vector identities
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A short reference list of vector calculus identities by Phil, dated 11.18.09, with a later derivation added (started 8.10.12). Sections cover scalar and vector triple products, gradient and Laplacian rules, divergence and curl identities, and radial-function identities including the Laplacian of 1/r and e^{ikr}/r. The closing derivation uses the epsilon-delta contraction. Many operator symbols were lost in extraction.
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Vector Identities PhL 11.18.09
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1. STRAIGHT
[A,B,C] ≡ A (B x C) = C (A x B) = B (C x A) = εijkAiBjCk = det [ (A)(B)(C) ]
= + cyclic = - anticylic = - swap (0 if two same) = ABCsin(B,C) = volume of piped A,B,C
A x (B x C) = (AC)B - (AB)C A x (B x C) + cyclic = 0
A x (A x C) = (AC)A - A2C A x (A x A) = 0 A x A = 0
(A x B) (C x D) = (AC) (BD) – (AD) (BC)
(A x B) (A x D) = A2 (BD) – (AD) (BA)
(A x B) (A x B) = A2B2 – (AB)2
(A x B) x (C x D) = [A,B,D] C - [A,B,C] D
(A x B) x (A x D) = [A,B,D] A
A x [ B x (C x D) ] = B [A,C,D] - (AB) (C x D)
A x [ A x (C x D) ] = A [A,C,D] - A2(C x D)
A x [ A x (A x D) ] = - A2(A x D)
A [B x (C x D) ] = (AC)(BD) - (AD)(BC)
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2. Gradient and Laplacian
(φv) = φψ + ψφ
2(φψ) = φ2ψ + ψ2φ + 2 φ ψ
(AB) = (A)B + (B)A + A x ( x B) + B x ( x A) p 153 MM
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3. Divergence
(φA) = A φ + φ (A) => (xiA) = Ai + xi (A)
(φ) = 2φ
(xA) = 0
(Ax) = 0 => (r x ) = 0
(φψ) = φψ + φ 2ψ
(A x B) = B ( x A) – A ( x B)
( φ x A) = φ ( x A)
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3. Curl
x (φA) = φ x A + φ ( x A)
x (φ) = 0
x ( x A) = (A) - 2A
x (φψ) = φ x ψ
x (A x B) = A (B) – B (A) + (B)A – (A)B
x (A x ) = ?? x ( r x ) = r 2 - (1 + r ∂r)
x ( x f(θ,φ) ) = 2f - (1/r) f
A ( x B) = (A x ) B
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4. Radial
(f(r)) = f'(r) => (r) = (1/r) = –/r2
(eikr/r) = (ik/r - 1/r2) eikr
x [f(r)r] = 0 => x = 0 x r = 0
[f(r)r] = 3 f(r) + r f '(r) => r = 3 = 2/r
(r x ) = 0
2(1/r)= –4π δ3(r) 2(r) = 0 2(eikr/r) = -k2 (eikr/r)
2[ f(r) r ] = (2f) r + 2 f
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Derivations (started 8.10.12)
A [B x (C x D) ] = AiεijkBj(C x D)k = AiεijkBjεkabCaDb
= εijkεkabAiBjCaDb
= εijkεabkAiBjCaDb
= (δiaδjb - δibδja) AiBjCaDb
= (AC)(BD) - (AD)(BC)