An affine tensor is a tensor that corresponds to certain allowable linear coordinate transformations, ,
where the determinant of
is nonzero. This transformation takes the rectangular coordinate system into the coordinate system having oblique axes. In this way an affine tensor can
be seen as a special kind of Cartesian tensor.

These tensors have the Jacobians,

(1)

(2)

(3)

(4)

The transformation laws for affine contravariant (tangent) tensors are

(5)

(6)

(7)

and so on, and the transformation laws for affine covariants (covectors) tensors are

(8)

(9)

(10)

and so on.

The transformation laws for mixed affine tensors are