6.9.5 Tensor Contravariant Type Contribution

Tensor Contravariant Type Contribution describes how these tensors transform under coordinate changes, essential in tensor algebra and geometry.

Tensor Contravariant Type Contribution is the specific part played by a tensor's contravariant 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 covariant 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 contravariant type contribution isolates exactly what the contravariant half of that pair supplies, so that it can be added to, combined with, or compared against the covariant contribution in a controlled way.

Contribution to the Type Pair

The First Coordinate of the Type

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

type = (p, q), contravariant contribution = p

Independence From the Covariant Contribution

The contravariant contribution can take any nonnegative integer value independently of whatever value the covariant 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 contravariant contribution zero and any covariant order, or contravariant contribution five and covariant 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 contravariant contribution is exactly the summand supplied by the upper indices. Increasing the contravariant order by one increases the total rank by one, regardless of what the covariant order happens to be, since the two summands add independently.

total rank = p + q

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 upper index supplies exactly one factor of the direct Jacobian matrix, and the contravariant contribution is precisely the count of such factors present. The lower indices independently supply an equal number of inverse-Jacobian factors according to the covariant 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.

\prod_{i = 1}^{p} \frac{\partial x^{'} (a i)}{\partial x^{b i}}

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 contravariant contribution enters this formula through the exponent p, so that each additional unit of contravariant order multiplies the dimension of the tensor space by another factor of n.

dim = n^{p + q}

Contribution to the Space as a Tensor Product Factor

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

Contribution When Combining Tensors

Contribution Under Tensor Product of Two Tensors

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

Contribution Under Contraction and Index Lowering

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