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

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 ?

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    $\begingroup$ The meaning of "the equations can be solved for $u$ and $v$, with $(u,v)$ in $D$" is "there exists $(u,v) \in D$ such that $p_1 = x_1(u,v)$, $p_2 = x_2(u,v)$ and $p_3 = x_3(u,v)$". $\endgroup$ – posilon May 3 '16 at 9:51
  • $\begingroup$ @posilon/ Thank you so much. $\endgroup$ – Chris kim May 3 '16 at 13:03
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    $\begingroup$ You 're welcome. I'm posting my previous comment as an answer, so that the question can be closed. $\endgroup$ – posilon May 8 '16 at 3:31

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 $$p_1=x_1(u,v), \quad p_2=x_2(u,v), \quad p_3=x_3(u,v).$$


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