Contravariant Tensor -- from Wolfram MathWorld

A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). To examine the transformation properties of a contravariant tensor, first consider a tensor of rank 1 (a vector) dr=dx_1x_1^^+dx_2x_2^^+dx_3x_3^^, (1) for which dx_i^'=(partialx_i^')/(partialx_j)dx_j. (2) Now let A_i=dx_i, then any set of quantities A_j which transform according to A_i^'=(partialx_i^')/(partialx_j)A_j, (3) or, defining a_(ij)=(partialx_i^')/(partialx_j), (4) ...