12.2: Vector Operators
( \newcommand{\kernel}{\mathrm{null}\,}\)
This section contains a summary of vector operators expressed in each of the three major coordinate systems:
- Cartesian (x,y,z)
- cylindrical (ρ,ϕ,z)
- spherical (r,θ,ϕ)
Associated basis vectors are identified using a caret (ˆ ) over the symbol. The vector operand A is expressed in terms of components in the basis directions as follows:
- Cartesian: A=ˆxAx+ˆyAy+ˆzAz
- cylindrical: A=ˆρAρ+ˆϕAϕ+ˆzAz
- spherical: A=ˆrAr+ˆθAθ+ˆϕAϕ
Gradient
Gradient in Cartesian coordinates:
∇f=ˆx∂f∂x+ˆy∂f∂y+ˆz∂f∂z
Gradient in cylindrical coordinates:
∇f=ˆρ∂f∂ρ+ˆϕ1ρ∂f∂ϕ+ˆz∂f∂z
Gradient in spherical coordinates:
∇f=ˆr∂f∂r+ˆθ1r∂f∂θ+ˆϕ1rsinθ∂f∂ϕ
Divergence
Divergence in Cartesian coordinates:
∇⋅A=∂Ax∂x+∂Ay∂y+∂Az∂z
Divergence in cylindrical coordinates:
∇⋅A=1ρ∂∂ρ(ρAρ)+1ρ∂Aϕ∂ϕ+∂Az∂z
Divergence in spherical coordinates:
∇⋅A= 1r2∂∂r(r2Ar) +1rsinθ∂∂θ(Aθsinθ) +1rsinθ∂Aϕ∂ϕ
Curl
Curl in Cartesian coordinates:
∇×A= ˆx(∂Az∂y−∂Ay∂z) +ˆy(∂Ax∂z−∂Az∂x) +ˆz(∂Ay∂x−∂Ax∂y)
Curl in cylindrical coordinates:
∇×A= ˆρ(1ρ∂Az∂ϕ−∂Aϕ∂z) +ˆϕ(∂Aρ∂z−∂Az∂ρ) +ˆz1ρ[∂∂ρ(ρAϕ)−∂Aρ∂ϕ]
Curl in spherical coordinates:
∇×A= ˆr1rsinθ[∂∂θ(Aϕsinθ)−∂Aθ∂ϕ] +ˆθ1r[1sinθ∂Ar∂ϕ−∂∂r(rAϕ)] +ˆϕ1r[∂∂r(rAθ)−∂Ar∂θ]
Laplacian
Laplacian in Cartesian coordinates:
∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2
Laplacian in cylindrical coordinates:
∇2f=1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂ϕ2+∂2f∂z2
Laplacian in spherical coordinates:
∇2f= 1r2∂∂r(r2∂f∂r) +1r2sinθ∂∂θ(∂f∂θsinθ) +1r2sin2θ∂2f∂ϕ2