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Calculus: The Polar Coordinate System


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About this Lesson

  • Type: Video Tutorial
  • Length: 12:32
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 134 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Parametric Equations, Polar Coordinates (18 lessons, $27.72)
Calculus: Understanding Polar Coordinates (4 lessons, $6.93)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Parametric Equations and Polar Coordinates
Understanding Polar Coordinates
The Polar Coordinate System Page [1 of 2]
It's cold out here. I don't know what it's doing where you are, but boy, you can see that it's fact, it's starting to snow. It's so cold! When you're thinking of snow and when you're thinking of the December season and stuff, it really sort of makes you think about the North Pole and all of the things that go on up in the North Pole. I think it's really about to come down, so why don't we actually go inside and take a look at polar form.
First of all, let me shovel myself out of my thing here. I want to really talk about this thing called polar coordinates. This really is sort of a very cold, cold subject, so we have to warm it up. How do you warm up polar coordinates? First of all, what are polar coordinates? Are they coordinates up at the Pole? The answer is exactly...exactly yes. When you think about good, old-fashioned how do you look at the plane? How do you look at the plane? Well, you can do like this. That's one way. Another way is to just think about the axes. You've got the x-axis and you've got the y-axis, and to describe any point - any point - on the plane, what do you do? Well, you've got to give me two pieces of information: You've got to tell me how far over to go in the x-direction and then how far up and down to go in the y-direction, and then that completely solidifies the point. I go over that much and up that much - (x,y) - and that's called Cartesian Coordinates, and that's the way we usually think about the plane.
But there is a whole other way of thinking about the plane, and it's completely different and turns out to be very useful in certain types of situations. What way is that? Well, instead of going up and up like this - over, up, and so forth in a rectangular way - let's do something different. Let's actually see if we can warm things up. Let's bring out the sun. If we bring out the sun - put it out here - then it starts to warm things up. Do you see that all of the snow is going away? So, in fact, we have this pole, which is going to be the very, very center of this new system, and everything is going to start and emanate from this thing right here. If you think of a point, say, for example, this point right here, instead of identifying it by going over and then up in a rectangular fashion, what I'm going to do is say, "Hey, look, why don't you just measure how far you are from this pole - from the starting position?" So measure this length - I'll call that r - "and then tell me how much I've got to pitch up." So tell me the angle that I have to go from here. Then I measure this angle, which I'll call q. So notice that before I had (x,y), and I had sort of a rectangular thing going, and here what I'm going to do is I'm going to go over q radians, and then go out r. So I can write this point as (r,q), and this is polar form. The idea is I put the sun right at the center - right at the pole, right at the origin - and then I let these rays shoot out like this, and then all I've got to do is figure out the angle and how far out to go on a ray, and that's going to identify a point.
Let's take a look at some examples together. Here are some actual examples together just to get a sense of what's going on here. Let's now plot polar. So if you plot polar, here's what it looks like. Remember, (x,y) in the Cartesian thing is always going to be (r,q) in polar form. So for example, if I say, "Plot (3,)," that means that the angle I'm going to go is going to be radians. We always, always, always start here. So we start here. And if the angle is positive, we're going to move in a counter-clockwise direction. So, in fact, this q is really a directed angle. It's a directed angle in the sense that if I say , then I go this way, like that. If I were to say -, then actually I would go in a clockwise direction, but always starting here. So the sign tells me if I'm going to go counter-clockwise if the sign is positive, or clockwise if the sign is negative. So it really is a directed angle. So this says go positive , so I always start here, and I go radians. I go like this, and that kind of hurts. Now what do I do? Now what I've got to do is go out from here a certain distance, and the distance is three. So actually now start here and march out three units in this direction. So one, two, three, and that's the point. So that point is really (3,). Now you might say, "Gee, that looks sort of weird." If you think it still looks weird, it's because you're still in the Cartesian rigid, rectangular system. Free yourself of the shackles of the Cartesian system and embrace this new idea of polar coordinates, where I first swing over the angle - in this case an angle of - and then I zoom out three. Isn't that great?
Let's try some more. Here's another one. How about this one? How about (3,3)? Where would that point be? Let's think about that. So I start here. I've got 3. What does 3 mean? Well, 3 means I swing around - one - come back to where we started - that's 2 - but I keep going - 3. So I'm going to be going out this way, and I go out three units. Wait a minute - one, two, three. Well, that's the same point. That's really interesting. So this point can also be thought of as (3,3). It's the same point, it's just that I spun around once and then stopped there. It's like a spinner, right? Instead of just landing here like this, I could have gotten there by spinning around once and then landing there. So an important little lesson about polar coordinates is that, in fact, there are many different ways of representing the same point. These two pair of coordinates actually represent the same point, so you've got to be careful about that.
Let's try another example. Let's take a look at this. How about (1,)? All right, now what would that look like? What I have to do there is first of all, the angle is . What does that mean? That means that instead of going counter-clockwise - the positive direction - I'm going to go in a negative direction, which is clockwise, . So that's 90 degrees, so that's way down here. Then I go one unit in this direction, so I go one unit down. So there's the point. That's the point: (1,). Great.
Let's try one last one. (-3,2). Now let's see what happens here. For here, what I do is I start, I see a positive angle, so I go over one , two , so I go back to where I started. But now what is -3 in this direction? Well, this is a positive direction, so -3 must be in the opposite direction. So again we're seeing that even r is actually a directed distance, which means that if you end up on this wing here but you see a negative value, it means you have to go the opposite direction - you have to go against the tide there - negative three units. And where is that? That's the same point. So here we're seeing another representation of the same point. (-3,2) also represents the same point that we've seen already two other times. And, again, we're seeing this one now because I go over - it takes me back to where I started - and then when it says -3, instead of going three units this way, since it says -3, I go three units the opposite way. So those are sort of how you plot points, and really the great lesson is that there are many different ways of expressing points. That's going to be sort of critical in what we're doing here.
Let's just think about what some very, very quick, simple graphs would look like. So what would some simple graphs look like? So, simple polar graphs. One thing we could look at is the following: r = 2. Now what would the graph be of r = 2? What is this saying? It's saying all of the points that have r equal to two. That means r represents the distance from the origin. So this is saying all of the points whose distance from the origin is two. Well, what kind of object is that? That is a circle, radius two, because those are all of the points whose distance from the origin is two. So, in fact, the graph of r = 2 is a circle -- centered at the origin, radius 2. In general, if I said r = 17, you can figure out what that is. That would be a circle centered at the origin that has radius 17. So, in fact, r = a represents a circle centered at the origin.
Let's take a look at something else that's really sort of simple. How about this one: q = . That's asking for all of the points in polar coordinates for which the angle is . Now what's ? Well, for you and me is like 60 degrees. That means that I want to find out all of the points in the plane whose angle from here is going to be 60 degrees. So you go up 60 degrees - that's about here - and so this angle here - q - is .
Is that the complete graph? Well, let's see. For any positive r, that's great; but what if I had a negative r? What if I had a (-3,)? Well, then I go for the -3, putting me in the opposite direction. So, in fact, I have to include this entire line, and that's the graph of q = .
So what's the moral? The moral is if you just have a function r equals a constant, you're going to get a circle centered at the origin because all of the radius is fixed. If you see q equals a constant, then, in fact, you're going to get some straight line through the origin whose pitch is actually the measure of q . So those are two very, very simple graphs; and what we'll look at are some even more exotic graphs and how you graph things that are really, really exotic. So, welcome to the world of polar coordinates. See you at the next lecture.

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