** Lab #11**

This worksheet does not need to be handed in, but you are responsible
for the material.
The purpose of the lab is to understand how a change of coordinates
transforms points, curves, tangent vectors, and area.

Recall that the polar transformation,
takes a point in the
-plane and sends it to a point in the -plane. It
follows that **P** takes a parametric curve in the
-plane and maps it to a curve, , in the
-plane. Similarly, the Jacobian of the transformation, **DP**,
takes a velocity vector to the parametric curve, call it ,
and sends it to a velocity vector, , of the image
curve .

** ACTIVITY 1: Inventing a Polar Curve**
Think of any continuous function on the interval .
(For example, you might choose .)
For later reference, be sure to write down the function that you used!
Before coming to class,
each student should carefully sketch the image under **P** of the graph of
the chosen function on a clean sheet of paper (or, better yet, get a
computer printout). This curve will be in the -plane; we
will use this curve in a later activity.

** ACTIVITY 2: Sketching Preimages of Curves**
For each of the two curves in Figure ,
sketch a curve in the -plane that is mapped to a
corresponding curve in Figure . The curve in the
-plane is called the * preimage*.
Check your guesses with a
graphing calculator or with maple (see ` ?plot,polar`).
(Hint: The first
curve in Figure is the image of a graph, the second curve is
the image of a parametric curve.)

** ACTIVITY 3: Another Preimage of a Curve**
By this point, your instructor will have collected your
curve from the first activity. When you get to this activity, your
instructor will give you
a curve that someone else in the
class sketched.
Can you figure out an approximate preimage of this curve?

** ACTIVITY 4: Images of Horizontal and Vertical Tangent Vectors**
Suppose that a parametric curve in the -plane has a horizontal
tangent vector, . Then
the image of this vector under
**DP** is a vector of the form . Notice that **T** is a
linear multiple of the vector
which is the direction of increasing .
- For the curve in Figure .A, mark the locations on the
curve that are the images of the
values at which . Use your result from
Activity 2 to explicitly compute the vectors
**T** at those points.
- If a parametric curve in the -plane has a
* vertical*
tangent vector **v**, write down the direction of
. Mark locations on Figure .B for which the
preimage of **T** is vertical.

** ACTIVITY 5: Preimages of Horizontal and Vertical Tangent Vectors**
- What equation must a tangent vector at ) satisfy if it is
mapped by
**DP** to a horizontal vector? Verify your conjecture for
Figure .A, using the fact that the curve through has the tangent vector . (Why this point and vector?)
- What equation must a tangent vector at ) satisfy if it is
mapped by
**DP** to a vertical vector?
Verify your conjecture for Figure .A, using
the fact that the curve through
has the tangent vector
.

** ACTIVITY 6: Transforming Area**
What is the approximate area of a small square of side length **s**
based at the point ? Test your conjecture by applying the
matrix to the vectors and for , and **2**.

** ACTIVITY 7: Bonus Activity**
Prior to February 14th (Valentine's Day), figure out the parametrization
of a curve in the -plane whose image under **P** is a
heart-shaped curve (with two cusp points).

**Figure:** Two curves that are images under **P** of curves
in the -plane.

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**Copyright:** 1996 by the Regents of the University of Minnesota.

Department of Mathematics. All rights reserved.

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hesse@math.umn.edu

Last modified: Feb 20, 1997

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