vectortensordescribes

Vectortensordescribes is a term used to denote a rule that assigns to each input vector a tensor that describes directional properties at a point. In this sense, it is a mapping from a vector space, often a tangent space on a manifold, to a tensor space, frequently the space of second-order tensors. The resulting tensor is called the descriptor of the input vector.

In a formal framing, let V be a vector space and T be a chosen tensor space on a domain. A vectortensordescribes is a (smooth or continuous) map f: V -> T that associates to every vector v in V a tensor f(v). The most common target is a rank-2 tensor, such as a symmetric matrix, which can encode anisotropic or directional information linked to the direction and magnitude of v. A simple example is f(v) = v ⊗ v, the outer product, which yields a dyadic descriptor that emphasizes the direction of v. More general descriptors may depend nonlinearly on v or incorporate additional parameters.

Applications of vectortensordescribes appear in mathematics and applied sciences where direction-dependent properties are important. They can model directional stiffness in continuum mechanics, diffusion orientation in imaging modalities, or gradient-structured data in machine learning. The concept serves to connect vector information with tensorial descriptions, offering a compact way to encode how a vector determines associated tensorial structure.

Relation to related ideas includes tensor fields and tensor-valued fields, with vectortensordescribes emphasizing the input vector as the source of the output tensor, rather than the point alone.