Is invariance of a multi-linear form required for co/contra variance? I'm reading the book: The Absolute Differential Calculus by Levi-Civita to get an idea of the history behind the development of tensor calculus.
On page 71 he states:

An m-fold covariant is an m-fold system which is transformed in the same way as the coefficients of a multilinear form in point variables; an m-fold contravariant is one which is transformed in the same way as the coefficients of a multilinear form in dual variables.

The few pages beforehand has required the multilinear form to be invariant, so has this been incorrectly left out in the above definition?
 A: In a nutshell
"tensors on a manifold $M$ are global objects which are invariant by definition; their components transform under coordinate change following laws called covariance & contravariance "
*"the $m$-fold systems (pag. 65 and pag.71 in loc.cit.) are not invariant under change of coordinates: they transform covariantly or contravariantly because they are the coefficients of tensors on the manifold $M$, not tensors themselves".*
Let us clarify this point.


*

*General statements tensors on manifolds


Tensors on manifolds, when given locally, have coefficients with precise transformation rules under coordinate transformations: the tensor itself is invariant, as it is "globally" given as a section of a certain fiber bundle over the manifold $M$ under analysis: its local espression depends on coordinates, though. 
For example, the multi-vector field (or contravariant tensor field) $X\in\Gamma(\wedge^m TM)$ is locally given by
$$X=X_{i_1,\dots,i_m}(x_1,\dots,x_n)\frac{\partial }{\partial x_{i_1}}\wedge\dots\wedge\frac{\partial }{\partial x_{i_m}}\in \wedge^m T_pM$$
at point $p\in M$ in local coordinates $(x_1,\dots,x_n)$; it transforms under change of coordinates $x_{\bullet}\mapsto y_{\bullet}=y_{\bullet}(x_{\bullet})$
by remembering that
$$\frac{\partial}{\partial x_r}=\frac{\partial y_s}{\partial x_r}\frac{\partial}{\partial y_s}. $$
In summary, in coordinates $(x_1,\dots,x_n)$ we have
$X=X_{i_1,\dots,i_m}(x_1,\dots,x_n)\frac{\partial }{\partial x_{i_1}}\wedge\dots\wedge\frac{\partial }{\partial x_{i_m}}$; in the coordinates
$(y_1,\dots,y_n)$ the tensor $X$ (it is globally invariant!) is given by
$X=\tilde{X}_{j_1,\dots,j_m}(y_1,\dots,y_n)\frac{\partial }{\partial y_{j_1}}\wedge\dots\wedge\frac{\partial }{\partial y_{j_m}}$, with
$$\tilde{X}_{j_1,\dots,j_m}(y_1,\dots,y_n)=X_{i_1,\dots,i_m}(x_1,\dots,x_n)
\frac{\partial y_{j_1}}{\partial x_{i_1}}\cdots\frac{\partial y_{j_m}}{\partial x_{i_m}}.$$
Dualizing one considers differential forms and coordinate transformations inducing
$$dy_s=\frac{d y_s }{d x_r}d x_r $$
at the level of differentials.
