I was reading the book Introduction to Manifold by Loring W Tu. And I am confused with a remark Tu made in his book. I need a little bit of clarification.
In Chapter 5 (differential forms), he wrote " Because integration of function on Euclidean space depends on a choice of coordinates and is not invariant under a change of coordinate, it is not possible to integrate functions on manifold. "
Can anyone tell me what does he mean by integration of function is not invariant under change of co ordinate in Euclidean space?
There is a comment on that in Lee's "Introduction to Smooth Manifolds": consider an $n$-dimensional cube $C$ in $\mathbb R^n$, and let $f:C\to\mathbb R$ be the constant function $f(x)=1$. Then, what should happen is $\displaystyle \int_CfdV=\rm{vol}(C)$, but the volume is not invariant under changes of coordinates.