9.21.5 Tensor Basis Expansion Notation

Tensor Basis Expansion Notation represents tensors using basis vectors, enabling systematic algebraic manipulation and analysis.

Tensor Basis Expansion Notation is the notation used to write a tensor explicitly as a sum of basis tensor products, each weighted by its corresponding component, so that the tensor's full basis-dependent representation is displayed as a single explicit expression rather than left as an abstract object.

Expansion of a Vector

Full Summation Form

A vector (v) is expanded in a basis ({e_i}) as an explicit sum of each basis vector scaled by its matching contravariant component.

v = \sum_{i = 1}^{n} v^{i} e_{i}

Einstein Convention Form

Because the pattern of a repeated upper-lower index pair recurs throughout tensor algebra, the summation symbol is dropped under the Einstein convention, leaving the expansion in its more common compact form.

v = v^{i} e_{i}

Both forms represent exactly the same sum; the compact form relies on the reader recognizing that (i) appears once up and once down and is therefore summed.

Expansion of Covectors and Higher-Rank Tensors

Covector Expansion

A covector (\omega) expands analogously, using the dual basis ({e^i}) and covariant components.

ω = ω_{i} e^{i}

Rank-Two Tensor Expansion

A rank-two tensor expands as a double sum over tensor products of basis elements, with one basis factor for each index.

T = T^{ij} e_{i} \otimes e_{j}

The two indices (i) and (j) are each summed independently over all (n) values, producing (n^2) terms in the fully written-out sum, even though the compact expansion notation displays only a single expression.

General Mixed-Rank Expansion

A general ((p, q)) tensor expands with (p) basis vector factors and (q) dual basis covector factors, each carrying its own summed index.

T = T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} e_{i_{1}} \otimes \dots \otimes e_{i_{p}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{q}}

Reading an Expansion Term by Term

Identifying the Basis Product

Every term of an expansion is a single tensor product of basis vectors and dual basis covectors, in an order that matches the order of indices on the tensor's component symbol. Reordering the basis factors without reordering the corresponding indices on the component changes which term of the sum is being described.

Identifying the Weight

The coefficient multiplying each basis tensor product is exactly one entry of the component array, singled out by fixing every index to a specific value. Expanding the tensor in full, rather than leaving it as (T^{ij} e_i \otimes e_j), means writing out every one of these (n^2) (or more generally (n^{p+q})) individual weighted terms.

Visual Illustration

Purpose of Explicit Expansion Notation

Tensor basis expansion notation makes the abstract statement "a tensor is a linear combination of basis tensor products" into a concrete, checkable expression. It is the notational bridge between the coordinate-free definition of a tensor and its coordinate-dependent representation as a component array, and it is what justifies substituting components directly into a sum whenever a specific numerical computation is required.