Curl

The circulation of a vector field V about the following small (δ << 1) square:


in rectangular coordinates is approximately:

Vx(a, b - δ, 0)(2δ) + Vy(a + δ, b, 0)(2δ) - Vx(a, b + δ, 0)(2δ) - Vy(a - δ, b, 0)(2δ).

With respect to (a, b), it is approximately:

(2δ)(Vx - ∂yVxδ + Vy + ∂xVyδ - Vx - ∂yVxδ - Vy + ∂xVyδ)

or:

(∂xVy - ∂yVx)(2δ)2.

Such calculations can be used to find the curl of V.

The components of the curl of V in rectangular coordinates are:
  • (∇ ×V)x = ∂yVz - ∂zVy
  • (∇ ×V)y = ∂zVx - ∂xVz
  • (∇ ×V)z = ∂xVy - ∂yVx
The components of the curl of V in cylindrical coordinates are:
  • (∇ ×V)ρ = ∂φVz / ρ - ∂zVφ
  • (∇ ×V)φ = ∂zVρ - ∂ρVz
  • (∇ ×V)z = ∂ρ(ρVφ) / ρ - ∂φVρ / ρ
The components of the curl of V in spherical coordinates are:
  • (∇ ×V)r = ∂θ(Vφsinθ) / (r sinθ) - ∂φVθ / (r sinθ)
  • (∇ ×V)φ = ∂r(rVθ) / r - ∂θVr / r
  • (∇ ×V)θ = ∂φVr / (r sinθ) - ∂r(rVφ) / r

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