1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: One-to-One in R2

  1. Aug 13, 2008 #1
    1. The problem statement, all variables and given/known data
    Let f = (f1,f2) be the mapping of R2 into R2 given by f1=excos(y), f2=exsin(y).
    (1) What is the range of f?
    (2) Show that every point of R2 has a neighborhood in which f is one-to-one.
    (3) Show that f is not one-to-one on R2.

    3. The attempt at a solution
    (1) Since ex is nonzero and cos and sin cannot be 0 at the same time, then the origin is not in the range, so the range is R2\{0,0}.

    (2) Calculating the Jacobian we get:
    J = e2xcos2y + e2xsin2y= e2x, which is nonzero, so every point has a neighborhood in which f is one-to-one.

    (3) This is where I'm really stuck. I assume we need to find a point that is mapped by two different ordered pairs but I don't know how to go about finding those points.

    Any help on #3 or comments on the previous parts is appreciated. Thanks.
  2. jcsd
  3. Aug 13, 2008 #2


    User Avatar
    Science Advisor

    Think about [itex]y_2= y_1+ 2\pi[/itex].
  4. Aug 13, 2008 #3
    Oh, duh. I guess it didn't occur to me to hold x constant, so the points (0,pi) and (0,3*pi) both map to (-1,0), hence f is not one-to-one. Thanks.

    Does the rest look okay?
  5. Aug 13, 2008 #4
    (1) and (2) looks fine to me.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook