Page 527 - Electromagnetics

x = r sin θ cos φ (D.82)
y = r sin θ sin φ (D.83)
z = r cos θ (D.84)

2
2
r = x + y + z 2 (D.85)

2
x + y 2
−1
θ = tan (D.86)
z
y
−1
φ = tan (D.87)
x
Vector algebra
ˆ r = ˆ x sin θ cos φ + ˆ y sin θ sin φ + ˆ z cos θ (D.88)
ˆ
θ = ˆ x cos θ cos φ + ˆ y cos θ sin φ − ˆ z sin θ (D.89)
ˆ
φ =−ˆ x sin φ + ˆ y cos φ (D.90)
ˆ ˆ
A = ˆ rA r + θA θ + φA φ (D.91)

A · B = A r B r + A θ B θ + A φ B φ (D.92)

ˆ

ˆ r θ φ
ˆ

(D.93)
A × B = A r A θ A φ

B r B θ B φ
Dyadic representation
ˆ ˆ
¯ a = ˆ ra rr ˆ r + ˆ ra rθ θ + ˆ ra rφ φ +
ˆ ˆ ˆ ˆ ˆ
+ θa θr ˆ r + θa θθ θ + θa θφ φ +
ˆ
ˆ
ˆ
ˆ
ˆ
+ φa φr ˆ r + φa φθ θ + φa φφ φ (D.94)
ˆ
ˆ
ˆ
ˆ

¯ a = ˆ ra + θa + φa = a r ˆ r + a θ θ + a φ φ (D.95)
r
θ
φ
ˆ
ˆ

a = a rr ˆ r + a rθ θ + a rφ φ (D.96)
r
ˆ
ˆ
a = a θr ˆ r + a θθ θ + a θφ φ (D.97)

θ
ˆ
ˆ
a = a φr ˆ r + a φθ θ + a φφ φ (D.98)

φ
ˆ
ˆ
a r = a rr ˆ r + a θr θ + a φr φ (D.99)
ˆ
ˆ
a θ = a rθ ˆ r + a θθ θ + a φθ φ (D.100)
ˆ
ˆ
a φ = a rφ ˆ r + a θφ θ + a φφ φ (D.101)
© 2001 by CRC Press LLC