Can someone please help me with the following definition:
$B$ is a bounded open set in $R^n$ and $g$ : $\bar{B} \rightarrow R^n$ is $C^1$ . We say $a$ is a regular value of $g$ if $\det{(g'(x))}\neq 0$, whenever $x \in B$ and $g(x)=a$.
I don't understand the meaning of determinant of a derivative of a function. If $f'$ was a linear function then it would make sense as you could represent it as a matrix but otherwise I don't understand what is going on.
The derivative $g'(x)$ is sometimes called Jacobi matrix for a reason: This object is a $n\times n$ matrix containing all the partial derivatives $\frac{\partial g_i}{\partial x_j}$.