Transformation of Coordinate Systems

[minderbinder]

Homework Statement

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

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

Homework Equations

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

The Attempt at a Solution

I'm at a loss of where to begin...

 

[minderbinder]

Thought about this some more, and I think the solution should be:

(u, v) = f(x, y) = ($$\frac{y}{x^2}$$, 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?

 

[compliant]

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?