6.8.5 Tensor Covariant Type Contribution

Tensor Covariant Type Contribution is the specific part played by a tensor's covariant order in fixing the tensor's overall type, its total rank, its transformation law, and the dimension of the vector space in which it lives, considered separately from the contribution made by the contravariant order. Because a tensor's type is the pair formed from its contravariant and covariant orders together, neither number alone describes the tensor completely, and the covariant type contribution isolates exactly what the covariant half of that pair supplies, so that it can be added to, combined with, or compared against the contravariant contribution in a controlled way.

Contribution to the Type Pair

The Second Coordinate of the Type

A tensor's type is conventionally written as an ordered pair, contravariant order first and covariant order second, and the covariant type contribution is precisely the value occupying that second position. Changing only the covariant order while leaving the contravariant order fixed moves a tensor to a different type, even though the two types may still share the same first coordinate, and this shift is entirely attributable to the covariant contribution changing.

Independence From the Contravariant Contribution

The covariant contribution can take any nonnegative integer value independently of whatever value the contravariant contribution takes, since the two counts arise from entirely separate sets of index positions, upper and lower respectively, with no constraint linking one to the other. A tensor may have covariant contribution zero and any contravariant order, or covariant contribution five and contravariant order zero, or any other combination, and every combination defines a distinct and equally legitimate tensor type.

Contribution to Total Rank and Transformation Law

Additive Contribution to Total Rank

The total rank of a tensor, understood as the overall number of indices it carries, is the sum of its contravariant order and its covariant order, and the covariant contribution is exactly the summand supplied by the lower indices. Increasing the covariant order by one increases the total rank by one, regardless of what the contravariant order happens to be, since the two summands add independently.

Contribution to the Number of Transformation Factors

In the full transformation law relating a tensor's components in one coordinate system to its components in another, every lower index supplies exactly one factor of the inverse Jacobian matrix, and the covariant contribution is precisely the count of such factors present. The upper indices independently supply an equal number of direct-Jacobian factors according to the contravariant order, and the two counts of factors, direct and inverse, multiply together in the complete law without interacting, each governed entirely by its own respective contribution.

Contribution to the Underlying Vector Space

Dimension Supplied to the Tensor Space

The space of all tensors sharing a fixed type has a dimension determined jointly by the contravariant and covariant orders together with the dimension of the base vector space: for a base space of dimension n, the tensor space of type having contravariant order p and covariant order q has dimension n raised to the power p plus q. The covariant contribution enters this formula through the exponent q, so that each additional unit of covariant order multiplies the dimension of the tensor space by another factor of n.

Contribution to the Space as a Tensor Product Factor

A tensor of covariant order q, contravariant order zero, can be regarded as living in the q-fold tensor product of the dual vector space with itself, and the covariant contribution names exactly how many copies of the dual space enter this product. When contravariant order is also present, the full tensor space is the tensor product of the p-fold product of the vector space with the q-fold product of the dual space, and the covariant contribution supplies precisely the second of these two factors.

Contribution When Combining Tensors

Contribution Under Tensor Product of Two Tensors

When two tensors are combined by tensor product, the covariant contribution of the resulting tensor is the sum of the covariant contributions of the two factors, entirely independently of how their contravariant orders combine. A tensor of covariant contribution two combined with a tensor of covariant contribution three yields a product tensor of covariant contribution five, this addition proceeding without any reference to whatever contravariant orders the two factors happened to carry.

Contribution Under Contraction and Index Raising

Contracting a lower index against an upper index removes one unit from the covariant contribution together with one unit from the contravariant contribution of whichever tensor or tensor product is involved. Raising a lower index using the inverse metric decreases the covariant contribution by one while simultaneously increasing the contravariant contribution by one, so that the total rank supplied by the two contributions together remains unchanged even as the balance between them shifts.