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Homework Help: Sketch the region R=T(S) in xy-space

  1. Mar 26, 2008 #1
    1. The problem statement, all variables and given/known data

    Consider the rectangle S=[0,1]x[0,[tex]\pi[/tex]/2]
    Sketch the region R=T(S) in xy-space.


    2. Relevant equations
    T(r,[tex]\theta[/tex]) = (rcos[tex]\theta[/tex],rsin[tex]\theta[/tex])

    3. The attempt at a solution
    how is the given a rectangle in polar coordinates? it seems to me to be a quarter circle
     
    Last edited: Mar 26, 2008
  2. jcsd
  3. Mar 26, 2008 #2

    Dick

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    S is a 'rectangle' in cartesian [r,theta] space. The image of T in cartesian [x,y] space is, indeed, a quarter circle. If you regard T as a transformation between polar r-theta coordinates and cartesian x-y coordinates then it is really a quarter circle in both. I'll admit the difference is a bit confusing if you are used to thinking of r,theta as polar coordinates. If it helps try thinking of T(x,y)=(x*cos(y),x*sin(y)) both in cartesian coordinates. That really does take a rectangle into a quarter circle.
     
    Last edited: Mar 26, 2008
  4. Mar 27, 2008 #3

    tiny-tim

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    Welcome to PF!

    Hi stvnseagal! Welcome to PF! :smile:

    I think the question is just trying to confuse you …

    Technically, "rectangle" means all its angles are right-angles - which they are! :smile:

    It's just that there's only three of them!

    :smile: Don't worry! :smile:
     
  5. Mar 27, 2008 #4

    HallsofIvy

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    It is a rectangle in "r, [itex]\theta[/itex]" space, not in x,y space. Since the two vertices (0, 0) and (0, [itex]\pi/2[/itex]) both have r= 0, T transforms both of them into the single point (0,0) in x,y space. (1, 0) in r, [itex]\theta[/itex] space is transformed into (1, 0) in xy space and (0, 1) in r, [itex]\theta[/itex] space is transformed into (0, 1) in xy space.

    Tiny Tim, "technically" a rectangle has four vertices! As I said, the "rectangle" part on applies to r, [itex]\theta[/itex] space.
     
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