Curl of a Vector Field

Topics: Vector Field - Calculus

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(definition)

The curl of a vector field $F(x,y,z)=⟨F_1(x,y,z),F_2(x,y,z),F_3(x,y,z)⟩$ is the vector field:

$$\text{curl}F:=\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}\right)\text{i}+\left(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x}\right)\text{j}+\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\text{k}$$

Alternative Notation

More simply, with a bit of notation abuse, we can say that the curl of a vector field $F$ is:

$$\text{curl}F=\nabla \times F$$

…where $\nabla =⟨\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}⟩$; see cross product.

That is:

\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \end{vmatrix}$$

Curl tells us the rate of rotation of the vectors in a vector field.

When the curl of a vector field is $\vec{0}$, we say that the vector field is irrotational.