Transformation of a square

Question

Having fun with some integrals, I caught myself thinking about transforming of regions. So I have the following questions.

Suppose we have the square determined by inequalities $0<x<1, 0<y<1$ and a transformation rule $u=xy,v=x+y$.

The question is: what form will this square have in new coordinates $(u,v)$?

I tried to express $x$ and $y$ in terms of $u$ and $v$ and got the following: $$x=\frac{v-\sqrt{v^2-4u}}{2}$$ $$y=\frac{v+\sqrt{v^2-4u}}{2}$$

And I don't know what my next step should be.

Answer 1

Perhaps you can get some insight from mapping some specific sets

$(x, y_0)\mapsto (x y_0, x + y_0)$

where $0\leq y_0 \leq 1$ is a constant number. In the $xy$-plane this a horizontal line. In the $uv$-plane a couple of things can happen, if $y_0 =0$, we obtain a vertical line running through the origin. If $y_0\not = 0$ then $u=x y_0$ and $v = x + y_0$, equivalently

$$ u = y_0(v-y_0) $$

These are straight lines with slope $1/y_0$ and intercept $y_0$.

From this we then know we are bounded in the $uv$-plane from above by the line $v = y + 1$ which is the result of setting $y_0 = 1$. A simlar analysis can be done for points of the form $(x_0,y)$.

$(x,\alpha x)\mapsto (\alpha x^2, (\alpha + 1)x)$

In the $xy$-plane these are straight lines going through the origin with slope $\alpha > 0$. Since we want to be inside the unit square we need to put some constraints on the domain of $x$: $ x \leq 1 / (\alpha+ 1)$. In the $uv$-plane this is mapped to $u = \alpha x^2$ and $v = (\alpha + 1) x$, or equivalently $$u = \frac{\alpha}{(1 + \alpha)^2} v^2$$

These are parabolas opening along the $u$-axis. The maximum value of the coefficient happens with $\alpha=1$, in that case the parabola has the form $v=2u^{1/2}$.

From this we then learn we are bounded in the $uv$-plane from below by the parabola $v=2u^{1/2}$

Answer 2

Consider the four edges in turn and compute the corresponding $(u,v)$:

• $(x,0)\to(0,x)$, a vertical line segment from $(0,0)$ to $(0,1)$;

• $(x,1)\to(x,x+1)$, an oblique line segment from $(0,1)$ to $(1,2)$;

• $(0,y)\to(0,y)$, a vertical line segment from $(0,0)$ to $(0,1)$;

• $(1,y)\to(y,1+y)$, an oblique line segment from $(0,1)$ to $(1,2)$.

As you see, these loci are overlapping. If you draw the curves for $x$ or $y$ held constant, you will notice that they are line segments. Their envelope will describe a curve similar to an astroid, I guess.