Title: Polar Coordinates
1Polar Coordinates
- Definition Conversions and Integration
2Where is it
Coordinate systems are used to locate the
position of a point.
- In polar coordinates
- We break up the plane with circles centered at
the origin and with rays emanating from the
origin. - We locate a point as the intersection of a circle
and a ray.
- In rectangular coordinates
- We break up the plane into a grid of horizontal
and vertical line lines. - We locate a point by identifying it as the
intersection of a vertical and a horizontal line.
3Locating points in Polar Coordinates
The first coordinate r 2 indicates the
distance of the point from the origin.
The second coordinate /6 indicates the
distance counter-clockwise around from the
positive x-axis.
4Locating points in Polar Coordinates
Note however that every point in the plane as
infinitely many polar representations.
5Locating points in Polar Coordinates
Note however that every point in the plane as
infinitely many polar representations.
6Locating points in Polar Coordinates
Note however that every point in the plane as
infinitely many polar representations.
And we can go clockwise or counterclockwise
around the circle as many times as we wish!
7Converting Between Polar and Rectangular
Coordinates
It is fairly easy to see that if (xy) and (r q)
represent the same point in the plane
These relationships allow us to convert back and
forth between rectangular and polar coordinates
8Integration in Polar Coordinates
- Non-rectangular Integration Elements
9Small Changes in r and q
Suppose we consider a small change from r . . .
to r dr
10Small Changes in r and q
Suppose we consider a small change from r . . .
to r dr This gives us a thin ring around
the origin.
11Small Changes in r and q
Suppose we consider a small change from q . . .
to q dq
12Small Changes in r and q
Suppose we consider a small change from q . . .
to q dq This gives us a pie-shaped wedge
that is subtended by the angle dq.
13Small Changes in r and q
Intersecting the thin ring . . . and the
pie-shaped wedge . . . we get . . .
14Small Changes in r and q
Intersecting the thin ring . . . and the
pie-shaped wedge . . . we get . . .
15Small Changes in r and q
In order to integrate a function given in polar
coordinates (without first converting to
rectangular coordinates!) we need to know the
area of this little piece.
dr
Why
dq
16Integration in Polar Coordinates
In order to integrate a function given in polar
coordinates we will first chop up our region
into a bunch of concentric circles and rays
emanating from the origin.
(rq f(rq))
(rq)
Now do this for each little wedge and add up
the volumes of the towers.
Problem the volume of the tower is the area
of the base times the height. But the base is
not a rectangle so its area is not dr dq!
17Area of the Small Bit
A area of sector of a circle
q
18Area of a Small Bit
In order to integrate a function given in polar
coordinates (without first converting to
rectangular coordinates!) we need to know the
area of this little piece.
dr
r
dq
19Area of a Small Bit