What do you mean by contraveriant and covariant tensors.

A tensor is covariant if its components transform in the same way as the basis vectors under a coordinate transformation. A tensor is contravariant if its components transform in the opposite way to the basis vectors.

More formally:

Let Tⁱ_j be a tensor component in one coordinate system, and T'^i'_j' in a transformed system. Let A^i'_i be the transformation matrix for basis vectors (covariant indices) and its inverse Aⁱ_i' for the transformation of their duals (contravariant indices).

Then for a covariant tensor:

T'^i'_j' = Aⁱ_i' A^j'_j Tⁱ_j

And for a contravariant tensor:

T'^i'_j' = A^i'_i A^j_j' Tⁱ_j

Note that mixed tensors have both covariant and contravariant indices and transform accordingly for each index type.