Math Is Fun Forum

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

show transformation maps a region to another region

Hello all,

Question is:

Show that (x,y) --> (x^2 - y^2, 2xy) maps the first quadrant {x>0, y>0}into the upper half plane {y>0}.

Now if i were to "brute force" this then I can easily do it. By "brute force" i mean I consider any arbitrary (x', y') in the upper half plane and show this point can be expressed as (x^2 - y^2, 2xy) where x>0, y>0 (first quadrant). This shows that every point in the upper half plane can be obtained. Then show that if (x,y) is chosen in the first quadrant, then the transformation cannot give a point in the lower half half plane.

So done. However, its not beautiful.

In the book they hint to use polar coordinates and then in the answers in the back they say:

(x,y) --> (x^2 - y^2, 2xy) becomes:
(r, 0) --> (r^2, 2*theta)

and I just dont see how?

Thanks for any help.

Last edited by LuisRodg (2011-07-30 05:35:46)

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: show transformation maps a region to another region

Hi LuisRodg,

I'm not sure whether my explanation is okay, anyway...

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: show transformation maps a region to another region

Thanks so much. I had worked up to the part:

And i didnt know how to continue. However, you said the above implied the following:

And I still dont know how? Yes, I see that the r is becoming r^2 and the angle of sin and cos is becoming double, but is there a way to keep working algebraicly until the last statement appears?

Thanks.

Last edited by LuisRodg (2011-07-30 06:22:45)

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: show transformation maps a region to another region

I don't know, it just appears obvious to me, cannot think of any more steps!

Is there any example in your problem source?

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Bob

Administrator

Registered: 2010-06-20

Posts: 10,127

Re: show transformation maps a region to another region

hi all,

That all seems ok to me.  But, to make the proof fully rigorous, you need to consider the domains for cos theta and sin theta, and the corresponding ranges.  OR talk about theta -> two theta meaning the 'first plus second quadrant' is now the range of the function.  It's probably already obvious to you, but you need to spell it out, I think.

Bob

Last edited by Bob (2011-07-30 20:27:00)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob