Scalar (math)

A scalar is a mathematical quantity that is characterized solely by specifying a numerical value (in physics, possibly with a unit ).

In the mathematical branch of linear algebra , scalar denotes an element of the basic field of a vector space , usually a real number . In contrast to this, the elements of a vector space are called vectors . Correspondingly, the base body is also called a scalar body . The multiplication of a vector by a scalar is called scalar multiplication or scaling . The resulting vector is called a scalar multiple of . ${\ displaystyle v}$ ${\ displaystyle \ lambda}$ ${\ displaystyle \ lambda \ cdot v}$ ${\ displaystyle v}$

In contrast to the scalar multiplication, the scalar product is a link that assigns a scalar as a value to two vectors .

The term scalar goes back to the Latin word scala (ladder) in the sense of a uniform division (see scale ).

Scalars in physics

In physics, scalars are used to describe physical quantities that are direction-independent. Examples of scalar physical quantities are the mass of a body, its temperature , its energy and also its distance from another body (as the amount of the difference between the position vectors ). In other words: a scalar physical quantity does not change when the position or orientation changes. If, on the other hand, a direction is required for the complete description of the variable, as is the case with the force or the speed, a vector is used, if several directions are dependent on a tensor (more precisely: tensor 2nd or higher level).

The speed of a particle has the direction in which the particle is moving. Since the direction changes with rotation, the speed is not a scalar, but a vector. But the amount of speed does not change with rotations and is a scalar.

Whether a quantity is a scalar depends on the transformation group under consideration . The energy is a scalar with regard to rotations, but in the relativity theory it is a component of a four-vector .

A subgroup of the scalars are the pseudoscalars , which reverse the sign under a plane reflection .

Extensions and delimitation of similar terms

Square matrices , which (understood as a linear mapping of a vector space onto itself) correspond to a multiplication of each vector by a fixed scalar , are assigned the property scalar . They are diagonal matrices whose entries on the diagonal are all the same . ${\ displaystyle \ lambda}$ ${\ displaystyle \ lambda}$

In a module over a ring , the multiplication of a module element by an element of the base ring is also called scalar multiplication. The term scalar for the elements of the base ring is only partly used in this case.

literature

H. Fischer, H. Kaul: Mathematics for Physicists. Volume 1, 7th edition, Vieweg u. Teubner 2011, ISBN 978-3-8348-1220-9

Web links

Wiktionary: Scalar - explanations of meanings, word origins, synonyms, translations

Individual evidence

↑ DWDS - Digital Dictionary of the German Language. Retrieved July 22, 2020 .

[1] DWDS - Digital Dictionary of the German Language. Retrieved July 22, 2020 .