Page 522 - Electromagnetics

Appendix D

Coordinate systems

Rectangular coordinate system
Coordinate variables

u = x, −∞ < x < ∞ (D.1)
v = y, −∞ < y < ∞ (D.2)
w = z, −∞ < z < ∞ (D.3)

Vector algebra
A = ˆ xA x + ˆ yA y + ˆ zA z (D.4)

A · B = A x B x + A y B y + A z B z (D.5)

ˆ x y z ˆ
ˆ

(D.6)
A × B = A x A y A z

B x B y B z

Dyadic representation
¯ a = ˆ xa xx ˆ x + ˆ xa xy ˆ y + ˆ xa xz ˆ z +
+ ˆ ya yx ˆ x + ˆ ya yy ˆ y + ˆ ya yz ˆ z +
+ ˆ za zx ˆ x + ˆ za zy ˆ y + ˆ za zz ˆ z (D.7)

¯ a = ˆ xa + ˆ ya + ˆ za = a x ˆ x + a y ˆ y + a z ˆ z (D.8)
x y z

a = a xx ˆ x + a xy ˆ y + a xz ˆ z (D.9)
x
a = a yx ˆ x + a yy ˆ y + a yz ˆ z (D.10)

y
a = a zx ˆ x + a zy ˆ y + a zz ˆ z (D.11)

z
a x = a xx ˆ x + a yx ˆ y + a zx ˆ z (D.12)
a y = a xy ˆ x + a yy ˆ y + a zy ˆ z (D.13)
a z = a xz ˆ x + a yz ˆ y + a zz ˆ z (D.14)

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