where $$\nabla_v$$ is the del operator in velocity space. I've read that $$\nabla_v =...
  41. Alexanddros81

    The pendulum is released from rest with θ = 30deg

    Homework Statement 13.54 The pentulum is released from rest with θ = 30deg. (a) Derive the equation of motion using θ as the independent variable. (b) Determine the speed of the bob as a function of θ. The solutions given in the textbok are a) ##\ddot θ = -4.905sinθ rad/s^2## b) ##6.26\sqrt...
  42. Alexanddros81

    Calculate the angular velocity and angular acceleration (pendulum)

    Homework Statement 13.53 The tension in the sting of the simple pendulum is 7.5N when θ=30deg. Calculate the angular velocity and angular acceleration of the string at this instant. Homework EquationsThe Attempt at a Solution Is this correct?
  43. Draconifors

    Triple integral using cylindrical coordinates

    Homework Statement The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems. For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
  44. T

    Volume of a sphere in cylindrical coordinates

    Homework Statement A sphere of radius 6 has a cylindrical hole of radius 3 drilled into it. What is the volume of the remaining solid. The Attempt at a Solution [/B] I am able to solve this using cylindrical coordinates but I'm having trouble when I try to solve it in spherical coordinates...
  45. C

    A Laplace Eq. in Cylindrical coordinates (no origin)

    Hi, I need to solve Laplace equation:##\nabla ^2 \Phi(x,r)=0## in cylindrical domain ##r_1<r<r_2##, ##0<z<+\infty##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2} \\ &\frac{\partial...
  46. C

    A Laplace Eq. in Cylindrical coordinates

    Hi, I need to solve Laplace equation:## \nabla ^2 \Phi(x,r)=0 ## in cylindrical domain ##0<r<r_0##, ##0<x<L## and ##0<\phi<2\pi##. The boundary conditions are the following ones: ## \left\{ \begin{aligned} &C_{di}\Phi(x,r_0)=\epsilon \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_0} \\...
  47. M

    Shell balances in cylindrical coordinates

    I have a question regarding writing a shell balance for a cylindrical system with transport in one direction (in any area of transport phenomena). When we set up the conservation equation(say steady state), we multiply the flux and the area at the surfaces of our control volume and plug them...
  48. B

    Recast a given vector field F in cylindrical coordinates

    Homework Statement F(x,y,z) = xzi Homework Equations N/A The Attempt at a Solution I just said that x = rcos(θ) so F(r,θ,z) = rcos(θ)z. Is this correct? Beaucse I am also asked to find curl of F in Cartesian coordinates and compare to curl of F in cylindrical coordinates. For Curl of F in...
  49. L

    A Laplacian cylindrical coordinates

    Laplacian in cylindrical coordinates is defined by \Delta=\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial}{\partial \rho}+\frac{1}{\rho^2}\frac{\partial^2}{\partial \varphi^2}+\frac{\partial^2}{\partial z^2} I am confused. I I have spherical symmetric function f(r) then \Delta...