Page 442 - Advanced engineering mathematics

422 CHAPTER 12 Vector Integral Calculus

since u 2 is tangent to a.Onside c, F is approximately
c
−F 2 (q 1 ,q 2 ,q 3 )h 2 (q 1 ,q 2 ,q 3 + dq 3 )dq 3 .
The net contribution from sides a and c is approximately
∂
− (F 2 h 2 )dq 2 dq 3 .
∂q 3
Similarly, from sides b and d, the net contribution is approximately
∂
(F 3 h 3 )dq 2 dq 3 .
∂q 2
Then
1 ∂ ∂
(∇× F) · u 1 = (F 3 h 3 ) − (F 2 h 2 ) dq 2 dq 3 .
h 2 h 3 dq 2 dq 3 ∂q 2 ∂q 3
Obtain the other components of ∇× F in the same way. We obtain
1 ∂ ∂
∇× F = (F 3 h 3 ) − (F 2 h 2 ) u 1
h 2 h 3 ∂q 2 ∂q 3
1 ∂ ∂
+ (F 1 h 1 ) − (F 3 h 3 ) u 2
h 1 h 3 ∂q 3 ∂q 1
1 ∂ ∂
+ (F 2 h 2 ) − (F 1 h 1 ) u 3 .
h 1 h 2 ∂q 1 ∂q 2
This can be written in a convenient determinant form:

1 h 1 u 1 h 2 u 2 h 3 u 3
∇× F = ∂/∂q 1 ∂/∂q 2 ∂/∂q 3 .

F 1 h 1 F 2 h 2 F 3 h 3
h 1 h 2 h 3
We will apply these to spherical coordinates, recalling that
h ρ = 1,h θ = ρ sin(ϕ),h ϕ = ρ.
If

F = F ρ u ρ + F θ u θ + F ϕ u ϕ ,
then divergence is given by
1 ∂ 1 ∂ 1 ∂
2
∇· F = (ρ F ρ ) + (F θ ) + (F ϕ sin(ϕ)).
2
ρ ∂ρ ρ sin(ϕ) ∂θ ρ sin(ϕ) ∂ϕ
The curl is obtained as

u ρ ρ sin(ϕ)u θ ρu ϕ

∇× F = ∂/∂ρ ∂/∂θ ∂/∂ϕ .

F ρ ρ sin(ϕ)F θ ρF ϕ
The gradient of a scalar function f (ρ,θ,ϕ) is
∂ f 1 ∂ f 1 ∂ f
∇ f = u ρ + u θ + u ϕ .
∂ρ ρ sin(ϕ) ∂θ ρ ∂ϕ
From this, we have the Laplacian
2
1 ∂ ∂ f 1 ∂ f 1 ∂ ∂ f
2 2
∇ f = ρ + + sin(ϕ) .
2
ρ ∂ρ ∂ρ ρ sin (ϕ) ∂θ 2 ρ sin(ϕ) ∂ϕ ∂ϕ
2
2
2
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October 14, 2010 14:53 THM/NEIL Page-422 27410_12_ch12_p367-424