Scalar
A quantity all values of which can be expressed by one (real) number. More generally, a scalar is an element of some field.
Comments
Generally, the setting is that of a (ground) field (more generally, a ring
) and a vector space
(of functions, vectors, matrices, tensors, etc.) over it (more generally, a module
). The elements of
(respectively,
) are called scalars. If
(respectively,
) is an algebra with unit element
, the elements
,
in
(respectively,
) are also called scalars. For example, one sometimes speaks of the (
)-matrices
as scalar matrices. The scalar multiples of an element
(respectively,
) are the elements
,
(respectively,
,
).
The term "scalar" comes from the original meaning as a quantity which can be completely specified by one (real) number.
A scalar field on a manifold is a function on
; that is, a scalar field, or field of scalars, is a tensor field (cf. Tensor bundle) of rank
. These are the scalars in the algebra of tensor fields on
over the ring of functions on
.
A scalar operator on, say, a complex Banach space is a scalar multiple of the identity operator.
Given a left module over a ring
and an
-algebra
, one forms the tensor product
. This is a module over
. The module
is said to be obtained from
by extension of scalars.
References
[a1] | P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 70 |
[a2] | K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 270; 290 |
Scalar. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scalar&oldid=48613