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Transformation of Coordinate Systems

  1. Jul 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Find a one-to-one C1 mapping [tex]f[/tex] from the first quadrant of the xy-plane to the first quadrant of the uv-plane such that the region where [tex]x^2 \leq y \leq 2x^2[/tex] and [tex]1 \leq xy \leq 3[/tex] is mapped to a rectangle. Compute the Jacobian det Df and the inverse mapping [tex]f^{-1}[/tex].

    The hint for the question states: Map all the regions where [tex]ax^2 \leq y \leq bx^2[/tex] and [tex]c \leq xy \leq d[/tex] to rectangles.

    2. Relevant equations

    I'm a little confused on what they mean by map to a rectangle.

    3. The attempt at a solution

    I'm at a loss of where to begin...
     
  2. jcsd
  3. Jul 1, 2009 #2
    Thought about this some more, and I think the solution should be:

    (u, v) = f(x, y) = ([tex] \frac{y}{x^2}[/tex], xy)

    I checked some coordinates and it appears to work. However, I got this solution through trial and error. Can someone point out to me a way to find the solution in a systematic way?
     
  4. Jul 1, 2009 #3
    let's start with the second inequality. xy goes from 1 to 3. this forms one dimension of a rectangle - along the xy-axis. but instead of using an xy-axis, you could use a u-axis or v-axis, if you let u or v equal to xy. Hint hint.

    now for the first inequality, if y goes from x^2 and 2x^2, is there a way to manipulate this so that the value of *some algebraic expression* goes from one integer to another? kinda like xy above?
     
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