tensoriel

Tensoriel, in mathematics and physics, is the French term for tensorial. An object described as tensoriel either is a tensor or transforms under a change of coordinates according to a tensor transformation law. In differential geometry, a tensorial quantity is independent of the chosen coordinate frame in the sense that its components change in a prescribed way to preserve the underlying multilinear structure.

Tensors are multilinear objects that generalize scalars, vectors, and linear maps. They are classified by rank (order) and by the variance of indices: covariant and contravariant. A scalar has rank 0, a vector has rank 1, a matrix is a second-order tensor, and higher orders extend similarly. In practice, tensors are represented by components in a basis or as coordinate-free multilinear maps.

Operations include the tensor product, which combines two tensors into a higher-rank tensor, and contraction, which reduces rank by summing over a pair of indices. A metric tensor raises and lowers indices, making contravariant and covariant components interchangeable in formulas. Symmetries, such as symmetry or antisymmetry, simplify expressions.

Tensorial objects are central in many areas. In physics, the stress-energy and curvature tensors describe matter, energy, and spacetime. In continuum mechanics, strain and stress tensors model deformations. In differential geometry, curvature tensors encode the geometry of manifolds. In computing, tensors organize multi-dimensional data and underpin algorithms in machine learning and computer vision.