Ranktwo

Rank-two tensor is a second-order tensor. In a finite-dimensional vector space V, it can be viewed as a bilinear form on V×V, or as a linear map from V* to V. With a basis, it is represented by a square matrix of components T_{ij}, and these components transform according to tensor transformation rules when the basis changes.

Rank-two tensors may be symmetric (T_{ij}=T_{ji}) or antisymmetric (T_{ij}=-T_{ji}). Symmetric tensors correspond to quadratic forms and metric-like objects; antisymmetric tensors relate to oriented areas and angular momentum. In many contexts, they are described as sections of the tensor product V*⊗V* (covariant) or V⊗V (contravariant), depending on index placement.

Applications occur in physics and engineering: the stress and strain tensors in continuum mechanics, the metric tensor in general relativity, and the inertia tensor in rigid body dynamics. In mathematics, rank-two tensors generalize matrices and bilinear forms, and their eigenvalues and invariants drive many structural results.

Notes: The term ranktwo is not widely defined as a standalone entity. In contexts where the term is capitalized or used as a proper name, it may refer to a project, username, or brand, but such usage is context-dependent.