Vector calculus in curvilinear coordinates Goals: Coordinate ...

Summary
Cylindrical polar coordinates (ρ, ϕ, z)
• Relation to cartesian coordinates (care is required to obtain ϕ in the correct quadrant)
x = ρ cos(ϕ) , y = ρ sin(ϕ) and z = z
ρ = √ x 2 + y 2 , ϕ = arctan(y/x) and z = z
• Basis vectors ê ρ = ∂ ρ r, ρê ϕ = ∂ ϕ r, ê z = ∂ z r
ê ρ = cos(ϕ)ê x + sin(ϕ)ê y , ê ϕ = − sin(ϕ)ê x + cos(ϕ)ê y and ê z = ê z
ê x = cos(ϕ)ê ρ − sin(ϕ)ê ϕ , ê y = sin(ϕ)ê ρ + cos(ϕ)ê ϕ and ê z = ê z
• Non-zero derivatives of basis vectors
∂ ϕ ê ρ = ê ϕ ,
∂ ϕ ê ϕ = −ê ρ
• ∇-operator
• Divergence
• Rotation
∇ × F(r) = ê ρ
[
∂ϕ F z
ρ
∇ = ê ρ ∂ ρ + êϕ
ρ ∂ ϕ + ê z ∂ z
∇ · F = ∂ ρ(ρF ρ )
ρ
+ ∂ ϕF ϕ
ρ
+ ∂ z F z
]
[
∂ρ (ρF ϕ )
− ∂ z F ϕ + ê ϕ [−∂ ρ F z + ∂ z F ρ ] + ê z − ∂ ]
ϕF ρ
ρ ρ
• Laplace
∆ = 1 ρ ∂ ρ(ρ∂ ρ ) + 1 ρ 2 ∂2 ϕ + ∂ 2 z
Spherical polar coordinates (r, θ, ϕ)
• Relation to cartesian and cylindrical coordinates (care is required to obtain ϕ in the correct quadrant)
x = r cos(ϕ) sin(θ) , y = r sin(ϕ) sin(θ) , z = r cos(θ) and ρ = r sin(θ)
r = √ x 2 + y 2 + z 2 , ϕ = arctan(y/x) and θ = arccos(z/r) = arctan(ρ/z)
• Basis vectors ê r = ∂ r r, rê θ = ∂ θ r, r sin(θ)ê ϕ = ∂ ϕ r
ê r = cos(θ)ê z + sin(θ)ê ρ , ê θ = − sin(θ)ê z + cos(θ)ê ρ and ê ϕ = ê ϕ
ê z = cos(θ)ê r − sin(θ)ê θ , ê ρ = sin(θ)ê r + cos(θ)ê θ and ê ϕ = ê ϕ
ê x = sin(θ) cos(ϕ)ê r + cos(θ) cos(ϕ)ê θ − sin(ϕ)ê ϕ and ê y = sin(θ) sin(ϕ)ê r + cos(θ) sin(ϕ)ê θ + cos(ϕ)ê ϕ
• Non-zero derivatives of basis vectors
∂ θ ê r = ê θ , ∂ θ ê θ = −ê r , ∂ ϕ ê r = sin(θ)ê ϕ , ∂ ϕ ê θ = cos(θ)ê ϕ and ∂ ϕ ê ϕ = −ê ρ
• ∇-operator
• Divergence
∇ = ê r ∂ r + êθ
r ∂ θ +
êϕ
r sin(θ) ∂ ϕ
∇ · F = ∂ r(r 2 F r )
r 2 + ∂ θ(sin(θ)F θ )
+ ∂ ϕF ϕ
r sin(θ) r sin(θ)
• Rotation
∇ × F =
[
êr
r sin(θ) [∂ θ(F ϕ sin(θ)) − ∂ ϕ F θ ] + êθ −∂ r (rF ϕ ) + ∂ ]
ϕF r
+ êϕ
r
sin(θ) r [∂ r(rF θ ) − ∂ θ F r ]
• Laplace
∆ = 1 r 2 ∂ r(r 2 ∂ r ) +
1
r 2 sin(θ) ∂ 1
θ(sin(θ)∂ θ ) +
r 2 sin 2 (θ) ∂2 ϕ