Single-Particle Electrodynamics - Assassination Science

Also,
(a×b)·(c×d) ≡ (a·c)(b·d) − (a·d)(b·c),
∇×∇ψ ≡ 0,
∇·(∇×a) ≡ 0,
∇×(∇×a) ≡ ∇(∇·a) − ∇ 2 a,
∇·(ψa) ≡ a·∇ψ + ψ∇·a,
∇×(ψa) ≡ (∇ψ)×a + ψ∇×a,
∇(a·b) ≡ (a·∇)b + (b·∇)a + a×(∇×b) + b×(∇×a),
∇·(a×b) ≡ b·∇×a − a·∇×b,
∇×(a×b) ≡ a(∇·b) − b(∇·a) + (b·∇)a − (a·∇)b.
(a·∇)x ≡ a.
(B.32)
(B.33)
(B.34)
(B.35)
(B.36)
(B.37)
(B.38)
(B.39)
(B.40)
(B.41)
B.4 Alternating functions
The three- and four-dimensional alternating functions are connected by
where
ε ijk ≡ ε 0ijk ≡ ε k0ji ≡ ε jk0i ≡ ε kji0 ,
ε 123 ≡ ε 0123 ≡ +1.
(B.42)
(B.43)
Products of two four-dimensional alternating functions, contracted over one,
two, three and four indices, are respectively given by
ε αβγδ ε ατµν ≡ −δτ β δµδ γ ν δ + δµδ β τ γ δν δ − δµδ β ν γ δτ δ + δν β δµδ γ τ δ − δν β δτ γ δµ δ + δτ β δν γ δµ,
δ
ε αβγδ ε αβµν ≡ −2δµδ γ ν δ + 2δν γ δµ,
δ
ε αβγδ ε αβγν ≡ −6δν,
δ
ε αβγδ ε αβγδ ≡ −24. (B.44)
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