Del in cylindrical and spherical coordinates

Del formula

^α The source that is used for these formulae uses θ for the azimuthal angle and φ for the polar angle, which is common mathematical notation. This page uses θ for the polar angle and φ for the azimuthal angle, which is common notation in physics. In order to get the mathematics formulae, switch θ and φ in the formulae shown in the table above.

Non-trivial calculation rules

1. div grad ⁡ f ≡ ∇ ⋅ ∇ f ≡ ∇ 2 f
2. curl grad ⁡ f ≡ ∇ × ∇ f = 0
3. div curl ⁡ A ≡ ∇ ⋅ ( ∇ × A ) = 0
4. curl curl ⁡ A ≡ ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A (Lagrange's formula for del)
5. ∇ 2 ( f g ) = f ∇ 2 g + 2 ∇ f ⋅ ∇ g + g ∇ 2 f

div ⁡ A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = A x ( x + d x ) d y d z − A x ( x ) d y d z + A y ( y + d y ) d x d z − A y ( y ) d x d z + A z ( z + d z ) d x d y − A z ( z ) d x d y d x d y d z = ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z

( curl ⁡ A ) x = lim S ⊥ x ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A z ( y + d y ) d z − A z ( y ) d z + A y ( z ) d y − A y ( z + d z ) d y d y d z = ∂ A y ∂ z − ∂ A z ∂ y

The expressions for ( curl ⁡ A ) y and ( curl ⁡ A ) z are found in the same way.

div ⁡ A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = A ρ ( ρ + d ρ ) ( ρ + d ρ ) d ϕ d z − A ρ ( ρ ) ρ d ϕ d z + A ϕ ( ϕ + d ϕ ) d ρ d z − A ϕ ( ϕ ) d ρ d z + A z ( z + d z ) d ρ ρ d ϕ − A z ( z ) d ρ ρ d ϕ d ρ ρ d ϕ d z = 1 ρ ∂ ( ρ A ρ ) ∂ ρ + 1 ρ ∂ A ϕ ∂ ϕ + ∂ A z ∂ z

( curl ⁡ A ) ρ = lim S ⊥ ρ ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S

div ⁡ A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = A r ( r + d r ) ( r + d r ) d θ ( r + d r ) sin ⁡ θ d ϕ − A r ( r ) r d θ r sin ⁡ θ d ϕ + . . . d r r d θ r sin ⁡ θ d ϕ = 1 r 2 ∂ ( r 2 A r ) ∂ r + . . .

( curl ⁡ A ) r = lim S ⊥ r ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A θ ( ϕ ) r d θ + A ϕ ( θ + d θ ) r sin ⁡ ( θ + d θ ) d ϕ − A θ ( ϕ + d ϕ ) r d θ − A ϕ ( θ ) r sin ⁡ θ d ϕ r d θ r sin ⁡ θ d ϕ = 1 r sin ⁡ θ ∂ ( sin ⁡ ( θ ) A ϕ ) ∂ θ − 1 r sin ⁡ θ ∂ A θ ∂ ϕ