12.9.2 Tensor Evaluation Argument Compatibility
Tensor Evaluation Argument Compatibility ensures consistent results by aligning input structures with algebraic rules in tensor algebra.
Tensor Evaluation Argument Compatibility is the requirement that every vector or covector supplied as an input during tensor evaluation must belong to the same vector space and dual space that the tensor itself is built from, and must be of the correct kind, vector or covector, expected by the particular slot it is intended to fill.
Components of the Compatibility Requirement
Matching Vector Space
For a tensor built over a vector space , any vector supplied as an argument to one of its vector slots must itself be an element of that same , not a vector from some other, unrelated vector space, even one of matching dimension.
Matching Dual Space
Likewise, any covector supplied to one of the tensor's covector slots must be an element of the dual space associated with that same underlying vector space , since the evaluation operation relies on the natural pairing between and to produce a scalar contribution from each contracted index.
Correct Kind for Each Slot
Beyond belonging to the correct space, each supplied argument must also be of the kind expected by its particular slot: a vector slot requires a vector, and a covector slot requires a covector. Supplying a covector where a vector is expected, or the reverse, does not correspond to any defined contraction, since the two objects pair with different types of tensor indices.
Why Compatibility Is Necessary
Contraction Requires a Well-Defined Pairing
The scalar contribution produced by evaluation, at each slot, comes from summing products of the tensor's components with the components of the supplied vector or covector, relying on the natural pairing between a vector space and its dual. This pairing exists specifically between and , and does not extend to an arbitrary unrelated vector space, even one that happens to share the same dimension.
Preventing Meaningless Contractions
Without argument compatibility, one might attempt to contract a tensor's index against a vector from an entirely different space, producing numbers that do not correspond to any meaningful geometric or algebraic pairing, since there is no canonical way to combine components from unrelated spaces.
Consequences of Incompatibility
Undefined Evaluation
If a supplied vector or covector fails to belong to the correct space, or is of the wrong kind for its slot, the evaluation operation is simply not defined at that slot, and consequently the overall evaluation of the tensor cannot proceed to produce a valid result.
Distinction from a Numerically Coincidental Match
Even if a mismatched vector happens to have the same number of components as required, without genuinely belonging to the correct vector space and pairing correctly with the tensor's indices, any apparent numerical result obtained would not represent a meaningful evaluation of the tensor.
Ensuring Compatibility in Practice
Verifying the Space Before Evaluation
Before evaluating a tensor on a given collection of vectors and covectors, it is necessary to confirm that every vector belongs to the tensor's underlying vector space and every covector belongs to its dual space, matching the specific slots they are intended to fill.
Compatibility Across Related Spaces
When vectors or covectors originate from a space related to by an explicit isomorphism, they must first be mapped into or using that isomorphism before argument compatibility, and therefore a valid evaluation, can be established.