Tuesday, March 26 ∗∗ Transformations of R2.Purpose: In class, we’ve seen several different coordinate systems on R2and R3beyond the usualrectangular ones: polar, cylindrical, and spherical. The lectures on Friday and Monday will cover thecrucial technique of simplifying hard integrals using a change of coordinates (Section 15.9). The pointof this worksheet is to familiarize you with some basic concepts and examples for this process.Starting point: Here we consider a variety of transformations T : R2→ R2. Previously, we have usedsuch functions to describe vector fields on the plane, but we can also use them to describe ways ofdistorting the plane:T1. Consider the transformation T (x, y) = (x − 2y,x + 2y).(a) Compute the image under T of each vertex in the below grid and make a careful plot ofthem, which should be fairly large as you will add to it later.To speed this up, divide the task up among all members of the group.yx(0,0)(−1,2) (2,2)(−1,−1) (2,−1)(b) For each pair A and B of vertices of the grid joined by a line, add the line segment joiningT (A) to T (B) to your plot. This gives a rough picture of what T is doing.Check your answer with the instructor.(c) What is the image of the x-axis under T ? The y-axis?(d) Consider the line L given by x + y = 1. What is the image of L under T ? Is it a circle, anellipse, a hyperbole, or something else?Hint: First, parameterize L by r: R → R2and then consider f(t ) = T (r(t)).(e) Consider the circle C given by x2+ y2= 1. What is the image of C under T ?(f) Add T (L), T (C) and T¡ ¢to your picture. Check your answer with the instructor.Note: The transformation T is a particularly simple sort called a linear transformation.2. Consider the transformation T (x, y) =¡y, x(1+ y2)¢. Draw the image of the picture below underT .xy(1,1)Hint: Parameterize each of the 5 line segments and proceed as in 1(d). To speed things, divideup the task.Check your answer with the instructor.3. In this problem, you’ll construct a transformation T : R2→ R2which rotates counter-clockwiseabout the origin by π/4, as shown below.T(a) Give a formula for T in terms of polar coordinates. That is, how does rotation affect r andθ?(b) Write down T in terms of the usual rectangular (x, y) coordinates. Hint: first convert intopolar, apply part (a) and then convert back into rectangular coordinates.Check your answer with the
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