Curl
Curl A measure of average rotation around points in a vector field.
$$\mathbf{curl}\ \vec{v}(x,y) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y},\ \text{where}\ \vec{v}(x,y) = \begin{bmatrix} P(x,y) \ Q(x,y) \end{bmatrix} $$
$$ \begin{cases} \mathbf{curl}\ \vec{v}(x,y) > 0 &\Rightarrow &\text{counter clockwise flow} \ \mathbf{curl}\ \vec{v}(x,y) < 0 &\Rightarrow &\text{clockwise flow} \ \mathbf{curl}\ \vec{v}(x,y) = 0 &\Rightarrow &\text{no rotation} \end{cases} $$
2D-curl 3-Dimensions $$ \vec{v}(x,y,z) = \begin{bmatrix} f_1(x,y,z) \ f_2(x,y,z) \ f_3(x,y,z) \end{bmatrix} $$ $$\mathbf{curl}\ \vec{v}(x,y,z) = \nabla \times \vec{v}(x,y,z) = \begin{bmatrix} \frac{\partial}{\partial x} \ \frac{\partial}{\partial y} \ \frac{\partial}{\partial z} \end{bmatrix} \times \begin{bmatrix} P(x,y,z) \ Q(x,y,z) \ R(x,y,z) \end{bmatrix} = \det \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ f_{1} & f_2 & f_3 \end{bmatrix} $$ $$ = \begin{align} \left(\frac{\partial f_{3}}{\partial y} - \frac{\partial f_{2}}{\partial z}\right) i + \left(\frac{\partial f_{1}}{\partial z} - \frac{\partial f_{3}}{\partial x}\right) j + \left(\frac{\partial f_{2}}{\partial x} - \frac{\partial f_{1}}{\partial y}\right) k \end{align} $$
0001/01/01 ·
Notes