Scalar Field
Definition
In physics, a scalar field is a function that assigns a scalar value to each vector (point) in space, i.e., a function of the following form.
$$ f: \mathbb{R}^{n} \to \mathbb{R} $$
Explanation
A function named a field indicates that the elements of its domain are vectors. Thus, a scalar field is a function that maps vectors to scalars.
| Function | Mapping | Example |
|---|---|---|
| Scalar field | vector $\mapsto$ scalar | temperature at point $(x, y, z)$ $T = T(x,y,z)$ |
| Vector field | vector $\mapsto$ vector | velocity of a body at point $(x, y, z)$ $\mathbf{v} = \mathbf{v}(x, y, z) = \begin{bmatrix} v_{x}(x,y,z) \\ v_{y}(x,y,z) \\ v_{z}(x,y,z) \end{bmatrix}$ |
| Tensor field | vector $\mapsto$ tensor | stress on a body at point $(x, y, z)$ $\sigma = \sigma(x, y, z) = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}$ |
Examples include the following.
- The gravitational potential in Newtonian mechanics
- In electromagnetism, the electric potential $V$. Differentiating it yields $E = -\nabla V$, the electric field.