Tenzor

Tenzor is a term used in several languages to denote the mathematical object commonly referred to in English as a tensor. In Turkish, for example, the standard spelling is tenzor, and the word appears in similar form in other languages that borrow the concept from multilinear algebra. Outside of language variations, tenzor generally refers to the same mathematical idea as a tensor: an algebraic object that generalizes scalars, vectors, and matrices.

Mathematically, a tenzor can be viewed in two equivalent ways. As an abstract multilinear map, it is a multilinear function that takes several vector or covector arguments and outputs a scalar or another tensor. As a component object, it is represented by a multi-dimensional array whose entries transform according to specific rules under a change of coordinates. The rank (or order) of a tenzor indicates the number of contravariant and covariant indices it carries; common examples include scalars (rank 0), vectors (rank 1), and matrices (rank 2). Covariant and contravariant components transform in complementary ways, and mixed tensors have both types of indices.

Core operations on tenzors include tensor product, contraction, and addition. The tensor product combines tensors to form higher-rank objects, while contraction reduces rank by summing over paired indices. Tenzors are often represented relative to a basis, with components that depend on the chosen coordinate system.

Applications span physics, engineering, and computer science. In physics, the metric tensor encodes spacetime geometry; in engineering, stress and strain are described by tensory quantities; in machine learning and data analysis, tensors index multi-dimensional datasets and are manipulated by specialized libraries. The term tenzor thus encodes a universal mathematical concept with language-specific spellings.