Cylindrical coordinates Definition and 234 Threads

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. D

    Triple Integral: Convert from Cartesian to Cylindrical Coordinates

    Homework Statement This is my last question about triple integrals in cylindrical coordinates. Evaluate the integral by changing to cylindrical coordinates: \int _{-3}^3\int _0^{\sqrt{9-x^2}}\int _0^{9-x^2-y^2}\sqrt{x^2+y^2}dzdydx Homework Equations In cylindrical coordinates...
  2. D

    Triple Integral in Cylindrical Coordinates

    Homework Statement Find the mass and center of mass of the solid S bounded by the paraboloid z=4x^2+4y^2 and the plane z=a\;\;(a>0) if S has constant density K. Homework Equations In cylindrical coordinates, x^2+y^2=r^2. The Attempt at a Solution In order to find the mass, I tried...
  3. D

    Triple Integral in Cylindrical Coordinates

    Homework Statement Evaluate \int \int \int_E x^2 \, dV where E is the solid that lies within the cylinder x^2+y^2=1, above the plane z=0, and below the cone z^2=4x^2+4y^2.Homework Equations In cylindrical coordinates, x^2+y^2=r^2 and x=r\cos{\theta}.The Attempt at a Solution I tried \int...
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    Writing equations in cylindrical coordinates (need work checked again please)

    Could someone tell me what I'm doing wrong? thanks! Homework Statement Write the equation is cylindrical coordinates 7x2 + 7y2 = 2y r = ? (has to be in the r = ? format) Homework Equations r2 = x2 +y2 x = rcos(θ) y = rsin(θ) The Attempt at a Solution 7x2 + 7y2 = 2y...
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    Transform a vector from Cartesian to Cylindrical coordinates

    Homework Statement Transform the vector below from Cartesian to Cylindrical coordinates: Q\,=\,\frac{\sqrt{x^2\,+\,y^2}}{\sqrt{x^2\,+\,y^2\,+\,z^2}}\,\hat{x}\,-\,\frac{y\,z}{x^2\,+\,y^2\,+\,z^2}\,\hat{z} Homework Equations Use these equations...
  6. D

    Triple integrals in spherical & cylindrical coordinates

    Homework Statement Set up triple integrals for the volume of the sphere rho = 2 in (a) spherical, (b) cylindrical, and (c) rectangular coordinates. Homework Equations Volume in cylindrical coordinates: Triple integral of dz r dr d(theta) over region D. Volume in spherical coordinates...
  7. Y

    Vector addition in cylindrical coordinates

    My question is about vector addition in cylindrical coordinates: Let A = 2x + y, B = x + 2y. In rectangular coordinates, AB = B-A = -x+y In cylindrical coordinates, x=rcosθ + θsinθ, y=rsinθ + θcosθ A =Axx + Ayy, B =Bxx + Byy Ar = Ax(x.r) + Bx(y.r)=2.236, Aθ = 0. So A = 2.236r Br =...
  8. Y

    How do you do vector addition in cylindrical coordinates?

    My question is about vector addition in cylindrical coordinates: Let A = 2x + y, B = x + 2y. In rectangular coordinates, AB = B-A = -x+y In cylindrical coordinates, x=rcosθ + θsinθ, y=rsinθ + θcosθ A =Axx + Ayy, B =Bxx + Byy Ar = Ax(x.r) + Bx(y.r)=2.236, Aθ = 0. So A = 2.236r Br...
  9. V

    Radial Acceleration in cylindrical coordinates

    Hey guys, Been given this formula for radial acceleration, I am not sure how they derived it. I have tried but the only forumla I know is a(radial) = v^2/r A_{r} = \ddot{r} - r\dot{\theta}^{2}EDIT - should be minus not plus
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    Vector Cross Product Formula in Spherical & Cylindrical Coordinates

    Hello all, it might be funny! but i am stuck to it! what is the vector cross product formula in spherical and cylindrical coordinates?! I know for Cartesian coordinate we have that nice looking determinant. but what about the other coordinates. I had looks to all the math books (like...
  11. P

    Laplace Cylindrical Coordinates (Separation of variables)

    Hello, The following equation: \frac{\partial^2 u}{\partial r^2}+\frac{1}{r} \cdot \frac{\partial u}{\partial r}+ \frac{\partial^2 u}{\partial z^2} = 0 is solved by separation of variables assuming a solution of the form: u=R(r)Z(z) In other cases the assumed solution is of the...
  12. G01

    Laplace's Equation in Cylindrical Coordinates

    Homework Statement A long copper pipe, with it's axis on the z axis, is cut in half and the two halves are insulated. One half is held at 0V, the other at 9V. Find the potential everywhere in space.Homework Equations \nabla^2V=0The Attempt at a Solution Alright. This is a laplace's equation...
  13. P

    MATLAB Surface Plot in Matlab (Cylindrical Coordinates)

    Hello Trying to plot in MATLAB the final solution equation u(r,z) of the steady state temperatures in the circular cylinder u(r,z) is defined in cylindrical coordinates and I'm confused trying to understand also how MATLAB plots a mesh. After some simplification The final solution looks...
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    Easy question about divergence in cylindrical coordinates

    Consider a cylindrical shell so that the cross sectional radius is some constant a. In the first term of the divergence expression in cylindrical coordinates: \frac{1}{r}\frac{\partial}{\partial r}(rA_{r}) When I multiply the radial component by r, do I go ahead and substitute r=a...
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    Green's Fucntions in Cylindrical coordinates

    Hey everyone, first time poster. I am looking for a general solution to poisson's equation in a cylindrical geometry for the electrostatic field. By general, I am thinking of two concentric, grounded, hollow cylinders of finite height, and the solution for the field using the green's function...
  16. S

    Vector potential in cylindrical coordinates

    Homework Statement What current density would produce the vector potential, A = k \hat{\phi} where k is a constant, in cylindrical coordinates? Homework Equations \nabla^2 A = -\mu_{0} J In cylindrical coordinates for radial and z symmetry \nabla^2 t = \frac{1}{s^2}...
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    Derivation of Energy Equation in Cylindrical Coordinates

    So, I have searched the globe to find the derivation of the conservation of energy equation in 3D cylindrical coordinates. I have looked in about every heat transfer, thermodynamic, thermofluid books in the library. I have spent about 4 hours on the internet trying to find something, but no...
  18. S

    Converting Cartesian Coordinates to Cylindrical Coordinates

    Homework Statement Say I have line, y = - 5 in cartesian coordinates. How do I express this in cylindrical coordinates? Also, if I have a point (0,-5) in cartesian coordinates, how to I express this position vector in cylindrical coordinates? Homework Equations y = r sin (phi)...
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    Derivation of Continuity Equation in Cylindrical Coordinates

    Help! I am stuck on the following derivation: Use the conservation of mass to derive the corresponding continuity equation in cylindrical coordinates. Please take a look at my work in the following attachments. Thanks! =)
  20. B

    Find Volume Using Cylindrical Coordinates

    Can someone explain to me how to find the volume of the following? I am asked to use cylindrical coordinates to find the volume of the solid enclosed by the sphere r^2+a^2 = a^2 and by the cone z = r cot φ where φ is some fixed angle between 0 and pi/2? I would have thought that the volume...
  21. J

    Vector question: dot product in cylindrical coordinates

    Hello. To find the work of a force, I have to perform a dot product between the force and a infinitesimal displacement. If they are in cylindrical coordinates, I can't manage to make the dot product. Please, could you help me? Thank you.
  22. R

    How to Graph r=2cos(theta) in Cylindrical Coordinates?

    I'm having some trouble understanding exactly how to graph this problem using cylindrical coordinates. The coordinates they give me is r=2cos(theta) How do I go about beginning to determine how to graph this with only the radius?
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    Spherical, Cylindrical Coordinates - quick easy questions

    EDIT: I figured it out. I'm sick, tired, and just confusing myself. I have a test in a few hours, and would like to clear some easy stuff up so I'm not thinking about it during the test. 1) Vector subtraction works in spherical and cylindrical coordinates right? That is to say, \vec {AB} ...
  24. S

    Conversion from Cartesian to Cylindrical Coordinates

    Hello. I am interested in learning the mathematical derivation from Cartesian coordinates Navier-Stokes equation to cylindrical coordinates Navier-Stokes equation. These equations have similar forms to the basic heat and mass transfer differential governing equations. I’ve tried looking...
  25. F

    Cylindrical Coordinates -Div/Curl/Grad

    We just started a section dealing with div/curl/grad in different orthogonal systems... before I get started doing problems involving these operations I wanted to make sure I am dealing with these operation correctly. Our first homework problem is as follows: In cylindrical coordinates compute...
  26. B

    Cone Edges in Cylindrical Coordinates

    Hi , I don't know how to get the edge of the cone in cylindrical coordinates. For example, we have a cone starting at the origin, of heigth 2 and the top is a circle of radius 1 (center at the origin). the edge of the cone is z=2r. but I don't know how they find it. Please can someone...
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    Parabolic cylindrical coordinates

    Can someone, please, show me an example of when you are better of with parabolic cylindrical coordinates than with cartesian coordinates when computing a triple integral over a solid?
  28. quasar987

    Proving Laplacian in Cylindrical Coordinates

    I'm supposed to prove the laplacian in cylindrical coord. is what it is. I tried tackling the problem in two ways and none work! I have no idea what's the matter. The first way is to calculate d²f/dr² , d²f/dO² and d²f/dz² and isolate d²f/dx² , d²f/dy² and d²f/dz². In cylindrical coord...
  29. S

    Gradient in cylindrical coordinates

    How can you derive the gradient of a vector R^3 to R in cylindrical coordinates?
  30. H

    Use the cylindrical coordinates to find the volume

    I'm having trouble figuring out this volume. Use the cylindrical coordinates to find the volume of the solid S bounded by z=x^2 + y^2 and z=12 - 2x^2 - 2y^2 I've included a pic of what I think the regions and the solid look like...
  31. D

    Fluid dynamics - dissipation using cylindrical coordinates

    I have this flow field in cylindrical coordinates of which I would like to calculate the dissipation as a function of these coordinates. Now in my fluid dynamics notes I found the following expression(s) for the dissipation: 2 \mu (e_{ij} -\frac{1}{3} \Delta \delta _{ij} )^2 = 2 \mu (...
  32. V

    Setting up a triple integral in cylindrical coordinates?

    The problem says to find the volume of material cut from the solid sphere, r^2 + z^2 \le 9 by the cylinder, r = 3\sin\theta I don't know how to graph the first equation, but I can do the second in polar coordinates. How do I go about converting to use cylindrical coordinates?
  33. D

    Having some difficulty with a Trip. Integral in Cylindrical Coordinates

    Hi everybody The integral in question is the triple integral of x dV over the region E, where E is enclosed by the planes z=0, and z=x+y+3, and the cylinders x^2 + y^2 = 4 and x^2 + y^2 = 9. Well--- so far in cylindrical coordinates I know the r limits will be from 2 to 3 since the...
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