I'm trying to understand the proof for variable substitution of multivariable integrals here, but I'm not really sure what is meant by $\Delta x \approx \left(\dfrac{\partial x}{\partial u}\right)_0\Delta u$, as it's said to be true by definition. But no definition is really given, and I'm thrown off by the subscripted $0$, which could mean anything. (evaluated at $0$? zeroth index?)
If anyone knows of a more comprehensible and rigorous proof, that would also be acceptable as an answer.
First look at the single variable case:
Suppose $f:U\to \mathbb R$ is differentiable at $a\in U$.
Then, $\left | \frac{f(a+\Delta x)-f(a)-f'(a)\Delta x}{\Delta x} \right |$ is small if $\Delta x$ is. This says that
$\Delta f = f(a+\Delta x)-f(a) \approx f'(x)\Delta x$ if $\Delta x$ is small. Thus, $\Delta f \approx f'(x)\Delta x$.
This is the usual tangent line approimation from Calulus I. In your case, $x=x(u,v)$ so if we apply the above argument to the $single variable$ function obtained by holding $v$ constant, we get precisely $\Delta x \approx \left(\dfrac{\partial x}{\partial u}\right)_0\Delta u$
To continue, now suppose you have a parameterization of an plane region in $\mathbb R^2$:
$\vec r=x(u,v)\vec i+y(u,v)\vec j.$
Then, just as in the single variable case, we can write
$\left | \frac{\left | \vec r(u+\Delta u,v)-\vec r(u,v)-\vec r_u\Delta u \right |}{\Delta u} \right |\Rightarrow \vec r(u+\Delta u,v)-\vec r(u,v)\approx \vec r_u\Delta u$
$\left | \frac{\left | \vec r(u,v+\Delta v)-\vec r(u,v)-\vec r_u\Delta v \right |}{\Delta v} \right |\Rightarrow \vec r(u,+\Delta v)-\vec r(u,v)\approx \vec r_u\Delta v$.
Thus, the area of the plane region approximated by the parallelogram whose vertices are $(\vec r(u,v),\vec r(u+\Delta u,v),r(u+\Delta u,v+\Delta v),r(u,v+\Delta v))$ is approximately equal to the magnitude of the cross product $r_u\Delta u\times r_v\Delta v$, from which we conclude that
$\vert r_u\Delta u\times r_v\Delta v\vert = \vert r_u\times r_v\vert\Delta u\Delta v$ and this is the familiar change of variables formula for double integrals.