Infinitesimal calculus MOC

Curl

The curl of a vector field ⃗𝐅 :ℝ3 →ℝ3 is a measurement of the field’s tendency of a field to twist. It is given by

curl⁡⃗𝐅=∇×⃗𝐅

where ∇ is the nabla operator (sometimes called grad). The formula for curl is actually derived by an infinitesimal circulation integral. Let 𝐶 be filled disc at 𝑝 with normal ˆ𝐮 and area 𝜇(𝐶). Then curl is given by

(∇×⃗𝐅)(𝑝)⋅ˆ𝐮=lim𝜇(𝐶)→01𝜇(𝐶)∮𝜕𝐶⃗𝐅⋅𝑑⃗𝐫

Properties

• If curl⁡⃗𝐅 =⃗𝟎 everywhere, then ⃗𝐅 is a Conservative vector field.
• Conversely, for any continuous, smooth ∇𝑓, curl⁡∇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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