Show that the function f(x,y) = (x² - y², 2xy) is 1-1 on the the set A where x > 0.
The problem gives a hint: If f(x,y) = f(a,b), then ||f(x,y)|| = ||f(a,b)||, where || || is the euclidean norm on R².
The Attempt at a Solution
I suppose this shouldn't be hard, but I don't know how to do it. So showing that f(x,y) is 1-1 means that if f(x,y) = f(a,b) then (x,y) = (a,b). So using the hint I have:
||(x² - y², 2xy)|| = ||(a² - b², 2ab)||
Expanding I have:
(x² - y²)² + (2xy)² = (a² - b²)² + (2ab²); I'll ignore the square roots
But here I've tried factoring it in tons of different ways to show that x = a (and somehow using that x, a > 0) and y = b. But I don't see it. The next problem says that the hint is also once again useful, so could maybe someone help me out? Oh and please don't use complex analysis as I do know this is z², but I'd like to do it using the hint. I'm studying for a term test for tomorrow.
Another hint would probably be enough...just give me something! lol