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. M

    Double integral with polar coordinates

    Homework Statement It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ. f(x,y)=1 (plane parallel to Oxy plane) They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it. Homework Equations x=rcosθ y=rsenθ r=√x2+y2 The Attempt at a...
  2. E

    Area of overlapping polar coordinates

    Homework Statement find the over lapping area of the following equations r=3sin(x) r=1+sin(x)Homework Equations area =1/2 ∫ f(x)^2 dxThe Attempt at a Solution first off I started by finding the intersecting angle by: 3sin(x)=1+sin(x) 2sin(x)=1 sin(x)=1/2 x=pi/6 and the peak is at pi/2 so I...
  3. C

    How Do You Calculate the Area of the Upper Crescent in Polar Coordinates?

    I am asked to consider the following graph: r2=a+sin(θ), where a=2 I have a picture of this plot, which I have attached, We are asked to find the area of the upper 'cresent' of the curve, contained at the top How would I go about calculating that? I've found that if I plot...
  4. J

    Double Integral Cartesian to Polar Coordinates

    Homework Statement Use polar coordinates to evaluate: ∫sqrt(2)0 ∫sqrt(4-y2)y 1/(1+x2+y2) dxdy Homework Equations The Attempt at a Solution I graphed it and I see r is the part of the elipse sqrt(4-y2) and goes from 0 to ∏/4. I'm not sure how to make the bounds for r or how to...
  5. J

    Laplace's equation w/ polar coordinates

    Homework Statement The lecture notes say that ∇ = urr + (1/r)ur + (1/r2)uθθ. I'm not sure how this comes about. The notes never explain it. Homework Equations (?) The Attempt at a Solution No attempts on the actual homework problem until this ∇ thing is cleared up.
  6. I

    Area and Volume integral using polar coordinates

    Hi I'm working on area and volume integrals. I was wondering, when you convert to do the integral in polar, cylindrical or spherical co-ordinates, is there a standard set of limits for the theta variable in each case? for example from 0 -pi for polar, 0-2pi for cylindrical? If not how...
  7. B

    Quick question about finding area for polar coordinates

    Homework Statement Find the area of the shaded region. r=sqrt(θ) Homework Equations A = integral from a to b 1/2r^2dθ The Attempt at a Solution I know how to solve the question, I just don't know what to use for a and b. I tried 0 and 2pi but I am getting the wrong answer...
  8. M

    Parametric equations and polar coordinates

    Homework Statement Find the area enclosed by the inner loop of the curve r=1-3sinθ Homework Equations A=o.5\int r^2 dθ The Attempt at a Solution I found the integral but i don't know how to find the interval at which i will be integrating from. I tried finding when r=0 and it turns...
  9. A

    MHB Best Way to Graph in Polar Coordinates

    what is the best way to graph in polar coordinates say r = 3 - 5 \cos \theta is it to plot several points then make a curve between them or ?
  10. S

    Ellipse and Kepler's Law in Polar Coordinates

    Greetings everyone, I am having difficulties grasping the polar form of the ellipse equation, and there seems to be more than one way to express an ellipse in this form, if I am not mistaken. For example on the following webpage http://farside.ph.utexas.edu/teaching/301/lectures/node155.html...
  11. M

    Integration of a vector field in polar coordinates

    Hello all, I am trying to understand how to integrate a vector field in polar coordinates. I am not looking to calculate flux here, just the sum of all vectors in a continuous region. However, there is something I am not doing properly and I am a bit lost at this point. Any help would be...
  12. R

    Use polar coordinates to find the volume of the given solid.

    Homework Statement 1. Use polar coordinates to find the volume of the given solid. 2. Inside the sphere x^2 + y^2 + z^2 = 16 and outside the cylinder x^2 + y^2 = 4. 2. The attempt at a solution My attempt as following: 2<=r<=4, and 0<=theta<=2pi So I do a double integral of...
  13. F

    Polar Coordinates to evaluate integrals

    Homework Statement Use Polar coordinates to evaluate were C denotes the unit circle about a fixed point Z0 in the complex plane The Attempt at a Solution I've only used polar integrals to convert an integral in sin and cos into one in therms of z, find the residues and then use the...
  14. S

    Finding the volume of the cone using cylindrical polar coordinates?

    The cone centre is the z-axis and has base ρ=1 and height z=1, I'm looking at the lecture notes and it says the limit φ=0 to 2pi, z=0 to 1, ρ=0 to (1-z). Could someone tell me where the (1-z) comes from please? Why is it not 0 to 1?
  15. G

    Find polar coordinates (r, θ) of the point.

    Homework Statement The Cartesian coordinates of a point are given. (3,-5) (i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. Homework Equations r^2=x^2+y^2 tanθ=(y/x) →...
  16. ElijahRockers

    Integrate by changing to polar coordinates.

    EDIT: I found my mistake. Theta goes from 90 to 270, not -90 to 90. Wrong side of the y axis. That changes last integration to -2 instead of 2, making final answer -126. Evaluate the given integral by changing to polar coordinates. \int\int_R 3(x+y) dA where R is the region that lies to the...
  17. M

    Polar coordinates and multivariable integrals.

    Homework Statement Im righting this down for my roommates since he's having tons of trouble trying to figure this out and I can't answer it. also sorry for having to hotlink it. http://i.imgur.com/afShz.jpg the equation is on the image since its very difficult to type it all out...
  18. D

    Integral calculus: plane areas in polar coordinates

    what is the area inside the graph of r=2sinθ and outside the graph of r=sinθ+cosθ? so i compute for the values of 'r',... but, i only got one intersection point which is (45°, 1.41). there must be two intersection points right? but I've only got one. what shall i do? i cannot compute for...
  19. T

    Solving Limit Using Polar Coordinates

    Staff was trying to understand a matter of calculation. I hope someone can explain me in detail how to solve this limit using polar coordinates: http://img36.imageshack.us/img36/2667/semttulokej.png
  20. ElijahRockers

    What are the steps for drawing vectors in polar coordinates?

    Homework Statement Let \hat r = <x_r , y_r> and \hat\theta = <x_\theta , y_\theta> Draw these vectors at points (x,y) = (1,0), (2,0), (3,0), (1,1), (0,1), (0,2). Here is the entire http://www.math.tamu.edu/~vargo/courses/251/HW5.pdf assignment so you can see what context it is in...
  21. B

    How Do You Convert Foucault Pendulum Equations to Polar Coordinates?

    Homework Statement For a Foucalt Pendulum: Relative to horizontal Cartesian x and y axes fixed to the Earth (with x as East) the equations of motion for horizontal motion are: x′′ + ω02x -2ωy′ = 0 and y′′ + ω02y + 2ωx′ = 0 [where x′, x′′, y′, y′′ are first and second time...
  22. N

    Polar coordinates of an electric field

    Homework Statement Three charges are arranged as presented below. Q1= 5.00E-9C, Q2= 6.00E-9C and Q3= -7.00E-9C. http://img15.imageshack.us/img15/9250/physicskf.png D) Find the direction of the above electric field using the polar coordinate system 0°< θ <360° Homework Equations...
  23. B

    Cauchy-Riemann Conditions in Polar Coordinates

    Homework Statement Using f(z) = f(re^iθ) = R(r,θ)e^iΩ(r,θ), show that the Cauchy-Riemann conditions in polar coordinates become ∂R/∂r = (R/r)∂Ω/∂θ Homework Equations Cauchy-Riemann in polar coordinates Hint: Set up the derivative first with dz radial and then with dz tangential...
  24. J

    Ball thrown across a carousel - fictitious forces in polar coordinates

    Homework Statement a carousel is spinning with a constant angular velocity ω. two people, A and B are standing across each other (with the center between them) at distance 2d (d is the radius of the carousel). A throws a ball to B, so B catches it after T seconds. describe the equations...
  25. T

    Convert the double integral to polar coordinates

    Homework Statement Evaluate the double integral by converting to polar coordinates. ∫∫ arctan y/x dA; R is the sector in the first quadrant between the circles 1/4= x^2+y^2 and x^2+y^2=1 and the lines y=x/√3 and y=x. Homework Equations arctan y/x= θ The Attempt at a Solution...
  26. G

    United States Calculus 2 - Calculus in Polar Coordinates

    Homework Statement Find the slope of the line tangent to the polar curve at the given point. At the point where the curve intersects the origin, find the equation of the tangent line in polar coordinates. r = 6 sinθ; (-3 7∏/6) Homework Equations The Attempt at a Solution...
  27. M

    Equation of a circle / polar coordinates

    I was looking at the equation of a circle in polar coordinates on wikipedia, http://en.wikipedia.org/wiki/Polar_coordinate_system and I understand that a is the radius of the circle, and that (r0, phi) is the center of the circle. But I don't see what the r and theta refer to :(.
  28. Q

    Integral with Polar Coordinates: Teta Angle Interval

    i just need to know something about the integral with polar coordinates , to know the interval of the teta angle of any domain the proffesor said that we put the pen on the x-axis and move it , i for one moment was not focusing and the doctor had to go to another class can anyone explain ? thank you
  29. N

    Double Integral Polar Coordinates to find area of region

    I'm studying for my final and tutors/my professor isn't available over the weekend. Could someone please spend a little time to help me? My problem is stated as: Let R be the right half of the circle x2+(y-1)2=1. Use a double integral polar coordinates to find the area of the region R. I...
  30. A

    Transformation from Cartesian to spherical polar coordinates

    Transformation from Cartesian to spherical polar coordinates In dimensions: x=r sinθ cos \varphi and y= r sin θ sin \varphi z=r cos θ Show one example of: ∂z\alpha/ ∂xμ . ∂xμ/ ∂z\alpha = δ\alpha\beta Now here is my answer: δyx=(∂y/∂r . ∂r/∂x) + (∂y/∂θ . ∂θ/∂x) + (∂y/∂\varphi...
  31. T

    Polar Coordinates and Conservative Vector Fields

    Homework Statement Let F = <-y/(x2+y2, x/(x2+y2>. Recall that F was not conservative on R2 - (0,0). In this problem, we show that F is conservative on R2 minus the non-positive x-axis. Let D be all of R2 except points of the form (-x,0), where x≥0. a) If (x,y) is included on D, show that...
  32. S

    Having trouble with double integrals in polar coordinates?

    I'm having trouble figuring out how to find what "r" is. I know r is the radius, but how do I go about finding it? Like what do I look for in a particular problem?
  33. G

    How Are Polar and Cartesian Graphs of \( r = e^{\theta} \) Related?

    Homework Statement "Consider the graph of r = e^{\theta} in polar coordinates. Then consider the graph of (\theta \cos{\theta}, \theta \sin{\theta}) where \theta \in \mathbb{R} on the Cartesian plane (x - y axis). How are the two graphs related? What relationship (if any) can we define between...
  34. K

    Spiral path in polar coordinates problem

    A bee goes from its hive in a spiral path given in plane polar coordinates by r = b*ekt , θ = ct, where b, k, c are positive constants. Show that the angle between the velocity vector and the acceleration vector remains constant as the bee moves outward. (Hint: Find v · a/va.) attempts...
  35. D

    Double Integral problem What am I suppose to do? Related to polar coordinates.

    The problem and my work is shown in the image below. However, I feel like I did something horrible wrong but I'm not sure where! I'm sorry if my handwriting is illegible. If you're having difficulties please leave a comment and I will not hesitate to type it out as a response. Any...
  36. D

    Double Integral, where did I go wrong? Related to polar coordinates.

    ∫∫cos(x^2 + y^2)dA, where R is the region that lies above the x-axis within the circle x^2 + y^2 = 9. Answer: .5pi*sin(9) My Work: ∫(0 ->pi) ∫(0 -> 9) cos(r^2) rdrdθ u = r^2 du = 2rdr dr = du/2r .5∫(0 ->pi) ∫(0 -> 9) cos(u) dudθ .5∫(0 ->pi) sin(u)(0 -> 9) dθ .5∫(0 ->pi)...
  37. T

    Polar Coordinates Improper Integral Proofs

    Homework Statement (a) we define the improper integral (over the entire plane R2) I=\int\int_{R^2}e^{-(x^2+y^2)}dA=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx=\lim_{a\rightarrow\infty}\int\int_{D_{a}} e^{-(x^2+y^2)} dA where Da is the disk with radius a and center the...
  38. E

    Easy Polar Coordinates question (Change of variables)

    I have a question regarding problem solving tips. When given an iterated integral and asked to convert it to polar coordinates, how do you select the bounds of theta - do you have to understand how the graph of r operates and therefore know where the theta bounds are based on the rectangular...
  39. D

    Particle moving in polar coordinates

    Equations given: r=A\theta \theta=\frac{1}{2}\alphat^{2} A=\frac{1}{\pi} meters per radian \alpha is a given constant Asks to show that radial acceleration is zero when \theta=\frac{1}{\sqrt{2}} radians. I have tried rearranging, plugging in, and deriving to try to solve this...
  40. R

    The tangent to an ellipse from polar coordinates

    I have an ellipse. Quite simple, ecc=0.60. And I'm doodling with calculus I learned 40 years ago. I can find the tangent to the ellipse, that is, the slope of the tangent, using cartestian coordinates. At the point where the tangent skims the top of the minor axis (b) the slope is 0 and and...
  41. C

    Arfken, parity operation on a point in polar coordinates

    Homework Statement Show that the parity operation (reflection through the origin) on a point (\rho, \varphi, z) relative to fixed (x, y, z) axes consists of the transformation: \rho \to \rho \varphi \to \varphi \pm \pi z \to -z Also, show that the unit vectors of the cylindrical polar...
  42. H

    Polar Coordinates Homework: Find the Polar Form of an Expression

    Homework Statement Hi, I have the coordinates of an "expression" for a point in a cartesian coordinate system. I'm trying to write it in a polar coordiante system (in function of r and theta) but I don't know how to find the answer a = y-component of the point b = x-component of the point...
  43. G

    Calculus II - Calculus in Polar Coordinates

    Homework Statement Find the points at which the following polar curves have a horizontal and vertical tangent line. (a) r = 3 + 6 cos(theta) Homework Equations The Attempt at a Solution x = r cos(theta) = (3 + 6 cos(theta)))cos(theta) = 3cos(theta) + 6 cos(theta)^2 y =...
  44. F

    Use polar coordinates to find the limit

    Hi! Is there somebody, who can help me with this exercise: "Use polar coordinates to find the limit. [If (r, θ ) are polar coordinates of the point (x,y) with r ≥ 0, note that r --> 0+ as (x,y) --> (0,0)]
  45. S

    Areas and Lengths in Polar Coordinates. Calculus 3

    Homework Statement "Find the area of the region that lies inside both curves (as an example), r=((sqrt(3)) cos(theta)) , r=sin(theta). This is Calculus 3. Areas and lengths in polar coordinates. Homework Equations Guys, I'm very confused because when the polar graphs are complicated we...
  46. A

    How can I integrate an acceleration vector in polar coordinates?

    Hi, Say I have an acceleration vector in polar coordinates: a = -30e_r where the unit vector e_r points in the same direction as the Cartesian unit vector j. How can I integrate that vector so that I have the velocity vector in polar coordinates? I know that if I have an acceleration vector...
  47. H

    Polar Coordinates: Find dy/dx Problem Solution

    Homework Statement Find dy/dx Problem in picture below Homework Equations The Attempt at a Solution [PLAIN]http://img28.imageshack.us/img28/7162/76013837.png The answer for this is dy/dx = -cos\theta sin\theta + (1-sin\theta)cos\theta/-cos^2\theta - (1-sin/theta)sin\theta I cannot figure...
  48. I

    Divergence in spherical polar coordinates

    I took the divergence of the function 1/r2\widehat{r} in spherical coordinate system and immediately got the answer as zero, but when I do it in cartesian coordiantes I get the answer as 5/r3. for \widehat{r} I used (xi+yj+zk)/(x2+y2+z2)1/2 what am i missing?
  49. 2

    Curl in spherical polar coordinates

    Hey, I've been stuck on this question for quite a while now: Homework Statement 1a. Write down an expression for the position vector r in spherical polar coordinates. 1b. Show that for any function g(r) of r only, where r = |r|, the result \nabla x [g(r)r] = 0 is true. Why does this...
  50. X

    Polar coordinates, maximum distance.

    Homework Statement The diagram (omitted) shows the curve C with polar equation r=e^(\theta), where 0\le\theta\le(pi/2). Find the maximum distance of a point of C from the line \theta=(pi/2), giving the answer in exact form. The Attempt at a Solution I'm not really sure how to attack...
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