Tensor (mathematics and physics)

Basic ideas and types

Tensors are classified by order (also called rank):

• Order 0: a scalar, a single number.
• Order 1: a vector, which can be seen as a list of components that transform linearly under change of basis.
• Order 2 and higher: objects that can map vectors to vectors or relate several vector arguments to a scalar in a multilinear way. A common order-2 example is a matrix, but not every matrix is treated as a tensor unless transformation rules are specified.

More formally, a tensor can be defined as a multilinear map on copies of a vector space and its dual, or as an element of a tensor product of vector spaces. The linear algebra concept of a tensor product constructs new spaces from existing ones, written symbolically as V₁ ⊗ V₂.

Coordinate representation and transformation

In coordinates, a tensor is represented by an array of numbers whose arrangement and index positions encode how each component transforms when the underlying basis changes. The defining feature of a tensor is this transformation law: components mix with combinations of basis change coefficients so the geometric quantity remains invariant. This distinguishes tensors from arbitrary arrays of numbers: one must specify how components behave under coordinate transformations to identify a tensor.

History and development

The modern tensor concept grew from 19th-century studies in differential geometry and the mechanics of continua. Mathematicians such as Riemann, Ricci, and Levi-Civita developed tools for expressing curvature and covariant differentiation, which later became central to Albert Einstein's general theory of relativity. Over the 20th century, tensor notation and theory were unified and abstracted, providing a flexible framework used across many fields.

Applications and examples

• Continuum mechanics: stress and strain tensors describe internal forces and deformations in materials; see elasticity.
• Fluid mechanics: tensors represent viscous stresses and rate-of-strain relations in fluids (fluid mechanics).
• Gravitation: the Einstein field equations relate the curvature tensor to energy and momentum in general relativity.
• Numerical computing and data science: the word "tensor" is also used for multidimensional arrays in software libraries, enabling efficient representation of high-dimensional data.

Distinctions and notable facts

Key points to remember: tensors are coordinate-independent geometric objects; their component arrays depend on a choice of basis but the object itself does not. There is a practical distinction between treating an array as a mere data structure and as a tensor with prescribed transformation rules. The tensor product operation creates higher-order tensors from lower-order ones and is central in both pure mathematics and applied contexts. For further foundational reading see textbooks on multilinear algebra and differential geometry, or introductory resources on linear maps, vectors, and the linear algebra background that underpins tensor theory.

For concise technical introductions and computational examples, consult specialized references and software documentation that treat tensors as both algebraic objects and practical data structures.

Related resources: mathematical overview, physical applications, elasticity, fluid mechanics, general relativity, etymology, scalars, linear maps, vectors, matrices, linear algebra.