10.5.5 Tensor Forward Basis Notation Convention

The Tensor Forward Basis Notation Convention standardizes tensor component notation using basis vectors for clear and consistent algebraic representation.

Tensor Forward Basis Notation Convention is the agreed-upon system of symbols used to distinguish an original basis from the new basis obtained through a forward change of basis, together with the placement of indices that indicates how each transformed quantity relates to that new basis. This convention exists so that anyone reading a tensor expression can immediately identify which basis a given index refers to, in which direction the change is being applied, and which matrix of coefficients governs that direction, without needing additional verbal explanation attached to every formula.

Purpose of the Convention

Disambiguating Old and New Bases

When a basis is replaced by a new one, both the old symbols and the new symbols must appear in the same body of work, often within the same equation. The notation convention assigns a distinguishing mark, typically a prime or an explicit numerical label, to every quantity expressed in the new basis, while leaving quantities in the old basis unmarked. This prevents the two sets of basis vectors from being confused with one another.

Fixing the Direction of Change

A change of basis can be described starting from the old basis and moving toward the new one, or the reverse. The forward direction is conventionally the one in which the new basis vectors are written as linear combinations of the old basis vectors. The notation convention fixes this direction explicitly so that the matrix of coefficients used in any formula is unambiguous.

e_{i}' = A_{i}^{j} e_{j}

Here the primed basis vector on the left is defined in terms of the unprimed basis vectors on the right, and the coefficient matrix carries the label associated with the forward direction of the change.

Structure of the Notation

Index Placement on the Coefficient Matrix

The forward change-of-basis coefficients carry one upper and one lower index, positioned so that the summation convention applies directly when the matrix multiplies a basis vector. The upper index matches the free index of the new basis vector being defined, and the lower index is summed against the old basis vector index.

A_{i}^{j}

Marking Transformed Components

Component arrays expressed relative to the new basis receive the same priming mark as the basis vectors they accompany, ensuring visual consistency between a transformed object and the basis in which it is expressed.

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

Reserving the Inverse Symbol for the Backward Direction

Because contravariant components transform with the inverse of the forward coefficient matrix, the convention reserves a distinct symbol, typically written with an explicit inverse superscript, for this backward-acting matrix, so that it is never confused with the forward matrix itself even though both matrices relate the same two bases.

Consistency Requirements

Uniform Application Across a Document

Once the convention is adopted for one change of basis, every subsequent change of basis within the same body of work must follow the identical priming, indexing, and directional scheme. Switching conventions midway, for instance using an unprimed symbol for the new basis in one section and a primed symbol in another, defeats the purpose of the convention entirely.

Compatibility with Summation Convention

The notation convention is designed to interact cleanly with the summation convention, so that repeated upper and lower indices across the coefficient matrix and the basis vector or component array automatically indicate the required summation, without additional summation symbols cluttering the expression.

Extension to Higher-Rank Objects

The same priming and indexing scheme extends directly to tensors of any rank, with each index of a mixed tensor acquiring its own prime when expressed in the new basis, and each index independently governed by either the forward matrix or its inverse depending on whether the index is covariant or contravariant.

Schematic Representation

The diagram shows the basis vectors moving forward under the coefficient matrix while the contravariant components move in the opposite sense under its inverse, both changes governed by the same underlying notation convention applied consistently to primed and unprimed symbols.