Scalar field

In mathematics and physics, a scalar field represents the spatial distribution of a scalar magnitude, associating a value to each point in space. In mathematics, the value is a number; in physics, a physical quantity. Scalar fields are used in physics, for example, to indicate the temperature or pressure distribution of a gas in space.

Definition

Mathematically, a scaling field in a region U Rn{displaystyle Usubset mathbb {R} ^{n} It's a function.

f:U→ → R{displaystyle f:Uto mathbb {R} }

at each point x한 한 U{displaystyle mathbf {x} in u}, assigned a number or scale f(x){displaystyle f(mathbf {x})}.

This function is also known as the point function or simply the scalar function.

Scalar fields in physics

In fluid mechanics pressure can be treated as a scalar field. The temperature distribution over a body is another scalar field. All these fields are classified as scalar fields for the sake of the necessary mathematical description. A construction that characterizes the scalar fields are the equipotential surfaces that are the sets of points on which the function takes the same value.

In relativistic physics, a scalar field is one for which the transformation law between the values measured by two different observers satisfy a tensor invariance relation. In this sense, the electric potential that in classical electromagnetism is treated as a scalar, in relativistic mechanics it is not a scalar but the temporary component of a four-vector potential that generalizes the classical vector potential.

In quantum physics, the term "scalar field" in a more restricted way, it is applied to describe the field associated with spin-zero particles (eg pions).

Scalar fields in differential geometry

Given a differentiable variety M{displaystyle {mathcal {M}}} and given an atlas of it:

{(Uα α ,φ φ α α )日本語Uα α M,φ φ α α :Uα α → → Rn!{displaystyle {(U_{alpha },phi _{alpha })Δquad U_{alpha }subset {mathcal {M}}, phi _{alpha }:U_{alpha }{to mathbb {R} ^{n}

A different scale field, :M→ → R{displaystyle Psi:{mathcal {M}}to R} is any function such that:

Русский Русский α α ,( φ φ α α − − 1):φ φ α α (Uα α ) Rn→ → R{displaystyle forall alphaquad (Psi circ phi _{alpha }{alpha }{phi _{alpha }(U_{alpha })subset mathbb {R} ^{n}to mathbb {R} }

It's a different scale field. Rn{displaystyle mathbb {R} ^{n}.