Results 1 to 2 of 2

Math Help - F(x,y)=(x^{2}-y^{2}, 2xy)

  1. #1
    Newbie
    Joined
    Mar 2009
    From
    São Paulo- Brazil
    Posts
    22

    F(x,y)=(x^{2}-y^{2}, 2xy)

    How do I show that the map F(x,y)=(x^{2}-y^{2}, 2xy) is injective for y>0?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Quote Originally Posted by Biscaim View Post
    How do I show that the map F(x,y)=(x^{2}-y^{2}, 2xy) is injective for y>0?
    You need to show that for every (x_1,y_1) and (x_1,y_1) where y1 > 0 and y2 >0
    if F(x_1,y_1)=F(x_2,y_2) then (x_1,y_1)=(x_2,y_2)

    x_1^{2}-y_1^{2} = x_2^{2}-y_2^{2}
    2x_1y_1 = 2x_2y_2

    y_1^{2}-y_2^{2} = x_1^{2}-x_2^{2} = x_1^{2}-\frac{x_1^2y_1^2}{y_2^2} = x_1^{2}\:\left(1-\frac{y_1^2}{y_2^2}\right) = x_1^{2}\:\frac{y_2^2-y_1^2}{y_2^2}

    (y_1^{2}-y_2^{2})\left(1+ \frac{x_1^2}{y_2^2}\right)=0

    y_1^{2}-y_2^{2}=0

    y_1-y_2=0 since they are both > 0
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum