Just got to the culminating chapter in Munkres’s Analysis on Manifolds and I’ve been thrown for a loop.
The author is in the process of defining the integral of a k-form η on A, an open set of R$^k$. We know that η can be written uniquely in the form η = f dx$_1$ $\land$ ··· $\land$ dx$_k$ , where $\land$ is the wedge product, f: R$^k$ --> R, and dx$_i$ equals the elementary 1-form in R$^k$.
Munkres defines the integral as follows:
$\int_A η$ = $\int_A f$.
But I don’t understand what motivates this definition.
What throws me is that η(x) = f(x) · (dx$_1$ $\land$ ··· $\land$ dx$_k$)(x) where x is in A, and (dx$_1$ $\land$ ··· $\land$ dx$_k$)(x) is an alternating k-tensor in { (x;v) | v an element of R$^k$ }.
Supposing just for the sake of argument that f = 1, shouldn't
$\int_A η$ = $\int_A (dx_1 \land ··· \land dx_k)$?
In which case, what the heck is
$\int_A (dx_1 \land ··· \land dx_k)$?
For x an element of A, (dx$_1$ $\land$ ··· $\land$ dx$_k$)(x) is simply the name of a function -- in this case an alternating k-tensor. This alternating k-tensor has no arguments and thus no value that can be summed ( = integrated).
I know I must have missed something fundamental. Can you please help me understand where I went wrong?
Thanks for your help.