12.1.3 Tensor Mapping Operation Scope

Tensor Mapping Operation Scope defines how tensors transform under linear mappings, establishing their behavior across different mathematical spaces and structures.

Tensor Mapping Operation Scope is the set of prerequisite conditions governing when a linear map between two vector spaces can be used to transport a tensor from one space to the other, specifying that purely contravariant tensors move forward along the map through pushforward, purely covariant tensors move backward through pullback, and mixed tensors require the map itself to be invertible before any transport is defined at all.

Foundational Setting

Transport as Distinct from Basis Change

Where the earlier tensor operations act within a single, fixed vector space, mapping operations instead relate tensors defined on two different vector spaces, connected by an explicit linear map. This is a related but distinct situation from an ordinary change of basis, since the two spaces involved need not even have the same dimension.

The Two Basic Directions of Transport

Given a linear map $f : V \to W$ , two natural, oppositely directed transport operations arise: pushforward, carrying objects from $V$ to $W$ , and pullback, carrying objects from $W$ back to $V$ .

Scope of Pushforward

Applying Pushforward to Contravariant Vectors

Pushforward applies directly to a vector in $V$ , producing a vector in $W$ simply by applying the map:

w^{i} = \sum_{j} f_{j}^{i} v^{j}

Why No Inverse Is Required Here

This operation falls within scope for any linear map $f$ , invertible or not, since pushforward for a purely contravariant vector requires only the map itself, not its inverse, to be applied.

Scope of Pullback

Applying Pullback to Covariant Covectors

Pullback applies to a covector in $W^{, producing a covector in V^{} by precomposing with the map: ω_{j} = \sum_{i} f_{j}^{i} η_{i} Also Requiring No Inverse Similarly, pullback for a purely covariant object falls within scope for any linear map, since it operates directly on the covector using the map's own components, without needing the map to be invertible or even a bijection between spaces of the same dimension. Scope of Mixed Tensor Transport Requiring Invertibility Transporting a mixed tensor, one with both upper and lower indices, requires applying pushforward to its contravariant indices and pullback to its covariant indices simultaneously. Doing so consistently in either direction requires the map f to be invertible, since transporting a mixed tensor from W back to V, for instance, requires the inverse map to act on the contravariant indices: T_{j}^{i} (V) = \sum_{k, l} {(f^{- 1})}_{k}^{i} f_{j}^{l} T_{l}^{k} (W) Falling Outside Scope Without Invertibility If f is not invertible, mixed tensor transport in this fully symmetric sense simply falls outside the operation's scope, even though the purely contravariant part could still be pushed forward and the purely covariant part could still be pulled back separately in their own single directions. Visual Overview Diagram of Directional Scope V W pushforward (vectors) pullback (covectors) Mixed tensor transport in either single direction requires f to be invertible. Summary of Key Traits Defining Characteristics Pushforward transports contravariant vectors along any linear map, invertible or not. Pullback transports covariant covectors along any linear map in the opposite direction, likewise without requiring invertibility. Transporting a mixed tensor as a single consistent object requires the map connecting the two spaces to be invertible. Without invertibility, only the purely contravariant and purely covariant parts of an object can be separately transported, each in its own single direction. Related content 12.1.1 Tensor Algebraic Operation Scope 12.1.2 Tensor Evaluation Operation Scope 12.1.4 Tensor Construction Operation Scope 12.1.5 Tensor Operation Verification Scope Location Codartium Formal Sciences Mathematics Algebra Tensor Algebras 12 Tensor Operations 12.1 Tensor Operation Scope 12.1.3 Tensor Mapping Operation Scope (You are) © 2025 Codartium — Made for everyone, free. About Contact Disclaimer Privacy Terms}$