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. fluidistic

    Differential x, cylindrical coordinates

    1. Homework Statement +attempt at solution+equations In Cartesian coordinates, x translate into x=r \cos \theta into cylindrical coordinates, y=r \sin \theta and z=z . However dx=\cos \theta dr - r \sin \theta d\theta. This is what I don't understand. Since x is a function of both...
  2. B

    Triple Integral Limits Help. Cylindrical Coordinates

    Homework Statement Find the volume of the solid bounded by the paraboloids z=x^2+y^2 and z=36-x^2-y^2. Answer is: 324\pi \\ Homework Equations r^2=x^2+y^2 x=rcos0 y=rcos0 The Attempt at a Solution 36-x^2+y^2=x^2+y^2\\ 36=2x^2+2y^2 18=x^2+y^2 r^2=18 V=\int_{0}^{2\pi} \int_0^{3\sqrt{2}}...
  3. S

    How to Calculate F_t and N for a Roller on a Cam Mechanism?

    Homework Statement A cam has a shape that is described by the function r = r_0(2 - cos \theta), where r_0 = 2.25 ft. A slotted bar is attached to the origin and rotates in the horizontal plane with a constant angular velocity (\dot{\theta} dot) of 0.85 radians/s. The bar moves a roller...
  4. B

    How do you express the center of a circle in cylindrical coordinates?

    This is something I have zero familiarity with. Anyways, I was given the equation: r=2asin(theta)+2bcos(theta) and had to prove that it was a circle, and then state its center in cartesian and cylindrical coordinates. After making the appropriate substitutions and completing the square...
  5. S

    Fluid Mechanics equations in Cartesian and Cylindrical coordinates?

    Homework Statement Not really a homework question, but more of a concept question which I'm unfamiliar with. So as we know, equations can be in any coordinate, but how do you convert them from one to another? For example, a few equations from fluid mechanics. the first equation is the vector...
  6. S

    Surface integral in cylindrical coordinates

    Hello everybody! Although this may sound like a homework problem, I can assure you that it isn't. To prove it, I will give you the answer: 40pi. So.. I'm self-studying some electrodynamics. I'm using the third edition of Griffiths, and I have a quick question. For those who own the book and...
  7. C

    Use cylindrical coordinates to find volume

    Homework Statement Use cylindrical coordinates to find volume... Homework EquationsInside: x2+y2+z2=16 Outside: z=sqrt(x2+y2) The Attempt at a Solution Cylindrical coordinates have always been a problem for me, so I initially tried to put them into spherical and then convert them over, but...
  8. S

    Spherical and cylindrical coordinates, not a problem

    Homework Statement do we only use spherical and cylindrical coordinates for triple integrals? or for double too? thanks for your replies in advance
  9. M

    Radial component of a velocity vector - cylindrical coordinates

    Hi there, I'm trying to determine the radial component of a velocity vector in a disk. The vector doesn't (necessarily) start from the centre of the disk and can be pointed in any direction. I've attached a .pdf with the schematics - it seems like a simple problem but it has me stumped...
  10. A

    First Order Differential Equation in Cylindrical Coordinates

    Consider cylindrical coordinates p = (x^2 + y^2)^.5  angle = arctan(y=x). Consider your curve to be specifi ed by z(p). Write down a ( first order) diff erential equation governing z(p) please help!
  11. G

    Potential of Dipole in Cylindrical coordinates

    Homework Statement I have been given the problem of finding the potential of a dipole in cylindrical coordinates. The only way that comes to my mind is to extract the dipole term from the multipole expansion of the potential of an arbitrary charge distribution in cylindrical coordinates. But I...
  12. B

    Triple integrating cylindrical coordinates?

    Homework Statement Integrate the function f(x,y,z)=−4x+3y over the solid given by the figure below, if P = (5,1,0) and Q = (-5,1,2). [PLAIN]http://img259.imageshack.us/img259/958/sfig1681g1.gif Homework Equations x=rcos(\theta) y=rsin(\theta) r=sqrt(x^2+y^2)The Attempt at a Solution i...
  13. P

    Divergence in cylindrical coordinates

    Homework Statement Calculate the divergence of the vector function f = a/s^2 (s hat) where s is the radial distance from the z axis, expressed in cylindrical coordinates. Homework Equations The Attempt at a Solution Using the divergence theorem I relate the volume integral of...
  14. D

    Converting vector in cartesian to cylindrical coordinates

    Homework Statement This seems like a trivial question (because it is), and I'm just not sure if I'm doing it right. I have vector in cartesian coordinate system: \vec{a}=2y\vec{i}-z\vec{j}+3x\vec{k} And I need to represent it in cylindrical and spherical coord. system Homework...
  15. Q

    Maximizing the height of a bullet in cylindrical coordinates

    Homework Statement A gun can fire shells in any direction with the same speed v0. Ignoring air resistance and using cylindrical polar coordinates with the gun at the origin and z measure vertically up, show that the gun can hit any object inside the surface z = \frac{v_{0}^{2}}{2g} -...
  16. M

    Simple Projectile in Cylindrical Coordinates

    Homework Statement A gun can fire shells in any direction with the same speed v_{0}. Ignoring air resistance and using cylindrical polar coordinates with the gun at the origin and z measured vertically up, show that the gun can hit any object inside the surface z = \frac{v^{2}_{0}}{2g} -...
  17. Telemachus

    Cylindrical coordinates to cartesian coordinates

    Homework Statement Hi there. Hi have in cylindrical coordinates that \theta=\displaystyle\frac{\pi}{3}, and I must make the graph, and take it into cartesian coordinates. How should I do? I've tried this way: \begin{Bmatrix}x=r\cos\displaystyle\frac{\pi}{3}\\y=r\sin\displaystyle\frac{\pi}{3}...
  18. P

    Hamiltonian in cylindrical coordinates

    Hi, I'm trying to find the Hamiltonian for a system using cylindrical coordinates. I start of with the Lagrangian L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2)-U(r,\theta,z) From that, using H=\sum p\dot{q}-L...
  19. C

    Hom. heat equation in cylindrical coordinates using Fourier & Laplace transforms

    I'm trying to solve the homogeneous heat equation of a semi-infinite cylinder in cylindrical coordinates for a semi-infinite cable (no theta dependence): \frac{\partial U}{\partial t}=D\left(\frac{\partial^{2} U}{\partial r^{2}}+\frac{1}{r}\frac{\partial U}{\partial r}+\frac{\partial^{2}...
  20. M

    Evaluate the integral by changing to cylindrical coordinates.

    I wish I knew how to type this out with the proper symbols but here it goes. It says to change the following to cylindrical coordinates and evaluate (x^2 + y^2)^(1/2) dz dy dx where -3<=x<=3, 0<=y<=(9-9x^2)^1/2, 0<=z<=9-x^2-y^2Homework Equations The Attempt at a Solution I got 162pi/5 Would...
  21. F

    Calculating a volume through cylindrical coordinates

    Just a question. Say you have a function, which in cylindrical coordinates it gives that \int\int\int \sqrt{x^2 + y^2} dx dy dz which is \int\int\int r^2 dr d/theta dz i want to find in cylindrical coordinates, in the area limited by the functions : x^2 + y^2 = z^2 z is greater or equal than...
  22. T

    How to find the unit vector in cylindrical coordinates

    So I'm trying to find out what the procedure is to convert a cartesian unit vector to a cylindrical unit vector. Any thoughts?
  23. K

    Calc 3- Triple Integral using cylindrical coordinates

    Use cylindrical coordinates to evaluate the triple integral , sqrt(x^2+y^2) where the region integrated is the solid bounded by the circular paraboloid z=9-16(x^2+y^2) and the xy-plane. I'm having trouble deciding what the bounds for r would be.
  24. fluidistic

    Vector field, cylindrical coordinates

    Homework Statement Describe the following vector field: \bold v (\bold x)=\frac{\bold a \times \bold x}{(\bold a \times \bold x)(\bold a \times \bold x)} with \bold a = \text{constant}. Calculate its divergence and curl. In what region is there a potential for \bold v? Calculate it. Hint...
  25. H

    Triple Integral Using Cylindrical Coordinates

    Homework Statement A conical container with radius 1, height 2 and with its base centred on the ground at the origin contains food. The density of the food at any given point is given by D(r) = a/(z + 1) where a is a constant and z is the height above the base. Using cylindrical polar...
  26. L

    Stress field in cylindrical coordinates

    Can anyone please explain the stress fields in cylindrical coordinates? What is the difference between \sigma_{rz} and \sigma_{\theta z}? What is the difference between stress in the r axis and stress in the \theta axis? Thanks
  27. D

    Equilibrium heat equation in 2D cylindrical coordinates

    Homework Statement Plate in the shape of the circular halo (inner radius a, outer radius b>a), the inner edge is being kept at a constant temperature T_0, and the outer at the temperature given by the function f(\phi)=T_0\cos(2\phi). Find the equilibrium distribution of the heat everywhere...
  28. J

    Converting a Vector Field from Cartesian to Cylindrical Coordinates

    Homework Statement I have a rather complicated vector field given in cartesian coordinates that I need to evaluate the line integral of over a unit square. I know to use Stoke's Theorem to do this, and I suspect that the integral would be greatly simplified if it were in cylindrical...
  29. ╔(σ_σ)╝

    Integration in cylindrical coordinates

    Homework Statement In cyclindrical coordinates we can represent points as (\rho,\phi,z) We define a vector in cyclindrical coordinates as follows A = A\rhoa\rho + A\phia\phi + Azaz I'm having some problem with subscripts. Anyway I don't understand this. If I am given a point say ( 5, 20...
  30. C

    Convert to cylindrical coordinates

    Evaluate by changing to cylindrical coordinates \int from 0 to 1 \int from 0 to (1-y^2)^1/2 \int from (x^2+y^2) to (x^2+y^2)^1/2 (xyz) dzdxdy I came to an answer of integral from 0 to pi integral from 0 to 1 integral from r^2 to r (rcos\thetarsin\thetaz) r dzdrd\theta Is this the correct answer?
  31. J

    Volume of a cone using cylindrical coordinates and integration

    Hi all! I was trying to figure out how to find the volume of a cone with radius R and height h using integration with cylindrical coordinates. I first tried to set the the integral as: \int_{0}^{2\pi}\int_{0}^{h}\int_{0}^{R}\rho d\rho dz d\phi ...but I think that this is setting up the...
  32. A

    Laplace equation in cylindrical coordinates

    Can anyone help with the solution of the Laplace equation in cylindrical coordinates \frac{\partial^{2} p}{\partial r^{2}} + \frac{1}{r} \frac{\partial p}{\partial r} + \frac{\partial^{2} p}{\partial z^{2}} = 0 with Neumann no-flux boundaries: \frac{\partial p}{\partial r}...
  33. M

    Triple integral with cylindrical coordinates

    Homework Statement Use cylindrical coordinates to evaluate the triple integral \int\int\int \sqrt{x^2+y^2} dV in region E where E is the solid bounded by the circular paraboloid z=9-(x^2+y^2) and the xy-plane. Homework Equations knowing that x = rcos\theta y= rsin\theta z=z...
  34. P

    Transform vector to Cylindrical Coordinates

    i need help transforming this equation into cylindrical coordinates... w = omega i = i hat j = j hat k = k hat r is a vector r(t) = Asin(wt)i + Bsin(wt)j + (Ct - D)k where w, A, B, C and D are constants. i, j, and k are throwing me off...i know they are components of x, y and z...and i know...
  35. S

    Stokes Theorem in cylindrical coordinates

    Homework Statement A vector field A is in cylindrical coordinates is given. A circle S of radius ρ is defined. The line integral \intA∙dl and the surface integral \int∇×A.dS are different. Homework Equations Field: A = ρcos(φ/2)uρ+ρ2 sin(φ/4) uφ+(1+z)uz (1) The Attempt at...
  36. M

    Line Integral in Cylindrical Coordinates

    Homework Statement Find the value of the (surface) integral \int curl \textbf{A} \bullet \textbf{a} if the vector \textbf{A}=y \textbf{i}+z \textbf{j}+x \textbf{k} and S is the surface defined by the paraboloid z=1-x^2-y^2 Homework Equations x=s\cos\phi y=s\sin\phi...
  37. M

    Helical Pathway Movement Using Vectors, Spherical & Cylindrical Coordinates

    would anybody like to discuss how to accurately follow a particle moving in a HELICAL PATHWAY using vectors, spherical and cylindrical coordinates? I'm not sure how to follow a geometric helical pathway using linear and parametric equations.
  38. N

    Elliptic Cylindrical Coordinates

    Is there a cylindrical coordinate system that is centered about the foci of an ellipse. It would include (r,theta,z) just like cylindrical coordinates only for an ellipse. If this coordinate system exists, what is the laplacian? Chris
  39. J

    Graphing the Surface y^2 + z^2 = 1 in Cylindrical Coordinates

    Homework Statement Identify and sketch the graph of the surface y^2 + z^2 = 1. Show atleast one contour perpendicular to each coordinate axis Homework Equations The Attempt at a Solution for the yz plane z = (1-y^2)^1/2 a circle of radius 2 centered at the origin xy, set z=0...
  40. J

    What is the equation of the resulting surface in cylindrical coordinates?

    Homework Statement z = 4y^2, x = 0, is rotated about the z axis. write the equation of the resulting surface in cylindrical coordinates Homework Equations The Attempt at a Solution not really sure what the x = 0 means so i ignored it i solved for y because that would be my...
  41. N

    When to use spherical and cylindrical coordinates?

    For example with a paraboloid, which do i use? I am also slightly confused with the limits in the integral. If doing a triple integral with drdθdΦ i understand the limits of the dr integral but when it comes to dθ and dΦ i don't understand why sometimes its 0 to 2π or 0 to π etc. For example...
  42. Zarlucicil

    Triple Integration of a Sphere in Cylindrical Coordinates

    Homework Statement The problem was to find the volume enclosed by a sphere of radius "a" centered on the origin by crafting a triple integral and solving for it using cylindrical coordinates. Homework Equations x^{2}+y^{2}+z^{2}=a^{2} : Equation for a sphere of radius "a" centered on...
  43. J

    Transforming divergence from cartesian to cylindrical coordinates

    Homework Statement Compute the divergence in cylindrical coordinates by transforming the expression for divergence in cartestian coordinates. Homework Equations F = F_x i + F_y j + F_z k div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z ... (divergence in Cartesian coordinates) I need to...
  44. S

    Circle to cylindrical coordinates

    Homework Statement Transform to cylindrical coordinates: x^{2}+y^{2}=R^{2} Doesn't look like a problem at all first... :smile: Homework Equations .. after all I know that is a circle (2d) and we can forget the z-axis (=0) and transform it to just polar coords. Also I know, that for...
  45. E

    Converting Cylindrical Coordinates to Rectangular: A Helpful Hint

    Convert r=2cos(theta) from cylindrical coordinates to rectangular coordinates I have tried squaring both sides so that it will be equal to x squared plus y squared, and then solving for a variable. No matter what I do though I am left with two variables so I feel like I am taking the...
  46. I

    Electric potential in cylindrical coordinates using separation of variables

    Hi, I am new to the forum. I've encountered a couple problems with separation of variables in cylindrical coordinates. Problem #1: a clindrical surface of radius R is oriented along the z-axis, and is split into two conducting half-cylinders. The potential satisfies the...
  47. S

    Differential Length (Cylindrical Coordinates)

    So we just were given some formulas and I am confused about this simple question Find the differential length or distance between the two points. P(2,pi/2,-1) and Q(5,3pi/2,5)I know this for cylindrical dL = dp (ap) + p dphi (aphi) + dz (az) So i would integrate I have a few questions...
  48. J

    Triple integral using cylindrical coordinates

    Homework Statement \int\int_{Q}\int(x^4+2x^2y^2+y^4)dV where Q is the cylindrical solid given by \{(x,y,x)| x^2+y^2 \leq a^2, 0\leqz\leq\frac{1}{\pi}\}Homework Equations When I convert to cylindrical I get f(r,\theta,z) = r^4\cos^2\theta + 2r^4\cos^2\theta\sin^2\theta + r^2\sin^2\theta, but I...
  49. S

    Finding Volume of Solid Cut by Cylindrical Coordinates: Is My Solution Correct?

    Homework Statement Use Cylindrical Coordinates. Find the volume of the solid that the cylinder r=acos\theta cuts out of the sphere of radius a centered at the origin. Homework Equations Sphere = x2+y2+z2=a3 The Attempt at a Solution I think that the limits are from -pi/2 to...
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