Curl（旋度） of a vector field

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#旋度 #Curl

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旋度（curl）是向量分析中的一个概念，表示向量场的旋转特性。它通过三阶行列式计算得出，对于流体动力学中的速度场，旋度衡量了局部的旋转效应。例如，一个旋度为零的向量场是无旋的，意味着流体不发生旋转。另一方面，非零旋度表示存在旋转。此外，旋度也可用于理解刚体旋转中速度场的性质，其值等于刚体关于旋转轴的角速度的两倍。

定义

The curl of a vector field $a(x,y,z)\boldsymbol{a}(x, y, z)$ is defined by
$\boldsymbol{a}=\nabla\times \boldsymbol{a}=(\frac{\partial a_z}{\partial y}-\frac{\partial a_y}{\partial z})\boldsymbol{i}+(\frac{\partial a_x}{\partial z}-\frac{\partial a_z}{\partial x})\boldsymbol{j}+(\frac{\partial a_y}{\partial x}-\frac{\partial a_x}{\partial y})\boldsymbol{k},$
where $a_x$ , $a_y$ and $a_z$ are the $x$ -, $y$ - and $z$ - components of $a\boldsymbol{a}$ . The RHS can be written in a more memorable form as a determinant:
$∇×a=∣ijk∂∂x∂∂y∂∂zaxayaz∣,\nabla\times \boldsymbol{a}=\begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ a_x & a_y & a_z \end{vmatrix},$
where it is understood that, on expanding the determinant, the partial derivatives in the second row act on the components of $a\boldsymbol{a}$ in the third row.
Clearly, $∇×a\nabla\times \boldsymbol{a}$ is itself a vector field. Any vector field $a\boldsymbol{a}$ for which $∇×a=0\nabla\times \boldsymbol{a}=0$ is said to be irrotational.

例子

例子 1

Find the curl of the vector field $a=x2y2z2i+y2z2j+x2z2k\boldsymbol{a}=x^2y^2z^2\boldsymbol{i}+y^2z^2\boldsymbol{j}+x^2z^2\boldsymbol{k}$ .

Solution

The Curl of $a\boldsymbol{a}$ is given by
$∇×a=∣ijk∂∂x∂∂y∂∂zx2y2z2y2z2x2z2∣=−2[y2zi+(xz2−x2y2z)j+x2yz2k].\nabla\times \boldsymbol{a}=\begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ x^2y^2z^2 & y^2z^2 & x^2z^2 \end{vmatrix}=-2[y^2z\boldsymbol{i}+(xz^2-x^2y^2z)\boldsymbol{j}+x^2yz^2\boldsymbol{k}].$

Physical properties

For a vector field $v(x,y,z)\boldsymbol{v}(x, y, z)$ describing the local velocity at any point in a fluid, $∇×v\nabla\times \boldsymbol{v}$ is a measure of the angular velocity of the fluid in the neighbourhood of that point. If a small paddle wheel were placed at various points in the fluid then it would tend to rotate in regions where $∇×v≠0\nabla\times \boldsymbol{v} \neq 0$ , while it would not rotate in regions where $∇×v=0\nabla\times \boldsymbol{v}=0$ .
Another insight into the physical interpretation of the curl operator is gained by considering the vector field $v\boldsymbol{v}$ describing the velocity at any point in a rigid body rotating about some axis with angular velocity $ω\omega$ . If $r\boldsymbol{r}$ is the position vector of the point with respect to some origin on the axis of rotation then the velocity of the point is given by $v=ω×r\boldsymbol{v}=\boldsymbol{\omega} \times \boldsymbol{r}$ . Without any loss of generality, we may take $ω\boldsymbol{\omega}$ to lie along the $z$ -axis of our coordinate system, so that $ω=ωk\omega=\omega \boldsymbol{k}$ . The velocity field is then $v=−ωyi+ωxj\boldsymbol{v}=-\omega y\boldsymbol{i}+\omega x\boldsymbol{j}$ . The curl of this vector field is easily found to be
$∇×v=∣ijk∂∂x∂∂y∂∂z−ωyωx0∣=2ωk=2ω.\nabla\times \boldsymbol{v}=\begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ -\omega y & \omega x & 0 \end{vmatrix}=2\omega \boldsymbol{k}=2\omega.$
Therefore the curl of the velocity field is a vector equal to twice the angular velocity vector of the rigid body about its axis of rotation.