Day 27 in MAT229

Today:

• Announcements:
  • Homework 7.8 returned
    • 3
    • 47
    • 51 -- this was nigh on a disaster...

  • Please read section 10.4 for next time, on polar areas and lengths

• Section 10.3: Polar Coordinates

  If you look over the pictures in this section, you will see that there are many beautiful curves that one can create using polar coordinates and polar plots.

  See, for example, p. 685 -- which should remind you of a spirograph, if any of you are that old....

  The polar coordinate system was introduced by Newton, an alternative to the Cartesian coordinate system. In the polar coordinate system every point in the plane is expressed by its distance and direction (angle) from the origin (called the pole). The polar axis plays the role formerly played by the positive x-axis. The polar coordinates are often given as , where

  • r is the distance from the pole (although we may allow r to be negative), and
  • is the angle between the polar axis and the ray passing through the point of interest.

  Here it is:

  

  We represent the point P in one of two ways -- in rectangular coordinates (x,y), or in polar coordinates . We can represent any point of the plane in either of those ways. Note, however, that we have a certain level of redundancy in the polar coordinate system.

  Look at the origin, for example.

  We allow r to be negative, too, which leads to further redundancy. Furthermore, we think of as an angle, and the plane is obviously then represented in a -periodic way.

  So now curves are given in terms of r and rather than x and y, via

  • 
  • 
  Alternatively, we can write r and in terms of x and y:
  • 
  • 
  Usually we think of polar coordinates as examples of parametric curves, in the following way: we will be given , and then can think of x and y as functions of :
  • 
  • 
  with as the parameter.

• Examples:
  • #1, p. 686
  • #3
  • #10
  • #16
  • #27
  • #67
  • #47
  • #56
  • #64