Curl (𝛻⨯𝑓)

Curl - Definition

2D Definition

If a vector field is given by a function:

• $\overline{v}(x,y)=v_1(x,y)\hat{i}+v_2(x,y)\hat{j}$
• $\overline{v}(x,y)=\left[\begin{array}{c}{v}_1(x,y)\\ v_2(x,y)\end{array}\right]$

Curl is given by the formula:

• $\text{2d-curl}\overline{v}=\frac{𝜕v_2}{𝜕x}-\frac{𝜕v_1}{𝜕y}$

3D Definition

If a vector field is given by a function:

• $\overline{v}(x,y,z)=v_1(x,y,z)\hat{i}+v_2(x,y,z)\hat{j}+v_3(x,y,z)\hat{k}$
• $\overline{v}(x,y)=\left[\begin{array}{c}{v}_1(x,y,z)\\ v_2(x,y,z)\\ v_3(x,y,z)\end{array}\right]$

Curl is given by the formula:

• $\nabla ⨯\overline{v}=(\frac{𝜕v_3}{𝜕y}-\frac{𝜕v_2}{𝜕z})\hat{i}+(\frac{𝜕v_1}{𝜕z}-\frac{𝜕v_3}{𝜕x})\hat{j}+(\frac{𝜕v_2}{𝜕x}-\frac{𝜕v_1}{𝜕y})\hat{k}$
• $\nabla ⨯\overline{v}=\left[\begin{array}{c}\frac{𝜕v_3}{𝜕y}-\frac{𝜕v_2}{𝜕z}\\ \frac{𝜕v_1}{𝜕z}-\frac{𝜕v_3}{𝜕x}\\ \frac{𝜕v_2}{𝜕x}-\frac{𝜕v_1}{𝜕y}\end{array}\right]$

where:

• $\nabla =\left[\begin{array}{c}\frac{𝜕}{𝜕x}\frac{𝜕}{𝜕y}\frac{𝜕}{𝜕z}\end{array}\right]$
• ⨯ is the cross-product of 𝑣̅ and 𝛻.

Intuition:

• $(\frac{𝜕v_3}{𝜕y}-\frac{𝜕v_2}{𝜕z})\hat{i}\text{ ← rotational component parallel to the }yz\text{-plane}$
• $(\frac{𝜕v_1}{𝜕z}-\frac{𝜕v_3}{𝜕x})\hat{j}\text{ ← rotational component parallel to the }xz\text{-plane}$
• $(\frac{𝜕v_2}{𝜕x}-\frac{𝜕v_1}{𝜕y})\hat{k}\text{ ← rotational component parallel to the }xy\text{-plane}$

In 2D to describe rotation, you need a single number: the angular velocity:

• a positive number indicates a counter-clockwise​ rotation
• a negative number indicates a clockwise rotation

The absolute value of the angular velocity gives the speed of rotation, typically in radians per second.

In 3D to describe rotation you need a vector:

• the magnitude of the vector is equal to twice the angular velocity
• the direction is determined by a super-important convention called the “right-hand rule”

Resources

• Steve Brunton’s The Curl of a Vector Field: Measuring Rotation
• https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/divergence-and-curl-articles/a/curl-warmup
• https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/curl-grant-videos/v/2d-curl-intuition