Field (physics)
Definition
In physics and mathematics, for a function $f : X \to Y$, when $X$ is a vector space, $f$ is called a field.
Explanation
- Different from the field in algebra.
In fact, the above definition is somewhat abstract. A field is a function that maps a point in some space, i.e., a vector, to another value; in simple terms it is a multivariable function. Here, “space” usually means Euclidean space, and depending on the case it can be Minkowski space or an arbitrary manifold. In the context of functions, the word “field” is not used alone (in that case it refers to the field of algebra); if a vector is mapped to a scalar it is called a scalar field, if a vector is mapped to a vector it is called a vector field, and if a vector is mapped to a tensor it is called a tensor field. More intuitively, a function that assigns a point of space to a scalar 🔒(26/04/13)physical quantity is called a scalar field (스칼라장 in Korean), and a function that assigns a point of space to a vector physical quantity is called a vector field (벡터장 in Korean).
| Function | Mapping | Examples |
|---|---|---|
| Scalar field | vector $\mapsto$ scalar | Temperature at point $(x, y, z)$ $T = T(x,y,z)$ |
| Vector field | vector $\mapsto$ vector | Velocity of an object 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 an object at point $(x, y, z)$ $\sigma = \sigma(x, y, z) = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}$ |
A 🔒(26/03/16)scalar field denotes the function $f: \mathbb{R}^{n} \to \mathbb{R}$.
A 🔒(26/03/28)vector field denotes the function $f: \mathbb{R}^{n} \to \mathbb{R}^{m}$.
A 🔒(26/03/30)tensor field denotes the function $f: \mathbb{R}^{n} \to \mathbb{R}^{m \times p}$.