I am reading Pavel Grinfeld's "introduction to tensor analysis and the calculus of moving surfaces" and have some confusion in the following pages:
The author says:"The definition of the integral (14.8) involves a limiting process, in which the domain (whether it is $\Omega$ , S, or U) is divided into small pieces and a finite sum approaches a limit. This procedure is nontrivial, but its independence from coordinates is evident."
How can integral over a domain be invariant? Consider the following:
Even though the value of function at each point in the domain is same in all coordinates, the infinetesimal pieces (which we divided from domain) is not the same in all coordinates.
In Cartesian coordinates, we divide our domain into equal infinitesimal pieces as shown below:
In Circular coordinates, we divide our domain into infinitesimal pieces which substend equal infinitesimal angles from the centre of circular coordinate system, as shown below:
The infinetesimal pieces (which we divided from domain) being not the same in all coordinates, how can we say the integral over the domain is invariant?
It is true that the infinitesimal pieces are different when you switch coordinates, but the integrand is the same. In your example for instance you forgot to account for the Jacobbian of the transformation
$$ {\rm d}\theta\frac{{\rm d}x}{{\rm d}\theta} = {\rm d}x $$
So that
$$ {\rm d}x\;f(x) = {\rm d}\theta\frac{{\rm d}x}{{\rm d}\theta}\;f(\theta) $$