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I need help with an aspect of the proof of Theorem 3.3.13 ... ...

Theorem 3.3.13 (together with a relevant definition) reads as follows:

In the above text from McInerney we read the following:

" ... ... The fact that \(\displaystyle \phi\) has rank \(\displaystyle n -1 \) follows by computing the Jacobian matrix at any point in \(\displaystyle U\) ... ... "

Can someone please demonstrate rigorously, formally and explicitly that \(\displaystyle \phi\) has rank \(\displaystyle n -1\) ... ...

My computations with respect to the Jacobian \(\displaystyle [D \phi(p) ]\) were as follows:

We have \(\displaystyle \phi ( x_1, \ ... \ ... \ x_{n-1} ) = ( x_1, \ ... \ ... \ x_{n-1}, f( x_1, \ ... \ ... \ x_{n-1}) ) \)

Now put ...

\(\displaystyle f_1( x_1, \ ... \ ... \ x_{n-1} ) = x_1\)

\(\displaystyle f_2( x_1, \ ... \ ... \ x_{n-1} ) = x_2\)

... ... ...

... ... ...

\(\displaystyle f_{n-1}( x_1, \ ... \ ... \ x_{n-1} ) = x_{n-1}\)

\(\displaystyle f_n( x_1, \ ... \ ... \ x_{n-1} ) = f( x_1, \ ... \ ... \ x_{n-1} )\)

Then ... the Jacobian ...

\(\displaystyle [D \phi(p) ] = \begin{bmatrix} \frac{ \partial f_1 }{ \partial x_1} & ... & ... & \frac{ \partial f_1 }{ \partial x_{n-1} } \\ ... & ... & ... & ... \\ ... & ... & ... & ... \\ \frac{ \partial f_{n-1} }{ \partial x_1} & ... & ... & \frac{ \partial f_{n-1} }{ \partial x_{n-1} } \\ \frac{ \partial f }{ \partial x_1} & ... & ... & \frac{ \partial f }{ \partial x_{n-1} } \end{bmatrix}\)

= \begin{bmatrix} 1 & 0 & 0 & ... & ... & 0 \\ 0 & 1 & 0 & ... & ... & 0 \\ ... & ... & ... & ... & ... & ... \\ ... & ... & ... & ... & ... & ... \\ \frac{ \partial f }{ \partial x_1} & ... & ... & ... & ... & \frac{ \partial f }{ \partial x_{n-1} } \end{bmatrix}

... now ... how do we show that the rank of \(\displaystyle [D \phi(p) ]\) is \(\displaystyle n-1\) ...?

Hope someone can help ...

Peter