6.10 Tensor Mixed Type Classification

Tensor Mixed Type Classification is the grouping of tensors according to whether they carry both contravariant and covariant indices simultaneously, that is, whether both the contravariant order and the covariant order of the type pair are strictly greater than zero, as opposed to the purely contravariant tensors that carry only upper indices or the purely covariant tensors that carry only lower indices. A mixed type tensor of contravariant order p and covariant order q accepts p one-form arguments and q vector arguments together, returning a single scalar that depends linearly on every one of these p plus q inputs, and its transformation law combines direct-Jacobian factors for the upper indices with inverse-Jacobian factors for the lower indices in a single multiplicative expression.

Defining the Mixed Type Category

The Requirement of Both Variances Present

A tensor belongs to the mixed type classification precisely when its type pair has both entries positive: at least one upper index and at least one lower index must be present simultaneously. This excludes the purely contravariant tensors, whose covariant order is zero, and the purely covariant tensors, whose contravariant order is zero, leaving as mixed type every tensor that genuinely combines the two kinds of index in a single object.

The Simplest Mixed Type

The simplest possible mixed type tensor carries exactly one upper index and one lower index, corresponding to type having contravariant order one and covariant order one. Such a tensor accepts one one-form and one vector and returns a scalar, and it is precisely the algebraic structure underlying a linear map from the vector space to itself, since fixing the vector argument and leaving the one-form slot open produces a vector, exhibiting the tensor as a machine that turns vectors into vectors.

Component Structure of Mixed Type Tensors

Simultaneous Upper and Lower Indices

The components of a mixed type tensor display both superscripted and subscripted indices on the same symbol, the superscripts numbering as many as the contravariant order and the subscripts numbering as many as the covariant order. Each upper index labels one one-form slot and each lower index labels one vector slot, and the two groups of indices are read independently of one another even though they appear on the same underlying array of numbers.

The Transformation Law as a Product of Two Kinds of Factor

Under a change of coordinates, each upper index of a mixed type tensor contributes one factor of the direct Jacobian and each lower index contributes one factor of the inverse Jacobian, all of these factors multiplying together and contracting with the original components summed over the appropriate dummy indices. No interaction occurs between the upper-index factors and the lower-index factors beyond this multiplication; each behaves exactly as it would in a purely contravariant or purely covariant tensor of the corresponding order.

Examples and Roles of Mixed Type Tensors

Linear Maps as Type (1,1) Tensors

Every linear transformation of a vector space into itself can be represented as a mixed type tensor of contravariant order one and covariant order one: feeding the tensor a vector into its lower slot and leaving the upper slot open, or equivalently contracting the tensor's lower index against the vector's components, produces a new vector, reproducing exactly the action of the linear map. The identity map corresponds to the tensor whose mixed components equal one whenever the upper and lower index agree and zero otherwise.

The Kronecker Delta as the Canonical Mixed Tensor

The Kronecker delta is the standard example of a type (1,1) tensor, equal to one when its upper and lower index coincide and zero otherwise in any basis, a property that remains true in every coordinate system precisely because its transformation law, one direct-Jacobian factor and one inverse-Jacobian factor, causes the two factors to compose into the identity matrix relating any coordinate system to itself.

Higher Mixed Types From Products and Raising or Lowering

Mixed type tensors of higher order arise naturally by forming tensor products of purely contravariant and purely covariant tensors, which combine their upper and lower indices without alteration, and also by raising a lower index or lowering an upper index of a tensor that was originally purely one variance, using the metric or its inverse. Both routes produce a tensor whose type pair has both entries positive, placing the result squarely within the mixed type classification regardless of which construction produced it.

Distinguishing Mixed Type From Symmetry-Mixed Tensors

Mixed type refers exclusively to the presence of both variances among a tensor's indices, and is a separate notion from a tensor lacking either full symmetry or full antisymmetry among indices of a single variance. A tensor can be purely covariant, carrying only lower indices, and still fail to be symmetric or antisymmetric in those indices; such a tensor is not of mixed type, since mixed type is determined solely by the type pair and not by any internal symmetry property of the indices it contains.