
1500 CURVILINEAR COORDINATES, VECTORS, OPERATORS, AND DIFFERENTIAL RELATIONS
Direct and inverse transformations of the components of a vector ~v = v
x
~
i + v
y
~
j + v
z
~
k =
v
ρ
~
i
ρ
+ v
ϕ
~
i
ϕ
+ v
z
~
i
z
:
v
ρ
= v
x
cos ϕ + v
y
sin ϕ,
v
ϕ
= −v
x
sin ϕ + v
y
cos ϕ,
v
z
= v
z
;
v
x
= v
ρ
cos ϕ −v
ϕ
sin ϕ,
v
y
= v
ρ
sin ϕ + v
ϕ
cos ϕ,
v
z
= v
z
.
Metric tensor components:
g
ρρ
= 1, g
ϕϕ
= ρ
2
, g
zz
= 1,
√
g = ρ.
◮ Basic differential relations.
Gradient of a scalar f :
∇f =
∂f
∂ρ
~
i
ρ
+
1
ρ
∂f
∂ϕ
~
i
ϕ
+
∂f
∂z
~
i
z
.
Divergence of a vector ~v:
div ~v ≡ ∇ ·~v =
1
ρ
∂(ρv
ρ
)
∂ρ
+
1
ρ
∂v
ϕ
∂ϕ
+
∂v
z
∂z
.
Gradient of a scalar f along a vector ~v:
(~v · ∇)f = v
ρ
∂f
∂ρ
+
v
ϕ
ρ
∂f
∂ϕ
+ v
z
∂f
∂z
.
Gradient of a vector ~w along a vector ~v:
(~v · ∇) ~w = (~v · ∇)w
ρ
~
i
ρ
+ (~v