1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Vector Calculus Question (I don't understand)

  1. Jul 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Let D* = [0,1]x[0,1] and define T on D* by T(u,v)=(-u^2+4u, v). Find the image D. Is T one-to one

    2. Relevant equations

    3. The attempt at a solution

    I have no idea... I don't know how to do it.
    The solution is [0,3] x [0,1]... yes it is one to one.
    am I supposed to say det A does not equal zero or use the u=u' v=v' approach?
    and... how do I find D? i don't know how to go about this.
    Could someone give me a similar example and solve that?
  2. jcsd
  3. Jul 17, 2010 #2


    Staff: Mentor

    This seems pretty straightforward to me, as the image vector doesn't have u and v tangled together. For fixed u, T maps v to v. For fixed v, T maps u to -u^2 + 4u, and this graph is a parabola.

    This is kind of a simple-minded way to look at this problem, but I think it will work.
  4. Jul 17, 2010 #3
    Okay... so I took your advice.

    so I thought about what you said

    so T(u,v) maps u => -u^2 + 4u
    and v to v

    so for u [0,1] is the interval... so that means [0, -1+4] = [0,3]

    for v [0,1] goes to [0,1]

    [0,3] x [0,1] so that means... it maps the square into a rectangle.

    But... how do I show that the mapping is one-to one??
  5. Jul 17, 2010 #4


    Staff: Mentor

    Let's call the outputs (w, z), so that T(u, v) = (w, z), with w = -u2 + 4u and z = v. Show that (w1, z1) = (w2, z2) ==> (u1, v1) = (u2, v2).

    It's also helpful to look at the portion of the parabola for which 0 <= u <= 1. Quadratic functions aren't normally one-to-one, but if the domain is limited in the right way, the limit domain version can be one-to-one.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Vector Calculus Question Date
Vector Calculus HW question Dec 18, 2015
Some vector calculus questions Oct 13, 2015
Vector calculus question Oct 17, 2014