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

- a   i   = n ∑ r=1  ∂ - x   i   ∂xr Ar ,

for each i.
A contravariant tensor of order 2 is a set of components denoted A^ij or [(A)]^ij , and satisfying:

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

for all i and j.
Higher-order contravariant tensors are defined similarly.