Define covariant and contra variant tensors.

A tensor is covariant if its components transform in the same way as the basis vectors under a change of coordinates. A tensor is contravariant if its components transform in the opposite way as the basis vectors.

More formally:

Let Tⁱ_j represent a tensor component in one coordinate system, and T'^i'_j' represent the corresponding component in a transformed coordinate system. Let A^i'_i be the transformation matrix relating the basis vectors. Then:

• Covariant: T'_i' = Aⁱ_i' T_i
• Contravariant: T'^i' = A^i'_i Tⁱ

Note that indices are used to indicate covariance (subscript) and contravariance (superscript).