In this page,(from Smooth Manifolds by John Lee), it is stated that we cannot define multiple integrals simply as integrals of functions in a coordinate independent way.
To show this, he takes $f(x)=1$ and defines the integral $\int f dv$ on the $n$-dimensional cube and shows that the volume is not invariant under coordinate change.
I have a couple of doubts regarding this:-
- What is an $n$-dimensional cube? Is it definition is $0\leq x_1\leq 1, 0\leq x_2\leq 1\cdots 0\leq x_n\leq 1$, where $x_i$ are coordinates in a particular coordinate system?
- Also, is it saying that if we take $n=3$ and instead of orthogonal coordinates, we take axes that make angles other than $90°$ with each other(skew coordinates)?. Now, here a cube will be a parallelopiped whose volume is different from the original one in Cartesian coordinates as if we follow the definition. (Note that the definition of cube says that in both coordinate systems, we are allowed to move only a unit distance via each parameter)
P.s. I have stated the unit cube to be "cube'.