10.22.5 Tensor Prime Coordinate Notation

Tensor Prime Coordinate Notation is the convention of attaching a prime mark to coordinate labels and indices in order to distinguish a new coordinate system from an original, unprimed coordinate system when describing a change of basis for tensor quantities. It provides a lightweight, purely typographic device for keeping two coordinate frames visually and symbolically separate throughout a transformation calculation.

Purpose of the Prime Device

Distinguishing Two Coordinate Systems With Minimal Notation

Rather than inventing entirely new letters for a second coordinate system, prime notation reuses the same index letters and simply appends a prime mark to those belonging to the new system. This keeps the correspondence between old and new indices visually obvious, since a primed index and its unprimed counterpart share the same base letter.

Compatibility With the Summation Convention

Because primed and unprimed indices are treated as distinct symbols for the purposes of the summation convention, an expression can freely mix primed and unprimed indices in the same term without ambiguity, as long as repeated indices, whether primed or unprimed, always occur once as a superscript and once as a subscript.

Basic Syntax

Priming a Coordinate

A coordinate belonging to the new system is written with the same letter as the corresponding old coordinate, with a prime attached to the index rather than to the coordinate symbol itself.

Priming an Index in a Tensor Component

The same prime is carried onto any tensor index that refers to the new coordinate system, so that a tensor component in the new system is written with primed indices while the same tensor in the old system is written with unprimed indices of the same letter.

Use in Jacobian Factors

Direct Factor Written With Prime Notation

The Jacobian factor relating the two systems places the primed index in the numerator position when differentiating a new coordinate with respect to an old one.

Inverse Factor Written With Prime Notation

Reversing the derivative places the primed index in the denominator, and this single visual swap is enough to indicate that the inverse relationship is being described.

Handling More Than Two Coordinate Systems

Double and Triple Priming

When a calculation requires a third coordinate system, a second prime mark is added, producing double-primed indices, and a fourth system would use a triple prime, though in practice most expositions avoid more than two prime levels for legibility.

Transitioning to Numbered or Lettered Alternatives

For chains of three or more coordinate systems, many texts switch from prime notation to a numbered bracket convention or to entirely distinct letters for each system, reserving prime notation specifically for the common two-frame case where its economy is most useful.

Precautions in Using Prime Notation

Prime Is Not a Derivative Symbol

Within tensor prime coordinate notation, the prime mark attached to an index carries no meaning of differentiation, unlike its common use in single-variable calculus to denote a derivative. Readers must interpret the prime purely as a coordinate-system label whenever it is attached to an index rather than to a function name.

Consistency Across an Entire Derivation

Once a prime convention is adopted to denote the new coordinate system, it must be applied uniformly to every index referring to that system throughout the derivation, since mixing primed and unprimed labels inconsistently for the same coordinate system reintroduces the very ambiguity the notation was designed to remove.

Compatibility With Basis Vector Labeling

Basis vectors associated with the new coordinate system are also given primed labels, matching the primed coordinate indices, so that the pairing between a coordinate index and its associated basis vector remains unambiguous even after the change of basis has been applied.