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 
.
.
(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.
,
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.)
-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 
.
.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.
-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.
) 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?)
) 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
.
? Test your conjecture by applying the
matrix 
to the vectors 
and 
for 
, and 2.
-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.