Page 525 - Electromagnetics

Dyadic representation
ˆ
¯ a = ˆρa ρρ ˆρ + ˆρa ρφ φ + ˆρa ρz ˆ z +
ˆ ˆ ˆ ˆ
+ φa φρ ˆρ + φa φφ φ + φa φz ˆ z +
ˆ
+ ˆ za zρ ˆρ + ˆ za zφ φ + ˆ za zz ˆ z (D.50)

ˆ
ˆ

¯ a = ˆρa + φa + ˆ za = a ρ ˆρ + a φ φ + a z ˆ z (D.51)
z
ρ
φ
ˆ
a = a ρρ ˆρ + a ρφ φ + a ρz ˆ z (D.52)

ρ
ˆ

a = a φρ ˆρ + a φφ φ + a φz ˆ z (D.53)
φ
ˆ

a = a zρ ˆρ + a zφ φ + a zz ˆ z (D.54)
z
ˆ
a ρ = a ρρ ˆρ + a φρ φ + a zρ ˆ z (D.55)
ˆ
a φ = a ρφ ˆρ + a φφ φ + a zφ ˆ z (D.56)
ˆ
a z = a ρz ˆρ + a φz φ + a zz ˆ z (D.57)
Differential operations
ˆ
dl = ˆρ dρ + φρ dφ + ˆ z dz (D.58)
dV = ρ dρ dφ dz (D.59)

dS ρ = ρ dφ dz, (D.60)
dS φ = dρ dz, (D.61)
dS z = ρ dρ dφ (D.62)

∂ f 1 ∂ f ∂ f
ˆ
∇ f = ˆρ + φ + ˆ z (D.63)
∂ρ ρ ∂φ ∂z

1 ∂
1 ∂ F φ ∂ F z
∇· F = ρF ρ + + (D.64)
ρ ∂ρ ρ ∂φ ∂z
ˆ

ˆρ ρφ ˆ z
1 ∂
∇× F = ∂ ∂φ ∂ (D.65)
∂ρ
ρ ∂z

F ρ ρF φ F z
2 2
1 ∂ ∂ f 1 ∂ f ∂ f
2
∇ f = ρ + + (D.66)
2
ρ ∂ρ ∂ρ ρ ∂φ 2 ∂z 2

2 2 2 ∂F φ F ρ ˆ 2 2 ∂ F ρ F φ 2
∇ F = ˆρ ∇ F ρ − − + φ ∇ F φ + − + ˆ z∇ F z (D.67)
2
2
ρ ∂φ ρ 2 ρ ∂φ ρ 2
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