Scalar (physics)

In

cm

". Examples of scalar quantities are

physical vectors in general (such as velocity).^[1]

A change of a vector space basis changes the description of a vector in terms of the basis used but does not change the vector itself, while a scalar has nothing to do with this change. In classical physics, like

Newtonian mechanics, rotations and reflections preserve scalars, while in relativity, Lorentz transformations or space-time translations preserve scalars. The term "scalar" has origin in the multiplication of vectors by a unitless scalar, which is a uniform scaling transformation

.

Relationship with the mathematical concept

A scalar in physics is also a

absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field for the vector space in which the electric field is described. As the vector space in this example and usual cases in physics is defined over the mathematical field of real numbers or complex numbers

, the magnitude is also an element of the field, so it is mathematically a scalar. Since the inner product is independent of any vector space basis, the electric field magnitude is also physically a scalar.

The mass of an object is unaffected by a change of vector space basis so it is also a physical scalar, described by a real number as an element of the real number field. Since a field is a vector space with addition defined based on

vector addition and multiplication defined as scalar multiplication

, the mass is also a mathematical scalar.

Scalar field

Since scalars mostly may be treated as special cases of multi-dimensional quantities such as

spinor fields, and tensor fields

.

Units

Like other

metric

in the sense that it is not just a real number while the metric is calculated to a real number, but the metric can be converted to the physical distance by converting each base vector length to the corresponding physical unit.

Any change of a coordinate system may affect the formula for computing scalars (for example, the

orthonormal), but not the scalars themselves. Vectors themselves also do not change by a change of a coordinate system, but their descriptions changes (e.g., a change of numbers representing a position vector

by rotating a coordinate system in use).

Classical scalars

An example of a scalar quantity is temperature: the temperature at a given point is a single number. Velocity, on the other hand, is a vector quantity.

Other examples of scalar quantities in physics are

gravitational force acting on a particle is not a scalar, but its magnitude is. The speed of an object is a scalar (e.g., 180 km/h), while its velocity

is not (e.g. a velocity of 180 km/h in a roughly northwest direction might consist of 108 km/h northward and 144 km/h westward). Some other examples of scalar quantities in Newtonian mechanics are electric charge and charge density.