covariant

A covariant tensor, denoted with a lowered index (e.g., a_μ) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor. To examine the transformation properties of a covariant tensor, first consider the gradient del ϕ congruent (dϕ)/(dx_1) (x_1)^^ + (dϕ)/(dx_2) (x_2)^^ + (dϕ)/(dx_3) (x_3)^^, for which (dϕ')/(dx_i^, ) = (dϕ)/(dx_j) (dx_j)/(dx_i^, ), where ϕ(x_1, x_2, x_3) = ϕ'(x_1^, , x_2^, , x_3^, ).

Cartesian tensor | contravariant tensor | four-vector | index lowering | Lorentz tensor | metric tensor | mixed tensor | tensor