Curl

The curl of a vector field $⃗ 𝐅 : ℝ 3 \to ℝ 3$ is a measurement of the field's tendency of a field to twist. It is given by

c u r l ⃗ 𝐅 = \nabla \times ⃗ 𝐅

where $\nabla$ is the nabla operator (sometimes called $g r a d$ ). The formula for curl is actually derived by an infinitesimal circulation integral. Let $𝐶$ be filled disc at $𝑝$ with normal $ˆ 𝐮$ and area $𝜇 (𝐶)$ . Then curl is given by

(\nabla \times ⃗ 𝐅) (𝑝) \cdot ˆ 𝐮 = l i m 𝜇 (𝐶) \to 0 1 𝜇 (𝐶) \oint 𝜕 𝐶 ⃗ 𝐅 \cdot 𝑑 ⃗ 𝐫

Properties

If $c u r l ⃗ 𝐅 = ⃗ 𝟎$ everywhere, then $⃗ 𝐅$ is a Conservative vector field.
Conversely, for any continuous, smooth $\nabla 𝑓$ , $c u r l \nabla 𝑓 = ⃗ 𝟎$ (this is easy to prove by Clairaut's theorem).

Practice problems

These practice problems are for both curl and Divergence.

2016. Calculus, p. 1149 (§16.5 exercises)

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