Page 178 - Intro to Tensor Calculus

173

Curl

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The contravariant components of the vector C =curl A are represented
i
C = ijk A k,j . (2.1.5)

In expanded form this representation becomes:

1
1 ∂A 3 ∂A 2
C = √ 2 − 3
g ∂x ∂x
1 ∂A 1 ∂A 3
2
C = √ 3 − 1 (2.1.6)
g ∂x ∂x

1
3 ∂A 2 ∂A 1
C = √ 1 − 2 .
g ∂x ∂x
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EXAMPLE 2.1-2. (Curl) Find the representation for the components of curl A in spherical coordinates
(ρ, θ, φ).
1
2
3
Solution: In spherical coordinates we have :x = ρ, x = θ, x = φ with g ij =0 for i 6= j and
2
2
2
2
2
2
g 11 = h =1, g 22 = h = ρ , g 33 = h = ρ sin θ.
1
3
2
2
4
2
The determinant of g ij is g = |g ij | = ρ sin θ with √ g = ρ sin θ. The relations (2.1.6) are tensor equations
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representing the components of the vector curl A. To find the components of curl A in spherical components
we write the equations (2.1.6) in terms of their physical components. These equations take on the form:

C(1) 1 ∂ ∂
= √ (h 3 A(3)) − (h 2 A(2))
h 1 g ∂θ ∂φ
C(2) 1 ∂ ∂
= √ (h 1 A(1)) − (h 3 A(3)) (2.1.7)
h 2 g ∂φ ∂ρ
C(3) 1 ∂ ∂
= √ (h 2 A(2)) − (h 1 A(1)) .
h 3 g ∂ρ ∂θ
We employ the notations
C(1) = C ρ , C(2) = C θ , C(3) = C φ , A(1) = A ρ , A(2) = A θ , A(3) = A φ
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to denote the physical components, and find the components of the vector curl A, in spherical coordinates,
are expressible in the form:
1 ∂ ∂
C ρ = (ρ sin θA φ ) − (ρA θ )
2
ρ sin θ ∂θ ∂φ
1 ∂ ∂
C θ = (A ρ ) − (ρ sin θA φ ) (2.1.8)
ρ sin θ ∂φ ∂ρ
1 ∂ ∂
C φ = (ρA θ ) − (A ρ ) .
ρ ∂ρ ∂θ