- Section 12.2 problems are due today.
- Change in plans: Section 12.3 problems: one of them will appear on the exam -- okay?
- Homework: one of these will appear on the exam: pp. 627-630, #4, 6, 8, 9, 10, 13, 14, 22, 25, 31, 50, 51
- We're going to be using a different coordinate system to
describe points in the plane -- no big deal, except that you're quite
likely unfamiliar with the new system (called polar
coordinates):
Coordinate r is called the radial coordinate and
is called the angular coordinate.
Notice something interesting: what are the radial coordinates of (x,y) and (-x,-y)?
(See the constraints on
- If horizontal and vertical lines are very easily expressed in the
usual Cartesian coordinates system, it's circles and lines centered at
the origin that are easily expressed in polar coordinates. Here are the
"grid lines" in polar coordinates:
- One of the problems of polar coordinates is their non-uniqueness:
We can sometimes use that to our advantage, however; for example, if we want to imagine a point travelling around a circle, then
the point makes multiple trips around the circle....
- We sometimes permit r to be negative, too, for
convenience. In terms of non-uniqueness we also have that
- One of the advantages of polar coordinates is that they allow us
to create curves that can't be represented as functions of Cartesian
coordinates y(x).
- Examples:
- Understanding polar coordinates:
- #1, p. 627
- #2, p. 627
- Converting between coordinate systems:
- #3, p. 627
- #5, p. 628
- #20, p. 628
- Converting equations:
- #16, p. 628
- #20, p. 628
- #33, p. 629
- Deriving equations in polar coordinates:
- a line passing through (x,y)=(a,b) with slope m
- Understanding polar coordinates:
- Let's convince ourselves of the derivation of the polar area formula:
- #16, p. 636.
- Now you try: #13, p. 636.