- Your final homework is due on Wednesday.
- Today we start polar coordinates.
You won't have a homework on these, but might be expected to reproduce some of those we'll examine together (or examples from the sections) for the final exam.
- We start with the the chain rule, and the derivative
dy/dx. We must express this derivative in terms of time, which
is the actual independent variable in parametric curves:
- Tangent lines and Arc Length
-
Suppose a particle is following the parametric curve
- the point is
and
- the slope is
evaluated at
.
- Examples:
- #1, p. 675
- #3 (we need a point and a slope, dy/dx)
- #20 (where is dy/dx=0 or
?)
- #41
in time.
The tangent line to the curve at
We can compute how far the particle has travelled during the interval
easily
using the dirt formula, 
(in its modified
form 
).
In this case, the rate is just the speed. So we compute the integral
This is actually just a re-expression of the arc length formula:
and
.
But arc length may be different from the distance the particle travelled: a particle can revisit many sections of the curve y(x) -- so once again we need to be careful to distinguish between the independent variable of interest (whether x or t).
Let's calculate how far a particle travelled when parameterized by
and 
, with 
What happens to our integral if we double the time interval?
- the point is
We can think of every point on the plane as being given as the intersection of a vertical and a horizontal line. These lines give rise to the xy-coordinates of a point.
Alternatively, you can think of every point in the plane being given by the
intersection of a circle centered at the origin, with given radius , and of a ray coming out from the origin
at a given angle
from the
positive x-axis (the "polar axis"). These coordinates give rise to the polar
representation of the plane.
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. It is especially useful for constructing or representing objects which exhibit radial symmetry (e.g. a circle).
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.
Furthermore, we think of as an angle, and the plane is
obviously then represented in a
-periodic way.
Suppose we allow r to be negative in the image above: then we may think
of the point above as the point
(for example), or
for that matter.
So now curves are given in terms of r and rather than x and y,
via
(but notice that and
are both given implicitly).
- #1, p. 686
- #3
- #10
- #16
- #27
- #67
- #47
- #56
- #64
Let's start with a circle of radius r, centered at the origin, and see how we might adapt the "familiar formulas" to more exotic shapes.
- Length: The arc length of a sector of the circle swept out
by angle
(with the familiar circumference formula being the case of sweeping out an angle of
.
So a small (infinitesimal) change
would sweep out a length of
.
- Area: The area of the sector is basically that of a half rectangle. Let's compute its dimensions.
- #1, p. 692
- 5
- 7
- 15
- 26
- 30
- 40
- 47
- 54