How Do You Determine the Ranges of u and v in Multivariable Transformations?

[kingwinner]

I was reading a statistics book, and part of the problem reduces to the calculus problem of doing the following:

1) Let u=x/y, v=y, with domain 0<x<y<1[/color], how to find the ranges of u and v after the transformation?


2) Let u=x/(x+y), v=x+y with domain x>1, y>1[/color], what values can u and v take on?


Is there a systematic way to do these?

Thank you for any help!

 

[kingwinner]

How can I determine step-by-step the corresponding region in the uv-plane after the transformation?

 

[kingwinner]

Please help...

 

[HallsofIvy]

I was reading a statistics book, and part of the problem reduces to the calculus problem of doing the following:

1) Let u=x/y, v=y, with domain 0<x<y<1[/color], how to find the ranges of u and v after the transformation?

The range of v should be obvious. Is it possible for u to be negative? Is it possible for u to be 0? So what is a lower bound for u? Since y can be as close to 0 as you please is there an upper bound on u?

Another, more general, way to do this is to look at the boundary lines. If x= 1, then u= 1/y and v= y so u= 1/v. Graph u= 1/v on the uv-plane. If y= 1, then u= x and v= 1. Graph v= 1 on the uv-plane. If x= 0, u= 0, v= y. Graph u= 0 on the uv-plane. If y= 0, u is infinite so that does not give a bound. What region is inside those boundaries?

2) Let u=x/(x+y), v=x+y with domain x>1, y>1[/color], what values can u and v take on?

Look at the boundary lines x= 1 and y= 1. On x= 1, you have u= 1/(y+1) and v= y+1. That is, u= 1/v. On y= 1, you have u= x/(x+1) and v= x+1. x= v-1 so u= (v-1)/v= 1- 1/v. Graph those curves on the u-v plane.

Is there a systematic way to do these?

Thank you for any help!

Yes,