6.17.2 Tensor p q Contravariant Count

Tensor p q Contravariant Count is the value p appearing in the type notation (p, q), recording exactly how many upper, contravariant indices a tensor possesses, and thereby specifying how many independent vector-shaped slots the tensor provides for accepting covectors or for supplying vector-valued output components. This count is one half of the complete type specification of a tensor, and understanding precisely what it measures, how it behaves under algebraic operations, and how it interacts with the covariant count q is essential to reading and manipulating tensors of arbitrary order correctly.

What the Contravariant Count Measures

Number of Upper Index Slots

For a tensor T^{i_1 ... i_p}_{j_1 ... j_q}, the contravariant count p is simply the number of superscript positions, i_1 through i_p. Each such slot is filled, when the tensor is used as a multilinear map, by an element of the dual space V*, so a type (p, q) tensor can be regarded as a multilinear map taking p covectors and q vectors and returning a scalar.

Contribution to Total Component Count

If the underlying vector space has dimension n, the contravariant count contributes a factor of n^p to the total number of independent components n^{p+q} of the tensor, since each of the p upper indices ranges independently over the n basis directions.

Transformation Behavior Tied to the Contravariant Count

Each Upper Index Uses the Inverse Transition Matrix

Under a change of basis with transition matrix A and inverse B = A^{-1}, every one of the p upper indices picks up its own independent factor of B:

This is the precise reason the count p is called "contravariant": each associated index transforms in the manner opposite to the basis vectors themselves, since the basis vectors transform with A while these components transform with B.

Consistency Check Using Vector Components

The simplest instance of this rule, p = 1 and q = 0, reduces exactly to the transformation law for the components of an ordinary vector, v'^i = B^i_k v^k, confirming that the general contravariant count formula is a direct extension of the familiar vector transformation law to any number of upper indices.

The Contravariant Count Under Algebraic Operations

Tensor Product Adds Counts

If S has contravariant count p and T has contravariant count r, their tensor product S ⊗ T has contravariant count p + r, since the upper indices of S and the upper indices of T are simply placed side by side without interaction.

Contraction Removes One Upper Index

Contracting one of the p upper indices of a type (p, q) tensor against one of its own lower indices, when q ≥ 1, produces a tensor with contravariant count p - 1, reflecting that the contracted upper index is summed away and no longer appears as a free index in the result. This is why the trace of a type (1,1) tensor, which has contravariant count 1, results in a type (0,0) scalar with contravariant count 0.

Raising an Index Increases the Count

Given a metric's inverse g^{ij}, contracting it against one lower index of a tensor raises that index, converting a covariant slot into a contravariant one and increasing p by one while decreasing q by one, leaving the total order p + q unchanged.

Diagram of the Contravariant Slots

Distinguishing the Contravariant Count from the Covariant Count

Independent Yet Complementary Roles

The contravariant count p and the covariant count q are set independently when defining a tensor type, and a tensor's full classification requires specifying both; knowing only that p = 1, for instance, does not distinguish between a vector (q = 0), a linear operator (q = 1), or a bilinear map producing vectors from pairs of covectors (q = 2).

Contravariant Count of Zero

When p = 0, the tensor has no upper indices at all, and it is purely covariant, meaning it consumes only vectors and cannot itself supply vector-valued output; this is the case for the metric tensor and for all bilinear or multilinear forms defined directly on V without reference to V*.