Multivariable Dirac Delta and Faddeev-Popov Determinant From this mathstack page and in particular Qmechanic's answer:

  
*
  
*There exists an $n$-dimensional generalization $$\tag{1} \delta^n({\bf f}({\bf x})) ~=~\sum_{{\bf x}_{(0)}}^{{\bf f}({\bf
x}_{(0)})=0}\frac{1}{|\det\frac{\partial {\bf f}({\bf x})}{\partial
 {\bf x}} |}\delta^n({\bf x}-{\bf x}_{(0)}) $$ of the substitution
  formula for the Dirac delta
  distribution under
  pertinent assumptions, such as e.g., that the function ${\bf f}:\Omega
 \subseteq \mathbb{R}^n \to \mathbb{R}^n$ has isolated zeros. Here the
  sum on the rhs. of eq. (1) extends to all the zeros ${\bf x}_{(0)}$ of
  the function ${\bf f}$.
  

Also from this page on the Faddeev-Popov procedure they say: 

For ordinary functions, a property of the Dirac delta function gives: $\delta(x-x_0) =
 \left|\frac{df(x)}{dx}\right|_{x=x_0}\delta(f(x))\,$ assuming $f(x)\,$
  has only one zero at $x=x_0\,$ and is differentiable there.
  Integrating both sides gives :$$1 =
 \left|\frac{df(x)}{dx}\right|_{x=x_0}\int\!dx\,\delta(f(x))\,$$.
  Extending over $n$ variables, suppose $f(x^i) = 0\,$ for some
   $x^i_0\,$. Then, replacing $\delta(x-x_0)\,$ with $\prod_i^n
 \delta^i(x^i-x^i_0)\,$ :$$1 = \left(\prod_i \left|\frac{\partial
 f(x^i)}{\partial x^i}\right|\right)  \int\!\left(\prod_i
 dx^i\right)\,\delta(f(x^i))\,$$. 
  Recognizing the first factor as the
   determinant of the diagonal matrix $\frac{\partial
 f(x^i)}{\partial x^i}\delta^{ij}\,$ (no summation implied), we can
  generalize to the functional version of
  the identity: :$$1 = \det\left|\frac{\delta G}{\delta
 \Omega}\right|_{G=0}
 \int\!\mathcal{D}\Omega\,\delta[G_a(\phi^\Omega)]\,$$, where
   $\Delta_F[\phi] \equiv \det\left|\frac{\delta F}{\delta
 g}\right|_{F=0}\,$ is the Faddeev-Popov
   determinant.

What I don't understand is that it seems their function $f$ seems to be $f:\Omega
 \subseteq \mathbb{R}^n \to \mathbb{R}$. How does the generalized Dirac formula $(1)$ work in this case? I don't really understad their notation in: 
$$1 = \left(\prod_i \left|\frac{\partial
 f(x^i)}{\partial x^i}\right|\right)  \int\!\left(\prod_i
 dx^i\right)\,\delta(f(x^i))\,$$ 
What does $$\frac{\partial
 f(x^i)}{\partial x^i}$$ mean here?
 A: The notation 
\begin{equation} 
\frac{ \partial f_i}{ \partial x ^i }
\end{equation} 
means the diagonal elements of the matrix:
\begin{equation} 
J _{ ij} = \frac{ \partial f _i }{ \partial x ^j }
\end{equation} 
where $f_i$ is the component of the vector $\vec{f} (x)$.

I found this very confusing a few weeks ago so. Here is the proof I wrote up for the identity based on the response I received to an earlier question of mine here:
Recall that if $ f (x) $ has one zero at $ x _0 $ then,
\begin{equation} 
\int d x \left| \frac{ df (x) }{ d x } \right|  _{ x = x _0  } \delta \left( f (x) \right) =   1
\end{equation}
 We want to generalize this to instead of having $ f (x) $ we have, $ {\mathbf{g}} ( {\mathbf{a}} ) $ for vectors of arbitrary size. To do this consider the Taylor expansion of $ {\mathbf{g}} $ around its root (we assume it only has one root, $ {\mathbf{a}} _0 $):
\begin{equation} 
g _i  ( {\mathbf{a}} ) = \overbrace{g _i ( {\mathbf{a}} _0 )}^0 + \sum _{ j} \frac{ \partial g _i }{ \partial a _j } \bigg|_{ a _0 } ( a _j - a _{ 0,j }) + ...
\end{equation} 
We want to insert this into a delta function, $ \delta ^{ ( n ) } ( {\mathbf{g}} ( {\mathbf{a}} ) ) $. This will only be nonzero near $ {\mathbf{a}} = {\mathbf{a}} _0 $. Thus we have,
\begin{align} 
\delta \left( {\mathbf{g}}  ( {\mathbf{a}} ) \right) & = \prod _i \delta \left( g _i ( {\mathbf{a}} ) \right) \\ 
&   =   \prod _i \delta \big( \sum _j J _{ ij} ( a _j - a _{ 0,j} )  \big)  
\end{align} 
where $ J _{ ij} $ is the Jacobian matrix defined by $ J _{ ij} \equiv \frac{ \partial g _{ i} }{  \partial a _j  } \big|_{ a _0 } $. We have,
\begin{align} 
\delta \left( {\mathbf{g}}  ( {\mathbf{a}} ) \right) & =   \delta \big( \sum _j J _{ 1j} ( a _j - a _{ 0,j} )  \big)  \delta \big( \sum _j J _{ 2j} ( a _j - a _{ 0,j} )  \big) ... 
\end{align} 
We now use the identity,
\begin{equation} 
\delta ( \alpha ( a - a _0 ) ) = \frac{ \delta ( a - a _0 ) }{ \left| \alpha \right| }
\end{equation} 
We choose to isolate each delta function in the equation above for a different $ a _j $:
\begin{align} 
\delta \big( {\mathbf{g}}  ( {\mathbf{a}} ) \big) & =   \frac{ \delta ( a _1 - a _{ 0,1 }  ) }{ \left| J _{ 1,1 } \right| }  \frac{ \delta ( a _2 - a _{ 0,2 }  ) }{ \left| J _{ 2,2 } \right| } ...
\end{align} 
If we take the Jacobian matrix to be greater then zero then we have the product:
\begin{equation} 
(  J _{ 1,1 } J _{ 2,2} .. ) ^{-1} = \frac{1}{ \det J } 
\end{equation} 
where we have used the fact that the determinant of $ J $ is independent of a unitary transformation. So we finally have,
\begin{align} 
\left( \int \prod _{ i} d a _i \right)  \delta ^{ ( n ) } \big( {\mathbf{g}}  ( {\mathbf{a}} ) \big) \det \big( \frac{ \partial g _i }{ \partial a _j  } \big)  & =   1
\end{align} 
where it is understood that the Jacobian matrix is evaluated at the root of $ {\mathbf{g}}  $.
We write the continuum generalization of this equation as,
\begin{equation} 
\int {\cal D} \alpha (x) \delta \left( G ( A ^\alpha ) \right) \det \left( \frac{ \delta G ( A ^\alpha ) }{ \delta \alpha } \right) = 1
\end{equation}  
A: Again assuming it only has a zero $x^i=x_0^i$ what you have is
$$
\delta(f(x^i)) = \frac{\delta(x^1-x_0^1)}{\left|\frac{\partial f}{\partial x^1}\right|_{x^i=x_0^i}} \frac{\delta(x^2-x_0^2)}{\left|\frac{\partial f}{\partial x^2}\right|_{x^i=x_0^i}}\cdots \frac{\delta(x^n-x_0^n)}{\left|\frac{\partial f}{\partial x^n}\right|_{x^i=x_0^i}} = \prod_{j=1}^n \frac{\delta(x^j-x_0^j)}{\left|\frac{\partial f}{\partial x^j}\right|_{x^i=x_0^i}} =  \frac{\prod_{j=1}^n\delta(x^j-x_0^j)}{\prod_{j=1}^n\left|\frac{\partial f}{\partial x^j}\right|_{x^i=x_0^i}}
$$
then
$$
\prod_{j=1}^n\left|\frac{\partial f}{\partial x^j}\right|_{x^i=x_0^i} \delta(f(x)) = \prod_{j=1}^n\delta(x^j-x_0^j)
$$
and integrating at both sides in all the variables $\int\left(\prod_{j=1}^n dx^j\right)$ you get
$$
\int\left(\prod_{j=1}^n dx^j\right)\prod_{j=1}^n\delta(x^j-x_0^j)=\int\prod_{j=1}^n dx^j\delta(x^j-x_0^j)=1
$$
and since the derivatives are evaluated in the zero and are just numbers
$$
\int\left(\prod_{j=1}^n dx^j\right)\prod_{j=1}^n\left|\frac{\partial f}{\partial x^j}\right|_{x^i=x_0^i} \delta(f(x)) = \prod_{j=1}^n\left|\frac{\partial f}{\partial x^j}\right|_{x^i=x_0^i}\int\left(\prod_{j=1}^n dx^j\right) \delta(f(x^i)) 
$$
so finally we get
$$
1== \prod_{j=1}^n\left|\frac{\partial f}{\partial x^j}\right|_{x^i=x_0^i}\int\left(\prod_{j=1}^n dx^j\right) \delta(f(x^i)) 
$$
The notation is a bit clumsy but I believe this is what you were looking for.
A: What confused me was the explanation from the tangentbundle homepage (second yellow box in OP). The generalization is straightforward, for simple zeros we have: 
$$\bigg\vert\frac{\mathrm{d}f(x)}{\mathrm{d}x}\bigg\vert_{x_0}\delta[f(x)] = \delta(x-x_0)$$
integrate
$$\int \mathrm{d}x\,\bigg\vert\frac{\mathrm{d}f(x)}{\mathrm{d}x}\bigg\vert_{x_0}\delta[f(x)] = 1$$
generalize
$$\int \mathrm{d}{\bf x\,\bigg\vert \mathrm{det}\frac{\partial f(x)}{\partial x}\bigg\vert_{x_0}\delta[f(x)]} = 1$$
generalize
$$\int \mathcal{D}\alpha(x)\,\bigg\vert \mathrm{Det}\frac{\delta G(A^\alpha)}{\delta \alpha}\bigg\vert_{A^\alpha_0}\delta[G(A^\alpha)] = 1,$$
where it is to be understood that $A^\alpha_0$ is such that $G(A^\alpha_0)=0.$ 
