Why the bases for space of alternating tensors contain elementary alternating tensors of increasing multi-index?

Question

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?

Answer 1

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.

Answer 2

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.