9.21.1 Tensor Basis Symbol Notation

Tensor Basis Symbol Notation defines how tensors are expressed using basis symbols, crucial for algebraic manipulation in mathematics and physics.

Tensor Basis Symbol Notation is the set of conventions governing which letters, index positions, and decorations are used to denote basis vectors, dual basis covectors, and their associated components when a tensor is written out in a specific coordinate system.

Core Symbols

Basis Vector Symbols

Basis vectors are conventionally written with a lowercase letter, most commonly (e), carrying a lower index that ranges over the dimension of the space.

{e_{1}, e_{2}, \dots, e_{n}}

The lower index on (e_i) is a labeling index, not a covariant tensor index in the transformation sense, but it is deliberately placed low because basis vectors transform with the change-of-basis matrix directly, matching the behavior of covariant objects.

Dual Basis Symbols

The dual basis covectors are written with the same letter, but with the index raised, and are frequently marked with a superscript to distinguish them visually from the basis vectors themselves.

{e^{1}, e^{2}, \dots, e^{n}}

Using the same base letter (e) for both the vector basis and its dual is a deliberate notational choice: it visually ties each dual basis covector (e^i) to the basis vector (e_i) it is biorthogonal to.

Primed and Decorated Symbols

Distinguishing Different Bases

When more than one basis is under discussion at once, a prime, tilde, hat, or bar is added to the basis symbol to distinguish the second basis from the first, while the index letters are often kept the same to signal that the two bases index the same space.

{e_{i}} versus {e_{i}'}

Some texts instead prime the index rather than the symbol, writing (e_{i'}), which shifts the marker onto the index slot but conveys the same meaning: a distinct basis with a distinct labeling index.

Hats for Orthonormal Bases

A hat placed over a basis vector symbol, such as (\hat{e}_i), conventionally signals that the basis is orthonormal, meaning each basis vector has unit length and all basis vectors are mutually orthogonal. This is a widely recognized symbolic shorthand rather than a separate mathematical object.

Symbol Pairing with Components

Matching Index Position

Because the summation convention pairs an upper index with a lower index, the symbol chosen for a basis vector's index must match, in name and position, the index used on the corresponding tensor component.

v = v^{k} e_{k}

Using a consistent letter, here (k), for both the component's superscript and the basis vector's subscript is the notational signal that a contraction (summation) is intended between exactly those two symbols.

Free Versus Dummy Indices

Symbol notation also distinguishes free indices, which appear once in an expression and denote a specific slot of the resulting tensor, from dummy indices, which appear as a matched upper-lower pair and are summed over. The letters chosen for dummy indices are arbitrary and may be renamed without changing the meaning of the expression, whereas free index letters must remain consistent across every term of an equation.

T^{i} = A_{j}^{i} v^{j}

Here (i) is free on both sides, while (j) is a dummy index that could equally be renamed (k) or (m) without altering the equation.

Visual Summary

Practical Importance

Consistent tensor basis symbol notation is what allows an equation written purely in terms of indices to be read unambiguously: the position of an index (upper or lower), the letter chosen, and any decoration such as a prime or hat together encode the basis being used, whether the index is summed, and whether the object is contravariant or covariant, all without needing to restate these facts in words each time.