9.13.4 Tensor Component Expansion Summed Basis Terms

Tensor Component Expansion Summed Basis Terms expresses tensor components through a basis, combining indices and coefficients in a structured algebraic framework.

Tensor Component Expansion Summed Basis Terms is the collection of individual addends that together make up a tensor's basis expansion, each term consisting of one component coefficient multiplying one basis tensor product. It refers to the terms themselves as discrete pieces of the sum, rather than to the sum as a whole or to the coefficients in isolation.

Anatomy of a Summed Term

Two Parts of Every Term

Each summed basis term consists of a scalar coefficient, drawn from the component array, and a basis tensor product, formed from one basis vector for every contravariant index and one dual basis covector for every covariant index. The term as a whole is the product of these two parts.

T_{j}^{i} e_{i} \otimes e^{j}

Fixed Index Values Within a Term

Unlike the general expansion formula, which uses the same index letters to stand for a range of values, a single summed basis term corresponds to one specific, fixed choice of index values, and thus to one specific coefficient and one specific basis tensor product.

Assembling the Full Expansion

Every Term Contributes Independently

The full expansion of a tensor is obtained by adding together every summed basis term that arises from every possible combination of index values permitted by the tensor's type. Each term contributes independently to the total, and the terms do not interact with one another within the sum itself.

T = \sum_{i, j} T_{j}^{i} e_{i} \otimes e^{j}

Total Count of Terms

For a tensor of type (p, q) built over an n-dimensional space, the total number of summed basis terms equals n raised to the power of p plus q, matching the total number of distinct index combinations available.

Terms with Zero Coefficient

A summed basis term whose coefficient equals zero contributes nothing to the total sum, since multiplying any basis tensor product by zero eliminates that term. Such terms are frequently omitted when the expansion is written out explicitly, leaving only the nonzero terms visible.

Behavior of Individual Terms

Linearity Across Terms

Because the expansion is a sum, operations that are linear with respect to addition, such as scaling the entire tensor by a constant, act on every summed basis term individually, multiplying each term's coefficient by that constant while leaving the basis tensor product of each term unchanged.

Rearrangement of Terms

The order in which summed basis terms are written does not affect the value of the total sum, since addition of tensors, like addition of numbers, does not depend on the order of the addends. Terms may therefore be grouped or reordered freely for convenience without altering the tensor represented.

Practical Significance

Isolating Individual Contributions

Examining a single summed basis term in isolation makes it possible to study the specific contribution of one basis tensor product to the overall tensor, which is useful when analyzing which parts of a tensor are significant and which are negligible or absent.

Term-by-Term Verification

When checking that two expansions represent the same tensor, it suffices to verify that the coefficient attached to each corresponding basis tensor product agrees between the two expansions, since two sums built from the same set of basis tensor products are equal exactly when their term-by-term coefficients match.