Differential geometry, equation, exercise problem

Question

There is a problem in the text " O'neill- Elementary differential geometry 2ed.

In my opinion, some expressions in (a) are not correct. What's the exact meaning of following statement? " the equations can be solved for $u$ and $v$, with $(u,v)$ in $D$.
Does this mean the existence of solution ?

Answer 1

The statement

The equations $$p_1=x_1(u,v), \quad p_2=x_2(u,v), \quad p_3=x_3(u,v)$$ can be solved for $u$ and $v$, with $(u,v)$ in $D$.

is equivalent to

There exists $(u,v) \in D$ such that $$p_1=x_1(u,v), \quad p_2=x_2(u,v), \quad p_3=x_3(u,v).$$