Exercise 9.18

Let the mapping be denoted by $\check{F}=(f_1,f_2)$. Then $\check{F}$ maps the points $(x,0)$ and $(-x,0)$ on the $x$-axis to $(x^2,0)$ on the positive $u$-axis, and for a fixed $y_0\ne 0$, $\check{F}$ maps the lines $(x,y_0)$ and $(x,-y_0)$ to the parabola $u=v^2∕{(4y_0)}^2-y_0^2$, which is symmetric with respect to the $u$-axis and passes through and otherwise lies to the left of the point $(-y_0^2,0)$. If $y_0>0$, then the points $(x,y_0)$, $x>0$, are mapped to the upper branch of the parabola, and the points $(x,y_0)$, $x<0$, are mapped to the lower branch, and the opposite is true if $y_0<0$.

Similarly, $\check{F}$ maps the point $(0,y)$ and $(0,-y)$ on the $y$-axis to $(-y^2,0)$ on the $u$-axis, and for a fixed $x_0\ne 0$, $\check{F}$ maps the lines $(x_0,y)$ and $(-x_0,y)$ to the parabola $u=-v^2∕{(4x_0)}^2+x_0^2$, which is symmetric with respect to the $u$-axis and passes through and otherwise lies to the right of the point $(x_0^2,0)$. If $x_0>0$, then the points $(x_0,y)$, $y>0$, are mapped to the upper branch of the parabola, and the points $(x_0,y)$, $y<0$, are mapped to the lower branch, and the opposite is true if $x_0<0$.

Other than mapping the origin in the $x$- $y$ plane to the origin in the $u$- $v$ plane, $\check{F}$ maps two distinct points in the $x$- $y$ plane to each point in the $u$- $v$ plane outside the origin. If we let $w=\sqrt{u^2+v^2}$ be the distance from the origin to the point $(u,v)$, then

are mapped by $\check{F}$ to the point $(u,v)$ if $v$ is positive, and

are mapped by $\check{F}$ to the point $(u,v)$ if $v$ is negative.

Since

the Jacobian of $\check{f}$ at $(x,y)$ is $4(x^2+y^2)$ which is nonzero except at the origin. If we exclude the origin in both planes, then $\check{F}$ is locally one-to-one, but globally two-to-one.

Letting $\check{b}=(3,4)$, then $\check{F}$ maps the point $ǎ=(2,1)$ to $\check{b}$. Again letting $w=\sqrt{u^2+v^2}$ be the distance from the origin to the point $(u,v)$, locally $\check{F}$ has the inverse function