Theorem 5-2 Spivak's Calulus on Manifolds 

My question is :
1) How to show  $H(f(y)=y$?$H(f(y))=(h^1(f(y)),\dots,h^k(f(y)))=?$
2)How we get $det g'(a,b)=det(D_jf^i(a))$ 
Thanks in advance!!
 A: \begin{align*}
H(f(y)) &= H(h^{-1}(\underbrace{y}_{\in \Bbb{R}^k},\underbrace{0}_{\in \Bbb{R}^{n-k}}))  \\
    &= (h^1(h^{-1}(y,0)), \dots, h^k(h^{-1}(y,0)))  \\
    &= (y^1, \dots, y^k)  \\
    &= y  \text{.}
\end{align*}
For the other, note that $\frac{\partial g}{\partial a} = \begin{pmatrix} (D_j f^i(y) ) \\ 0 \end{pmatrix}$ and $\frac{\partial g}{\partial b} = \begin{pmatrix} 0 \\ \Bbb{I}_{n-k} \end{pmatrix}$, so $g' = \begin{pmatrix} (D_j f^i(y) )_{i=1}^k & (D_j f^i(y) )_{i=k+1}^n \\ 0 & \Bbb{I}_{n-k} \end{pmatrix}$
\begin{align*}
\det g' &= \det (D_j f^i(y) )_{i=1}^k \det \Bbb{I}_{n-k} - \det (D_j f^i(y) )_{i=k+1}^n \cdot 0  \\
    &= \det (D_j f^i(y) )_{i=1}^k  \\
    &= \det (D_j f^i(y) )  \text{,}  
\end{align*}
where the last equality is perhaps most easily seen by the evaluation of the determinant by recursive minors starting with all the $1$s in the $\Bbb{I}_{n-k}$.
It might be helpful to mentally tag the various objects as they are made: 


*

*$U$ is an open neighborhood in $\Bbb{R}^n$ of $x \in M$,

*$V$ is an open neighborhood in $\Bbb{R}^n$, with the convenient property that 

*$h$ maps $U$ to $V$, flattening $M$ out so that the image of $M$ lies on the subspace with last $n-k$ coordinates equal to zero, and maps points of $U \smallsetminus M$ "above" and "below" that subspace,

*$W$ is that image of $U \cap M$ in $\Bbb{R}^k$ where we drop the $n-k$ superfluous zero coordinates,

*$f$ maps the $W$ patch of $\Bbb{R}^k$ to $M$ in $\Bbb{R}^n$,

*$H$ is the canonical projection of $h$ onto its first $k$ coordinates; this is another way to get from $U$ to $\Bbb{R}^k$, but it is far from invertible (it is not injective) and is not required to land exactly on $W$.  However, $W$ is mapped to itself pointwise by this projection, so $H$ must have $\mathrm{rank}\, H \geq \dim W = k$ on points of $U \cap M$ (and it can't be greater because $H$ maps to $\Bbb{R}^k$),

*$g$ maps the (filled) cylinder over $W$ by taking the $W$-part to its point of $U \cap M$ and just adding the last $n-k$ coordinates of its input to the last $n-k$ coordinates of that point of $U \cap M$.  This makes $g'$ a block matrix with $f'$ as the upper-left block, the identity matrix in the lower right, and zeroes in the other two blocks,

*...

