Define contravariant and co-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 to the basis vectors under a change of coordinates.

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 Aⁱ_i' be the transformation matrix relating the coordinate systems. Then:

• Covariant: T'^i'_j' = Aⁱ_i' A^j_j' Tⁱ_j
• Contravariant: T'^i'_j' = A^i'_i A^j_j' Tⁱ_j

Note that the indices in the transformation matrices are placed such that they match the position (upper or lower) of the indices on the tensor component.