Vector Field

Definition

In physics, a vector field is a mapping that assigns a vector value to each vector (point) of space, i.e., a function of the following form.

$$ f: \mathbb{R}^{n} \to \mathbb{R}^{m} $$

Explanation

There is also the term vector (value) function used in the same sense. A function called a field indicates that the elements of its domain are vectors. Thus, a vector field is a function that maps vectors to vectors.

함수	대응관계	예시
스칼라 필드	벡터 $\mapsto$ 🔒(26/04/27)스칼라	점 $(x, y, z)$에서의 온도 $T = T(x,y,z)$
벡터 필드	벡터 $\mapsto$ 벡터	점 $(x, y, z)$에서 물체의 속도 $\mathbf{v} = \mathbf{v}(x, y, z) = \begin{bmatrix} v_{x}(x,y,z) \\ v_{y}(x,y,z) \\ v_{z}(x,y,z) \end{bmatrix}$
텐서 필드	벡터 $\mapsto$ 텐서	점 $(x, y, z)$에서 물체가 받는 응력 $\sigma = \sigma(x, y, z) = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}$

Examples include the following.

(linear) velocity $$ \mathbf{v}(x, y, z) = (v_x(x, y, z), v_y(x, y, z), v_z(x, y, z)) $$
angular velocity $$ \boldsymbol{\omega}(x, y, z) = (\omega_x(x, y, z), \omega_y(x, y, z), \omega_z(x, y, z)) $$
electric field $\mathbf{E}$: $$ \mathbf{E}(x, y, z) = (E_x(x, y, z), E_y(x, y, z), E_z(x, y, z)) $$
magnetic field $\mathbf{B}$: $$ \mathbf{B}(x, y, z) = (B_x(x, y, z), B_y(x, y, z), B_z(x, y, z)) $$