Page 573 - Engineering Electromagnetics, 8th Edition

APPENDIX A Vector Analysis 555

in terms of the component differential lengths, h 1 du, h 2 dν, and h 3 dw,
1 ∂V 1 ∂V 1 ∂V
dV = h 1 du + h 2 dν + h 3 dw
h 1 ∂u h 2 ∂ν h 3 ∂w
Then, because

and dV =∇V · dL
dL = h 1 dua u + h 2 dνa ν + h 3 dwa w
we see that
1 ∂V 1 ∂V 1 ∂V
∇V = a u + a ν + a w (A.3)
h 1 ∂u h 2 ∂ν h 3 ∂w
ThecomponentsofthecurlofavectorHareobtainedbyconsideringadifferential
path first in a u = constant surface and finding the circulation of H about that path,
as discussed for rectangular coordinates in Section 7.3. The contribution along the
segment in the a ν direction is

1 ∂
H ν0 h 2 dν − (H ν h 2 dν)dw
2 ∂w
and that from the oppositely directed segment is
1 ∂
−H v0 h 2 dν − (H ν h 2 dν)dw
2 ∂w
The sum of these two parts is
∂
− (H ν h 2 dν)dw
∂w
or
∂
− (h 2 H ν )dν dw
∂w
and the sum of the contributions from the other two sides of the path is
∂
(h 3 H w )dν dw
∂ν
Adding these two terms and dividing the sum by the enclosed area, h 2 h 3 dν dw,we
see that the a u component of curl H is
1 ∂ ∂
(∇× H) u = (h 3 H w ) − (h 2 H ν )
h 2 h 3 ∂ν ∂w
and the other two components may be obtained by cyclic permutation. The result is
expressible as a determinant,

a u a ν a w

h 2 h 3 h 3 h 1 h 1 h 2

∂ ∂ ∂
(A.4)
∇× H =

∂u ∂ν ∂w

h 1 H u h 2 H ν h 3 H w