Polar coordinates Definition and 586 Threads

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°).
Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term polar coordinates has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.
Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.
The polar coordinate system is extended to three dimensions in two ways: the cylindrical and spherical coordinate systems.

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  1. B

    Help describing a region in polar coordinates

    Homework Statement If (r, θ) are the polar coordinates of a point then describe the region defined by the restrictions -1 < r < 0, π/2 < θ < 3π/2 Homework Equations No clue The Attempt at a Solution I tried drawing the curve in a polar grid by starting at π/2 and finishing at 3π/2. I was...
  2. F

    Asymptote of a curve in polar coordinates

    Homework Statement The curve ##C## has polar equation ## r\theta =1 ## for ## 0<\theta<2\pi## Use the fact that ## \lim_{\theta \rightarrow 0}\frac{sin \theta }{\theta }=1## to show the line ## y=1## is an asymptote to ## C##.The Attempt at a Solution **Attempt** $$\ r\theta =1$$ $$\...
  3. Mind----Blown

    Significance of terms of acceleration in polar coordinates

    How do i get an idea, or a 'feel' of the components of the acceleration in polar coordinates which constitute the component in the eθ direction? from what i know, a= (r¨−rθ˙^2) er + (rθ¨+ 2r˙θ˙) eθ ; (where er and eθ are unit vectors in the radial direction and the direction of increase of the...
  4. C

    I Limits of integration on Polar curves

    General question, how do you determine the limits of integration of a polar curve? Always found this somewhat confusing and can't seem to find a decent explanation on the internet.
  5. welssen

    Determine the angular momentum in polar coordinates

    Hi there, I've been trying to solve the following problem, which I found looks pretty basic, but actually got me really confused about the definition of angular momentum. Problem The trajectory of a point mass m is described by the following equations, in spherical coordinates: r(t) = r_0 +...
  6. D

    How Do You Calculate the Laplacian in Polar Coordinates?

    Homework Statement I am trying to calculate the laplacian in polar coordinates but I failed.Please see the attached Homework Equations The Attempt at a Solution My solution to this was uploaded in the attached.I was wondering what's wrong with the purple brackets since they shouldn't exist(...
  7. CheeseSandwich

    I Conceptual Question About Polar Coordinate System

    I am learning about the polar coordinate system, and I have a few conceptual questions. I understand that in Cartesian coordinates there is exactly one set of coordinates for any given point. However, in polar coordinates there is an infinite number of coordinates for a given point. I see how...
  8. Dilemma

    Finding the volume - Polar coordinates

    Hello everyone, 1. Homework Statement Question : Find the volume of the region which remains inside the cyclinder x 2 + y 2 = 2y, and is bounded from above by the paraboloid surface x 2 + y 2 + z = 1 and from below by the plane z = 0 Homework Equations The Attempt at a Solution This looks...
  9. maxhersch

    Double Integration in Polar Coordinates

    Homework Statement Integrate by changing to polar coordinates: ## \int_{0}^6 \int_{0}^\sqrt{36-x^2} tan^{-1} \left( \frac y x \right) \, dy \, dx ## Homework Equations ## x = r \cos \left( \theta \right) ## ## y = r \sin \left( \theta \right) ## The Attempt at a Solution So this is a...
  10. doktorwho

    Finding the radius of curvature of trajectory

    Homework Statement The functions are given: ##r(t)=pe^{kt}## ##\theta (t)=kt## ##v(r)=\sqrt2kr## ##a(t)=2k^2r## Find the radius of the curvature of the trajectory in the function of ##r## Homework Equations $$R=\frac{(\dot x^2 + \dot y^2)^{3/2}}{(\dot x\ddot y - \ddot x\dot y)}$$ There is also...
  11. doktorwho

    Solving for the trajectory in the polar coordinate system

    Homework Statement On the surface of a river at ##t=0## there is a boat 1 (point ##F_0##) at a distance ##r_0## from the point ##O## (the coordinate beginning) which is on the right side of the coast (picture uploaded below). A line ##OF_0## makes an angle ##θ_0=10°## with the ##x-axis## whose...
  12. toforfiltum

    Evaluating Cartesian integral in polar coordinates

    Homework Statement Transform given integral in Cartesian coordinates to one in polar coordinates and evaluate polar integral. ##\int_{0}^3 \int_{0}^x \frac {dydx}{\sqrt(x^2+y^2)}## Homework EquationsThe Attempt at a Solution I drew out the region in the ##xy## plane and I know that ##0 \leq...
  13. F

    Converting Laplacian to polar coordinates

    Homework Statement $$ U_{tt}=\alpha^2\bigtriangledown^2U$$ in polar coordinates if solution depends only on R, t. Homework EquationsThe Attempt at a Solution So, the books solution is $$U_{tt}=\alpha^2[U_{rr}+\frac{1}{r}U_r]$$. I am getting stuck along the way can't figure out this last step I...
  14. G

    I Problem with polar coordinates

    Hello, I have a question about polar coordinates. It is \vec r = \begin{pmatrix}r cos\phi \\ rsin\phi \\ z\end{pmatrix}=r\cdot \vec e_r + z\cdot \vec e_z and than is \ddot{\vec r} = (\ddot{r}-r\dot{\phi}^2)\vec e_r + (r\ddot{\phi} +2\dot{r}\dot{\phi})\vec e_{\phi} + \ddot{z}\vec e_z The...
  15. G

    Having a hard time writing equations of motion....

    Homework Statement [/B] Homework Equations The Attempt at a Solution [/B]
  16. S

    Equations of Motion of a Mass Attached to Rotating Spring

    1. Homework Statement A particle of mass m is attached to the end of a light spring of equilibrium length a, whose other end is fixed, so that the spring is free to rotate in a horizontal plane. The tension in the spring is k times its extension. Initially the system is at rest and the...
  17. deusy

    Acceleration of the end of a hinged rod in a pulley system

    Homework Statement As shown in image. 2. Homework Equations Moment of inertia of pulley = 1/2*M*R^2 Moment of inertia of rod (about end) = 1/3*M*L^2 Acceleration of end of rod in theta direction = L*α Acceleration of end of rod in radial direction = L*ω^2 The Attempt at a Solution...
  18. Drakkith

    Setting up a Double Integral in Polar Coordinates

    Homework Statement Consider the 'ice cream cone' bounded by z = 14 − x2 − y2 and z = x2 + y2 .(a) Find the equation of the intersection of the two surfaces in terms of x and y. (b) Set up the integral in polar coordinates. Homework EquationsThe Attempt at a Solution I got part a without...
  19. S

    Metric tensor and gradient in spherical polar coordinates

    Homework Statement Let ##x##, ##y##, and ##z## be the usual cartesian coordinates in ##\mathbb{R}^{3}## and let ##u^{1} = r##, ##u^{2} = \theta## (colatitude), and ##u^{3} = \phi## be spherical coordinates. Compute the metric tensor components for the spherical coordinates...
  20. P

    How Do Polar Coordinates Explain a Bead's Velocity on a Rotating Wheel?

    Note: All bold and underlined variables in this post are base vectors I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The...
  21. Mr Davis 97

    Magnitude of a vector in polar coordinates

    Homework Statement What is the magnitude of the velocity vector if ##\vec{v} = 4 \hat{r} + 6 \hat{\theta}## Homework EquationsThe Attempt at a Solution I know how do do this in Cartesian coordinates (use the Pythagorean theorem), but not so sure how to do it in polar coordinates.
  22. Mr Davis 97

    I Describing a position vector with polar coordinates.

    I have read that in polar coordinates, we can form the position vector, velocity, and acceleration, just as with Cartesian coordinates. The position vector in Cartesian coordinates is ##\vec{r} = r_x \hat{i} + r_y \hat{j}##. And any choice of ##r_x## and ##r_y## maps the vector to a position in...
  23. V

    Polar Coordinates: Position of Particle at T/8

    1. The question The position of a particle is given by r(t) = acos(wt) i + bsin(wt) j. Assume a and b are both positive and a > b. The plane polar coordinates of a particle at a time t equal to 1/8 of the time period T will be given by _ Homework Equations r(t) = acos(wt) i + bsin(wt) j. The...
  24. F

    I Example of computing geodesics with 2D polar coordinates

    I am trying to find and solve the geodesics equation for polar coordinates. If I start by the definition of Christoffel symbols with the following expressions : $$de_{i}=w_{i}^{j}\,de_{j}=\Gamma_{ik}^{j}du^{k}\,de_{j}$$ with $$u^{k}$$ is the k-th component of polar coordinates ($$1$$ is for...
  25. smodak

    I Length of bases in Polar coordinates

    According to this video the length of basis is r. It grows as we further from the origin . Why?
  26. S

    Line integral convert to polar coordinates

    Homework Statement I need to find the work done by the force field: $$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$ moving a particle from a to b along a path given by: $$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$ The Attempt...
  27. Eclair_de_XII

    Finding the equation of a tangent line in polar coordinates?

    Homework Statement "Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates." ##7.##...
  28. B

    MHB Polar coordinates to evaluate double integral

    I am trying to evaluate \int\int xy dxdy over the region R that is defined by r=sin(2theta), from 0<theta<pi/2. I am struggling on where to begin with this. I have tried converting to polar coordinates but am not really getting anywhere. Any guidance would be really appreciated (Crying)
  29. arpon

    I Infinitesimal area element in polar coordinate

    We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate...
  30. S

    Double Integral in polar coordinates

    Homework Statement Evaluate ∫∫D (3x + 4y 2 ) dA, where D = {(x, y) : y ≥ 0, 1 ≤ x 2 + y 2 ≤ 4} with the use of polar coordinates. Homework Equations The Attempt at a Solution I made a sketch of the circle. It's radius is = 1 and it's lowest point is at (0,0), highest at (0,2), leftmost point...
  31. chwala

    Understanding Scale Factors in Cylindrical Polar Coordinates

    Homework Statement Using the cylindrical polar co ordinates ##(ℝ,θ,z)## calculate the gradient of ##f=ℝ sin θ + z^2## the textbook says that the scale factors are ## h1=1, h2=ℝ & h3=1## how did they arrive at this?[/B]Homework EquationsThe Attempt at a Solution ##h1=|∂f/∂ℝ|= sin θ...
  32. Edison Bias

    I Integral problems with polar coordinates and variable substitution

    H! I wonder how to solve: I=\int_{-\infty}^{\infty}e^{-u^2}\frac{1}{1+Cu} du I have solved: \int_{-\infty}^{\infty}e^{-u^2}du which equals \sqrt{\pi} and I solved it with polar coordinates and variable substitution. Thankful for help! Edison
  33. S

    Acceleration only due to conservation of angular momentum

    I don't understand why the conservation of angular momentum can imply an acceleration, in absence of a force. Consider for istance planetary motion. The angular momentum \vec{L} of the planets is conserved and that means \mid \vec{L} \mid=mr^2 \dot{\theta}=mrv_{\theta} is conserved too...
  34. C

    Physics olympiad problem -- struggling with polar coordinates

    Homework Statement This is a physics olympiad problem; and I am still struggling with it. I will quote it here: " A particle moves along a horizontal track following the trajectory $$r=r_{0}e^{-k\theta}$$, where $$\theta$$ is the angle made by the position vector with the horizontal. Recall...
  35. S

    Is polar coordinate system non inertial?

    Studying the acceleration expressed in polar coordinates I came up with this doubt: is this frame to be considered inertial or non inertial? (\ddot r - r\dot{\varphi}^2)\hat{\mathbf r} + (2\dot r \dot\varphi+r\ddot{\varphi}) \hat{\boldsymbol{\varphi}} (1) I do not understand what is the...
  36. Flinze

    Finding Polar Coordinates for Vector B⃗ = -2.0ι^ + 3.0 j^

    Homework Statement B⃗ = -2.0ι^ + 3.0 j^. Find the polar coordinates r and theta. Homework Equations n/a The Attempt at a Solution r=sqrt((-2.0)^2+(3.0^2)) r = 3.6 theta = tan^-1(3/-2) = -56 degrees The answers seem to be wrong, can I get any guidance on this question please?
  37. S

    Minkowski metric in spherical polar coordinates

    Homework Statement Consider Minkowski space in the usual Cartesian coordinates ##x^{\mu}=(t,x,y,z)##. The line element is ##ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}## in these coordinates. Consider a new coordinate system ##x^{\mu'}## which differs from these...
  38. K

    MHB Put the 2D nonlinear system into Polar Coordinates

    Show that, in polar coordinates, the system is given by r′ = r(r^2 − 4) θ′ = 1x′1 = x1 − x2 − x1^3 x′2 = x1 + x2 − x2^3
  39. A

    Bessel functions and the dirac delta

    Homework Statement Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
  40. G

    Arc length in polar coordinates

    I know that an arc length in polar coordinates can be computed by integrating $$\int ds$$ using the formula ##ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta##. But, seeing that ##s=\rho\theta## and ##ds = \rho d\theta##, why is it wrong to calculate arc lengths with this expression for ##ds##?
  41. A

    Curvilinear Motion: Polar Coordinates (Engineering Dynamics)

    Homework Statement Homework EquationsThe Attempt at a Solution I have stared at this for hours and don't know where to start. I think I need to get r in terms of t but I don't really know how with the information given. I just need a good hint to get started.
  42. C

    Determine the angular speed theta_dot

    Homework Statement Homework Equations Vθ = r x θdot Vr = rdot Accelθ = r ⋅ θdot^2 + 2(r dot)(theta dot) Accelr = r double dot - (r ⋅(θdot)^2) The Attempt at a Solution I know that the answer is supposed to be But I'm not quite sure on how I'm supposed to get to an answer like that. I'm...
  43. H

    2 vectors with cylindrical polar coordinates

    Hi this isn't my homework, but it is taken from a worksheet for a Maths course(trying to refresh my rusty math), so I hope it fits in here. 1. Homework Statement two cylindrical polar vectors with same origin: P(2,55°,3); Q(4,25°,6) units in m Homework Equations a) Express in cartesian...
  44. M

    Velocity in spherical polar coordinates

    I am looking at this derivation of velocity in spherical polar coordinates and I am confused by the definition of r, theta and phi. http://www.usna.edu/Users/math/rmm/SphericalCoordinates.pdf I thought phi was the co latitude in the r,θ,∅ system and not the latitude. Of course the two are...
  45. Gbox

    Acceleration in Plance Polar Coordinates

    I am looking to understand more about ##a=(\ddot{r}-r(\ddot{\theta})^2)\hat{r}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\hat{\theta}## I understand the terms ##\ddot{r}## and ##r\ddot{\theta}## ,but why ##-r(\ddot{\theta})^2## has opposite direction to ##\hat{r}## and why ##2\dot{r}\dot{\theta}##...
  46. W

    Double Integral in Polar Coordinates Symmetry Issue

    Homework Statement Find the volume of the solid lying inside both the sphere x^2 + y^2 + z^2 = 4a^2 and the cylinder x^2 + y^2 = 2ay above the xy plane. Homework Equations Polar coordinates: r^2 = x^2 + y^2 x = r\cos(\theta) y = r\sin(\theta) The Attempt at a Solution So I tried this...
  47. X

    Expressing A Quantity In Polar Coordinates?

    Homework Statement Express the quantity ∂2/∂x2+∂2/∂y2 in polar coordinates. Homework Equations x=ρcosφ y=ρsinφ ρ=sqrt(x2+y2) The Attempt at a Solution This is my first post, so I apologize for any weird looking equations, etc. I know that this is not a difficult problem, but I just cannot...
  48. M

    Double integrals: cartesian to polar coordinates

    Homework Statement Change the Cartesian integral into an equivalent polar integral and then evaluate. Homework Equations x=rcosθ y=rsinθ I have: ∫∫r2cosθ dr dθ The bounds for theta would be from π/4 to π/2, but what would the bounds for r be? I only need help figuring out the bounds, not...
  49. shanepitts

    Transforming Cartesian to Polar Coordinates

    Homework Statement I am currently trying to calculate the moment and products of inertia of a ring rotating about the x-axis at the moment the ring lies in the xy plane. The problem is that the notations I have from textbook are denoted for Cartesian coordinates. i.e. Ixx=∑i mi(yi2+zi2), and...
  50. Dewgale

    Usage of Del in Spherical Polar Coordinates

    Hi all, I'm having some problems in grasping/properly understanding the usage of the del operator ( ##\nabla## ) in spherical co-ordinates, and I was wondering if someone could point me to some good resources on the subject, or take a bit of time to try to explain it to me. It just doesn't seem...
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