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.