Tuesday October 30 Transformations of R2 Purpose In class we ve seen several different coordinate systems on R2 and R3 beyond the usual rectangular ones polar cylindrical and spherical The lectures on Friday and Monday will cover the crucial technique of simplifying hard integrals using a change of coordinates Section 15 9 The point of 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 used such functions to describe vector fields on the plane but we can also use them to describe ways of distorting the plane T 1 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 of them 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 y 1 2 2 2 x 0 0 1 1 2 1 b For each pair A and B of vertices of the grid joined by a line add the line segment joining T 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 an ellipse a hyperbole or something else Hint First parameterize L by r R R2 and then consider f t T r t e Consider the circle C given by x 2 y 2 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 y 2 Draw the image of the picture below under T y 1 1 x Hint Parameterize each of the 5 line segments and proceed as in 1 d To speed things divide up the task Check your answer with the instructor 3 In this problem you ll construct a transformation T R2 R2 which rotates counter clockwise about the origin by 4 as shown below T a Give a formula for T in terms of polar coordinates That is how does rotation affect r and b Write down T in terms of the usual rectangular x y coordinates Hint first convert into polar apply part a and then convert back into rectangular coordinates Check your answer with the instructor
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