Show that a particular mapping of an open ball in $\mathbb{R}^k$ onto $\mathbb{R}^k$ is a diffeomorphism.

Question

This is a problem from Guilleman and Pollack: Page 5, question 4.

Let $B_a$ be the open ball $\{ x: |x|^2 < a \}$ in $\mathbb{R}^k$, where $|x|^2 = \sum_i x_i^2$.

Show that the map

\begin{equation} x \mapsto \frac{ax}{\sqrt{a^2 - |x|^2}} \end{equation}

is a diffeomorphism of $B_a$ onto $\mathbb{R}^k$.

My question is mostly about the language of this problem. The author says show that this is a map ${\bf onto}$ $\mathbb{R}^k$. Does this mean it is supposed to be a ${\bf surjective}$ map? If so, shouldn't the definition of the ball be such that $\{ x: |x|^2 < a^2 \}$, so that it is a ball of radius $a$, and points near the boundary get sent to infinity in $\mathbb{R}^k$?

Answer 1

This is listed as the first typo of the book by Ted Shifrin here, so at least Ted agree with you.