I am wondering what is the correct way to write integrals in index notation.
At first I thought
$$\int_M f \varepsilon_{ij}$$
would be the index equivalent of
$$ \int_M f dx \wedge dy $$
but I started to worry about orientation.
Is the orientation given by the sign of the permutation of the free indexes wrt alphabetical order?
It feels like one should indicate the orientation on the domain.
Maybe something like
$$\int_{M^{ij}} f \varepsilon_{ij}$$
I have never seen like that used though.
Is there anything wrong with this idea?
Even when using index notation people usually write $$\int_M f dM$$
or more generally things like $$\int_M A^{ij} B_i C_j dM$$ where the the expression being integrated is a possibly complicated but fully contracted scalar expression followed by the area/volume element.
That seem redundant though when integrating antisymmetric tensors.
Instead of writing
$$\int_M A_{ij}$$
or maybe
$$\int_{M^{ij}} A_{ij}$$
one needs to write
$$\int_M 1/2 A_{ij} \varepsilon^{ij} dM$$
Best Answer
This is confusing because the usual way of writing integrals in differential forms leaves something implicit: the tangent $k$-vector of the manifold of integration, and the notation itself tends to make people confuse $\mathrm dx$---the cotangent basis vector associated with the coordinate $x$---with $dx$---the symbol that tells us we're integrating with respect to $x$.
Suppose you have an integral in differential forms
$$\int_M f(x,y) \, \mathrm dx \wedge \mathrm dy$$
It really means this:
$$\int_M f(x,y) \, (\mathrm dx \wedge \mathrm dy)(T_{(M)}) \, dx \, dy$$
where $T_{(M)}$ is the tangent 2-vector for the manifold of integration $M$. For instance, a common choice of orientation would be that $T_{(M)} = e_x \wedge e_y$. And naturally, $(\mathrm dx \wedge \mathrm dy)(e_x \wedge e_y) = 1$. So the integral would reduce to $\int_M f(x,y) \, dx \, dy$.
This approach naturally generates the relevant surface/volume elements one might need. Suppose you have some surface two-form $\omega$ on a sphere that you want to integrate. You would set up the integral like so:
$$\int_S \omega(\theta, \phi; e_\theta \wedge e_\phi) \, d\theta \, d\phi$$
Note that $e_\theta \wedge e_\phi = r^2 \sin \theta \, \hat \theta \wedge \hat \phi$, and this would be appropriate to use if $\omega$ is written in terms of unit forms instead of $\mathrm d \theta$ and $\mathrm d \phi$.
You can write all of these above in index notation if you like. Just remember that one should always use the tangent $k$-vector associated with the coordinates of integration--$e_\theta \wedge e_\phi$ in that sphere example--though the orientation of that tangent $k$-vector is not necessarily determined by the ordering of the coordinates. This may introduce minus signs, if the problem specifies such.
Most of the time, though, people mean to say that a manifold is oriented the same way as the coordinates are ordered.