7.18.4 Tensor Component Order Dependence

Tensor Component Order Dependence is the fact that the number of components a tensor possesses changes with its order, meaning the total number of indices it carries, and does so according to an exponential relationship with the dimension of the underlying space, so that even a small increase in order produces a large increase in the raw number of components once the dimension exceeds one.

The Nature of the Dependence

Raw Entry Count as a Function of Order

For a tensor defined on a vector space of fixed dimension n, the Tensor Component Total Entry Calculation expresses the raw entry count as n raised to the power r, where r is the order of the tensor. Holding the dimension fixed and varying the order shows that the entry count grows exponentially in r, since each additional index multiplies the entry count by a further factor of n rather than adding a fixed amount.

Comparing Successive Orders

Increasing the order of a tensor by exactly one, while keeping the dimension fixed, multiplies the raw entry count by n:

so that the ratio between the entry counts of consecutive orders is always exactly n, regardless of which order is being compared to the next.

Illustration

Each bar is n times taller than the one before it, illustrating that the total entry count multiplies by the same factor with every additional order.

Effect of Order on Symmetry Considerations

More Indices Allow More Possible Pairings

As the order of a tensor increases, the number of distinct pairs of indices available for testing symmetric or antisymmetric behavior also increases, since more indices means more possible ways to select two of them for such a test. A rank-two tensor has only one possible pair of indices to examine, while a rank-four tensor has several possible pairs, each of which might independently follow a symmetric pattern, an antisymmetric pattern, or no fixed pattern at all.

Higher Order Permits More Elaborate Symmetry Structures

Beyond simple pairwise relationships, tensors of sufficiently high order can exhibit mixed symmetry patterns that involve three or more indices together in a single combined relationship, a possibility that does not arise at all for tensors of order one or two, since such patterns require enough indices to produce nontrivial permutations beyond a single exchange.

Practical Significance

Anticipating the Scale of Higher Order Tensors

Recognizing the Tensor Component Order Dependence of a tensor's component count allows the practical scale of the data involved to be anticipated as the order increases, since even a modest dimension can produce an unwieldy number of components once the order becomes large enough, due to the exponential nature of the relationship.

Order Is Independent of Dimension in How It Governs Growth

While Tensor Component Dimension Dependence describes how the entry count grows as the base of the exponentiation changes, Tensor Component Order Dependence describes how the entry count grows as the exponent itself changes, and the two effects combine multiplicatively rather than being interchangeable with one another.

Relationship to Other Tensor Concepts

Tensor Component Order Dependence describes how the output of the Tensor Component Total Entry Calculation varies as the order, or rank, of a tensor changes, holding the dimension of the underlying space fixed. It complements Tensor Component Dimension Dependence, and together the two describe the complete behavior of the exponential relationship that governs the raw entry count referenced throughout Tensor Component Enumeration.