have slightly different meanings in mathematics and
physics. In the mathematical fields of multilinear algebra and differential
geometry, a tensor is a multilinear function. In physics and engineering,
the same term usually means what a mathematician would call a tensor field:
an association of a different (mathematical) tensor with each point of a
geometric space, varying continuously with position. For example, the
Euclidean inner product (dot product) — a real-valued function of two
vectors that is linear in each — is a mathematical tensor. Similarly, on a
smooth curved surface such as a
torus, the
metric tensor (field) essentially defines a different inner product of
tangent vectors at each point of the surface. Just as a linear
transformation can be represented as a matrix of numbers with respect to
given vector bases, so a tensor can be written as an organized collection of
numbers. In physics, the numbers may be obtained as physical quantities that
depend on a basis, and the collection is determined to be a tensor if the
quantities transform appropriately under change of basis.