Tensors keep geometry honest when coordinates change
A tensor is an algebraic object that encodes multilinear links among vectors, scalars, and other tensors on a vector space. Its components form a multi-way array once you pick a basis, yet the geometric meaning is meant to survive a change of coordinates. That discipline is why physicists lean on them so hard.
Scalars and vectors are the simplest tensors; dual vectors, multilinear maps, and even the ordinary dot product sit in the same family. With respect to a basis, an n-dimensional vector becomes a one-dimensional list of n entries, while a linear operator appears as an n-by-n square array. Indices mark each component—Tij or T ij for an order-2 example—and whether an index rides upstairs or downstairs signals how it transforms. The total number of indices is the order, degree, or (sometimes) rank of the tensor, though “rank” also means something else for matrices.
Change the basis and every component must obey a transformation law. Contravariant vector components transform with the inverse of the change-of-basis matrix R; covariant covector components transform with R itself. A general (p, q)-tensor mixes p contravariant and q covariant indices, each carrying its own factor. The matrix of a linear operator is type (1,1): one index of each kind. When matching upper and lower indices contract, the R factors cancel, leaving geometric invariants such as the vector itself written as vi ei in any frame—the Kronecker delta doing the renaming.
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised the language in 1900, building on Bernhard Riemann, Elwin Bruno Christoffel, and others under the banner of absolute differential calculus. That toolkit reframed the intrinsic geometry of a manifold through the Riemann curvature tensor. In applications, a different tensor may sit at each point of a body—stress varying through a solid—so the whole field of tensors is called a tensor field, often shortened to “tensor” in physics talk.
Mechanics uses them for stress, elasticity, fluid flow, and moments of inertia; electrodynamics for the electromagnetic and Maxwell tensors, permittivity, and magnetic susceptibility; general relativity for the stress–energy and curvature tensors. Different formal definitions sound unlike, yet they describe one geometric idea at different levels of abstraction.
Source: Tensor
