2B29 Electromagnetic Theory. iii) Some useful Mathematical Tools VECTORS
a ⋅ (b × c) = b ⋅ (c × a) = c ⋅ (a × b) a × (b × c) = (a ⋅ c) b − (a ⋅ b) c (a × b) ⋅ (c × d) = (a ⋅ c) (b ⋅ d) − (a ⋅ d) (b ⋅ c)
VECTOR CALCULUS
* *
∇×∇ φ = 0 ∇ ⋅ (∇ × A) = 0 ∇ ⋅ (φ A) = φ ∇ ⋅ A + A ⋅ ∇φ ∇ × (φ A) = ∇φ × A + φ∇ × A ∇ ( A ⋅ B) = ( A ⋅ ∇ )B + (B ⋅ ∇ ) A + A × (∇ × B) + B × (∇ × A) ∇ ⋅ ( A × B) = (∇ × A) ⋅ B − (∇ × B) ⋅ A ∇ × ( A × B) = A(∇ ⋅ B) − B(∇ ⋅ A) + (B ⋅ ∇ ) A − ( A ⋅ ∇ )B ∇ × (∇ × A) = ∇ (∇ ⋅ A) − ∇ 2 A
where
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ˆ ˆ Bz + Ay + Az + Ay + Az + Ay + Az ( A ⋅ ∇ )B = ˆi Ax Bx + j Ax By + k Ax ∂y ∂z ∂y ∂z ∂y ∂ z ∂x ∂x ∂x
x 2 + y 2 + z 2 , e.g. as φ(r), F(r), then r d r dφ r dF r dF , i.e. ∇φ (r ) = , ∇ ⋅ F (r ) = ⋅ , ∇ × F(r ) = × , ∇≡ r dr r dr r dr r dr 1 d 2 d φ 1 d 2 (r φ ) d 2 φ 2 d φ ∇ 2φ = 2 = 2+ . r = r d r d r r d r2 dr r dr
For a function that depends only on the distance r = r =
EXPLICIT FORMS OF THE VECTOR OPERATORS (a)
Cartesian (x, y, z), volume element dτ = dx dy dz.
∂ψ ˆ ∂ψ ˆ ∂ψ ∂ Ax ∂ Ay ∂ Az +j +k + + and ∇ ⋅ A = ∂x ∂y ∂z ∂x ∂y ∂z ∂ A ∂ Ay ˆ ∂ Ax ∂ Az ˆ ∂ Ay ∂ Ax ∇ × A = ˆi z − − +k − +j ∂ z ∂ z ∂ x ∂y ∂y ∂x
∇ψ = ˆi
∇×A =
ˆi
ˆj
kˆ
∂ ∂x
∂ ∂y
∂ ∂z
Ax
Ay
Az
∇ ⋅ ∇ψ = ∇ 2ψ = 2B29 Mathematical Tools 2004
∂ 2ψ ∂ 2ψ ∂ 2ψ + + ∂ x2 ∂ y2 ∂ z 2 Section iii)
1
(b)
Cylindrical polar (ρ, φ, z), volume element dτ = ρ dρ dφ dz.
∂ψ ˆ 1 ∂ψ ∂ψ +φ + zˆ ∂ρ ρ ∂φ ∂z 1 ∂ ( ρ Aρ ) 1 ∂ Aφ ∂ Az ∇⋅A = + + ρ ∂ρ ρ ∂φ ∂z ∇ψ = ρˆ
ρˆ ∇×A =
1 ∂
ρ ∂ρ Aρ
(c)
ρφˆ ∂ ∂φ ρ Aφ
zˆ
∂ ∂z Az
Spherical polar (r, θ, φ), volume element dτ = r2 sinθ dr dθ dφ.
∇ψ = rˆ ∇⋅A =
∇ 2ψ =
∂ψ ˆ 1 ∂ψ ˆ 1 ∂ψ +θ +φ ∂r r ∂θ r sin θ ∂φ
1 ∂ (r 2 Ar ) 1 ∂ (sin θ Aθ ) 1 ∂ Aφ + + 2 r r sin θ r sin θ ∂φ ∂r ∂θ rˆ r θˆ r sin θφˆ 1 ∂ ∂ ∂ ∇×A = ∂φ r sin θ ∂ r ∂θ Ar r Aθ r sin θ Aφ
1 ∂ 2 ∂ψ 1 ∂ ∂ψ 1 ∂ 2ψ r sin θ + + r 2 ∂ r ∂ r r 2 sin θ ∂θ ∂θ r 2 sin 2 θ ∂φ 2 1 ∂ 2 ∂ψ 1 ∂ 2 (rψ ) ∂ 2ψ 2 ∂ψ = + r = r2 ∂ r ∂ r r ∂ r2 ∂ r2 r ∂ r
INTEGRAL RELATIONS If V is a volume with volume element dτ and S is a closed surface enclosing volume V and having a surface ˆ at dS then element dS and an outward normal n *
∫ ∇ ⋅ A d τ = v∫ A ⋅ nˆ d S
V
(Divergence theorem)
S
ˆ to S is defined by If S is an open surface and C is a contour bounding it with line element dA and the normal n the right-hand rule in relation to the sense of a line integral around C then *
∫ (∇ × A) ⋅ nˆ d S = v∫ A ⋅ dA S
2B29 Mathematical Tools 2004
(Stokes's theorem)
C
Section iii)
2