Vectortensor

Vectortensor is a nonstandard term often used informally to describe objects that combine vector and tensor structures. In mathematical literature, it is more precise to speak of vector-valued tensor fields or tensor-valued vector fields. Concretely, for a smooth manifold M and a fixed finite-dimensional vector space V, a vectortensor of type (k, l) with values in V is a smooth section of the tensor bundle T^k_l(M) tensored with V. Equivalently, it can be viewed as a family of (k, l)-tensors at each point of M, with components labeled by an index running over a basis of V.

Interpretations and representations vary with context. If V ≅ R^n, a vectortensor of type (k, l) with values in V can be seen as an n-tuple of ordinary (k, l)-tensor fields, or as a single tensor field whose components themselves are vectors in R^n. In coordinates, one writes components that carry two types of indices: the tensor indices for the (k, l) structure and an extra vector-space index for the V-valued part.

Operations on vectortensors are performed pointwise in the same way as for ordinary tensor fields, including addition, scalar multiplication, contractions, and tensor products, while respecting the V-valued structure. They arise naturally in several applications, such as differential geometry (vector-valued differential forms, stress- or velocity-field representations in continuum mechanics), and computer graphics or physics simulations where multiple vector fields are bundled together as a single object.

Because “vectortensor” is not a standard term, precise meaning is defined locally in a given work. The preferred terminology is usually vector-valued tensor field or tensor-valued vector field, accompanied by explicit type and base-space specifications.