1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?