Curl Operator

The divergence operator is a differentiation operator that operates on vector fields. It represents how much rotation a point experiences as it moves along a vector field. It uses the cross product with partial derivatives.

Definition §

Curl Operator ^definition

Let $\vec{F}$ be a vector field in Euclidean space $\vec{F}:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ where $\vec{F}(x_1,\dots ,x_n)=(f_1(x_1,\dots ,x_n),\dots ,f_n(x_1,\dots ,x_n))$. If $\vec{F}$ is $C^1$ (that is, all $f_i$ are $C_1$), then the curl of $\vec{F}$ is:

$$\begin{array}{rl}\nabla \times \vec{F}=\text{curl }\vec{F}& =\left(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n}\right)\times (f_1,\dots ,f_n)\end{array}$$

• If $\vec{F}:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$, then $\nabla \times \vec{F}:D\to {\mathbb{R}}^n$
• In the case of a real vector field $\vec{F}:{\mathbb{R}}^3\to {\mathbb{R}}^3$, if we have $\vec{F}(x,y,z)=(u(x,y,z),v(x,y,z),w(x,y,z))$ where $u,v,w:{\mathbb{R}}^3\to \mathbb{R}$ are scalar functions, we have:

$$\nabla \times \vec{F}=\det \left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ u& v& w\end{array}\right]=\left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z},\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x},\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)$$

Rotational Nature Of A Vector Field §

The curl operator can define the rotational nature of a vector field. Let $\vec{F}$ be a vector field:

• If $\nabla \times \vec{F}=0$ for all points $\vec{p}$, then $\vec{F}$ is an irrotational vector field
• For any non-zero curl, the direction of rotation is actually defined using the right curl rule:
  • If the $k$-component of the curl vector $\nabla \times \vec{F}$ is greater than zero, then the thumb points upwards, resulting in counter-clockwise rotation around the $z$-axis
  • Similarly, if the $k$-component of the curl vector $\nabla \times \vec{F}$ is less than zero, then the thumb points downwards, resulting in clockwise rotation around the $z$-axis
  • This can be generalise to both the $x$ and $y$-axis.

A way to visualise this is to imagine placing an object (a collection of points) in this vector field. If this object rotates as it moves, then the vector field has non-zero curl. Else, it is irrotational (zero curl):

#todo Add propellor to show ‘spin’ %%🖋 Edit in Excalidraw, and the dark exported image%%

Applications §

Fluids §

• A non-zero curl, $\nabla \times \vec{F}\ne 0$ describes rotational flow
• Similarly, a zero curl describes irrotational flow.

Forces §

• If $\vec{F}$ represents a force field, then $\nabla \times \vec{F}=\vec{0}$ defines $F$ to be a conservative force.
• A non-zero curl describes a non-conservative force.

Examples §

1: Positive Curl Around Z-axis

Evaluate the curl of the vector field given below:

$$\vec{F}(x,y)=(-y,x)$$

Solution

Using the curl formula, we have:

$$\begin{array}{rl}\nabla \times \vec{F}=\det \left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ -y& x& 0\end{array}\right]& =\left[\begin{array}{ccc}-\frac{\partial }{\partial z}(x)& -\left(-\frac{\partial }{\partial z}(-y)\right)& \frac{\partial }{\partial x}(x)+\frac{\partial }{\partial y}(y)\end{array}\right]\\ & =\left[\begin{array}{ccc}0& 0& 1+1\end{array}\right]=\left[\begin{array}{ccc}0& 0& 2\end{array}\right]\end{array}$$

By using the right curl rule (point thumb in positive $z$-axis), we know that this vector field rotates anti-clockwise around the $z$-axis.

Visuals

Resources §

• https://www.youtube.com/watch?v=gWbfdPUynz0