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Given two co-ordinate systems xⁱ and [(x)]ⁱ , a covariant tensor of order 1 is a set of components denoted A_i (with respect to xⁱ ) or [(A)]_i (with respect to [(x)]ⁱ ) , and satisfying:

- A   i  = ∑ r  ∂xr ∂ - x   i   Ar ,

for every i.
A covariant tensor of order 2 is a set of components denoted A_ij or [(A)]_ij , and satisfying:

- A   ij  = ∑ r  ∑ s  ∂xr ∂ - x   i   ∂xs ∂ - x   j   Ars ,

for ever i and j.
Higher-order covariant tensors can be defined similarly.