Why the bases for space of alternating tensors contain elementary alternating tensors of increasing multi-index? I have been reading Introduction to Smooth Manifolds by John M.Lee
The basis for the space of alternating covariant k-tensors $\Lambda^k(V^*)$, is given by the collection
{ $\epsilon^I$;  $\mathit{I}$  is an increasing multi-index of length k }
My question is why $\mathit{I}$ has to been an increasing multi-index?
Can someone show me why $\mathit{I}$ cannot be any multi-index of length k?
 A: All the indices have to be different (because it's alternating), and if two multiindices are permutations of each other then the corresponding vectors are the same up to a sign.
Demanding that the indices are increasing is a way to force them all to be different while simultaneously picking a single element out of all the equivalent permutations.
A: There is, however, an instance to have into account where the order of basic 2-form break the ascending indexing one.
The example is one of the Stokes' theorem, from the old vector calculus.
This, relates a 1-form $Pdx+Qdy+Rdz$ and a 2-form 
$$Ady\wedge dz+Bdz\wedge dx+Cdx\wedge dy,$$ 
where the coefficients 
$P,Q,R,A,B,C$ are smooth functions on the cartesian coordinates $x,y,z$ of $\mathbb R^3$.
It s well known that if
$$A=\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\ ,\
B=\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\ ,\
C=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$$ 
then a line integral is equal to a double one 
$$\int_{\partial\Omega}Pdx+Qdy+Rdz=
\iint_{\Omega}Ady\wedge dz+Bdz\wedge dx+Cdx\wedge dy,$$
where $\Omega$ is a bounded surface on the space with border $\partial\Omega$,
and 
under the comply of some simple assumptions over the functions on this game.
