9.5 Tensor Coordinate Basis System

Tensor Coordinate Basis System is the general framework by which a chosen basis of a vector space, together with its dual basis, is used to assign numerical coordinates to tensors of any order and any mixture of upper and lower slots, providing the complete set of rules governing how such coordinates are constructed, how they transform under a change of basis, and how they combine to represent tensor operations; it is the umbrella structure beneath which more specific coordinate schemes, including those built from tensor products of factor bases, are organized as particular cases.

The Two Ingredients of the System

A Primal Basis for Vector Slots

A tensor coordinate basis system begins with a primal basis {e_i} of the underlying vector space, used to assign numerical values to any tensor slot that accepts a covector, producing what are recorded as lower indices in the resulting coordinate array.

A Dual Basis for Covector Slots

The system pairs this primal basis with its unique dual basis {e^i}, satisfying e^i(e_j) = δ^i_j, used to assign numerical values to any tensor slot that accepts a vector, producing what are recorded as upper indices.

What the System Provides

A Complete Coordinate Array for Any Tensor Type

By pairing every slot of a tensor with the appropriate member of the primal or dual basis, the tensor coordinate basis system produces a full array of numbers indexed by as many upper and lower indices as the tensor has vector and covector slots respectively, regardless of the tensor's order.

A Transformation Rule Linking Different Bases

The system specifies exactly how coordinates change when the primal basis is replaced by another, related by a transition matrix and its inverse: upper indices transform using the inverse transition matrix and lower indices transform using the transition matrix itself, ensuring that coordinates computed in any two admissible bases remain consistently related.

Relation to More Specific Coordinate Schemes

Tensor Product Basis Coordinates as a Special Case

When the underlying space is itself built as a tensor product of factor spaces, the general tensor coordinate basis system specializes to a tensor product basis coordinate system, in which the primal and dual bases are themselves constructed by taking tensor products of the factor bases, and coordinates are addressed by multi indices tagged with a factor label.

Curvilinear Coordinate Bases as Another Case

When the space in question is the tangent space of a manifold equipped with curvilinear coordinates, the general system specializes to a coordinate basis of partial derivative vectors and differential covectors, with the same primal and dual pairing rules applying pointwise at every location on the manifold.

Diagram of the System's Structure

Consequences of the System

It Grounds All Component-Based Tensor Reasoning

Every operation performed on tensors in terms of indices — contraction, index raising and lowering, symmetrization — presupposes that a tensor coordinate basis system has already been fixed, since these operations act directly on the numerical arrays that the system produces and carry no meaning prior to that assignment.

It Guarantees Coordinate-Independent Conclusions When Applied Correctly

Because the system fully specifies how coordinates transform between bases, any tensor equation verified in coordinates produced by one admissible basis of the system is guaranteed to hold in coordinates produced by any other admissible basis, which is what allows coordinate calculations to stand in for basis-independent tensor statements.