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Cylindrical Coordinates
Cylindrical coordinates are related to rectangular coordinates as follows.
ρ =
x^2 + y^2 x = ρ cos θ tan θ = y x y = ρ sin θ z = z z = z
The cylindrical coordinate vectors are defined as
eρ :=
|∇ρ|
∇ρ
eθ :=
|∇θ|
∇θ
ez :=
|∇z|
∇z
Thus,
eρ = x √ x^2 + y^2
i + y √ x^2 + y^2
j = cos θ i + sin θ j
eθ = − √ y x^2 + y^2
i + √ x x^2 + y^2
j = − sin θ i + cos θ j
ez = k
The inverse relationship is as follows.
i = cos θ eρ − sin θ eθ j = sin θ eρ + cos θ eθ k = ez
It is worth noting that the above computations also imply the following.
∂ρ ∂x = cos θ ∂ρ ∂y = sin θ ∂θ ∂x
ρ sin θ ∂θ ∂y
ρ cos θ
The position vector R = xi + yj + zk is written
R = ρ eρ + z ez. (cylindrical coordinates)
If R = R(t) is a parameterized curve, then d dtR = dρdt eρ + ρd dteρ + dzdt ez. Since eρ = cos θ i + sin θ j, it follows that d dteρ = dθdt eθ. Thus,
dR dt
dρ dt
eρ + ρ dθ dt
eθ + dz dt
ez
Hence, dR = dρ eρ + ρ dθ eθ + dz ez and it follows that the element of volume in cylindrical coordinates is given by
dV = ρ dρ dθ dz
If f = f (x, y, z) is a scalar field (that is, a real-valued function of three variables), then
∇f = ∂f ∂x i + ∂f ∂y j + ∂f ∂z k.
If we view x and y as functions of ρ and θ and apply the chain rule, we obtain
∇f =
∂f ∂ρ
∂ρ ∂x
∂f ∂θ
∂θ ∂x
i +
∂f ∂ρ
∂ρ ∂y
∂f ∂θ
∂θ ∂y
j + ∂f ∂z
k.
Writing this in terms of ρ, θ, and the cylindrical coordinate vectors yields
∇f =
cos θ
∂f ∂ρ
ρ sin θ ∂f ∂θ
(cos θ eρ − sin θ eθ) +
sin θ ∂f ∂ρ
ρ cos θ ∂f ∂θ
(sin θ eρ + cos θ eθ) + ∂f ∂z ez.
Simplifying, we obtain the result
∇f = ∂f ∂ρ eρ +
ρ
∂f ∂θ eθ + ∂f ∂z ez.
If F = F(x, y, z) is a vector field (that is, a vector-valued function of three variables), then we can write
F = F 1 i + F 2 j + F 3 k = (cos θF 1 + sin θF 2 )eρ + (− sin θF 1 + cos θF 2 )eθ + F 3 ez
Thus, F = Fρ eρ + Fθ eθ + Fz ez, where
Fρ = cos θ F 1 + sin θ F 2 F 1 = cos θ Fρ − sin θ Fθ Fθ = − sin θ F 1 + cos θ F 2 F 2 = sin θ Fρ + cos θ Fθ Fz = F 3 F 3 = Fz
Now we can transform ∇ · F and ∇ × F into cylindrical coordinates. To transform ∇ · F, we compute as follows.
