Appendix B Useful identities Algebraic identities for vectors and dyadics A + B = B + A (B.1) A · B = B · A (B.2) A × B =?B × A (B.3) A · (B + C) = A · B + A · C (B.4) A × (B + C) = A × B + A × C (B.5) A · (B × C) = B · (C × A) = C · (A × B) (B.6) A × (B × C) = B(A · C) ? C(A · B) = B × (A × C) + C × (B × A) (B.7) (A × B) · (C × D) = A · [B × (C × D)] = (B · D)(A · C) ? (B · C)(A · D) (B.8) (A × B) × (C × D) = C[A · (B × D)] ? D[A · (B × C)] (B.9) A × [B × (C × D)] = (B · D)(A × C) ? (B · C)(A × D) (B.10) A · (ˉc · B) = (A · ˉc) · B (B.11) A × (ˉc × B) = (A × ˉc) × B (B.12) C · (ˉa · ˉ b) = (C · ˉa) · ˉ b (B.13) (ˉa · ˉ b) · C = ˉa · ( ˉ b · C) (B.14) A · (B × ˉc) =?B · (A × ˉc) = (A × B) · ˉc (B.15) A × (B × ˉc) = B · (A × ˉc) ? ˉc(A · B) (B.16) A · ˉ I = ˉ I · A = A (B.17) Integral theorems Note: S bounds V, Gamma1 bounds S, ?n is normal to S at r, ? l and ?m are tangential to S at r, ? l is tangential to the contour Gamma1, ?m × ? l = ?n, dl = ? l dl, and dS = ?n dS. Divergence theorem integraldisplay V ?·A dV = contintegraldisplay S A · dS (B.18) integraldisplay V ?·ˉa dV = contintegraldisplay S ?n · ˉa dS (B.19) integraldisplay S ? s · A dS = contintegraldisplay Gamma1 ?m · A dl (B.20) Gradient theorem integraldisplay V ?adV = contintegraldisplay S adS (B.21) integraldisplay V ?A dV = contintegraldisplay S ?nA dS (B.22) integraldisplay V ? s adS= contintegraldisplay Gamma1 ?madl (B.23) Curl theorem integraldisplay V (?×A) dV =? contintegraldisplay S A × dS (B.24) integraldisplay V (?×ˉa) dV = contintegraldisplay S ?n × ˉa dS (B.25) integraldisplay S ? s × A dS = contintegraldisplay Gamma1 ?m × A dl (B.26) Stokes’s theorem integraldisplay S (?×A) · dS = contintegraldisplay Gamma1 A · dl (B.27) integraldisplay S ?n · (?×ˉa) dS = contintegraldisplay Gamma1 dl · ˉa (B.28) Green’s ?rst identity for scalar ?elds integraldisplay V (?a ·?b + a? 2 b) dV = contintegraldisplay S a ?b ?n dS (B.29) Green’s second identity for scalar ?elds (Green’s theorem) integraldisplay V (a? 2 b ? b? 2 a) dV = contintegraldisplay S parenleftbigg a ?b ?n ? b ?a ?n parenrightbigg dS (B.30) Green’s ?rst identity for vector ?elds integraldisplay V {(?×A) · (?×B) ? A · [?×(?×B)]}dV = integraldisplay V ?·[A × (?×B)] dV = contintegraldisplay S [A × (?×B)] · dS (B.31) Green’s second identity for vector ?elds integraldisplay V {B · [?×(?×A)] ? A · [?×(?×B)]}dV = contintegraldisplay S [A × (?×B) ? B × (?×A)] · dS (B.32) Helmholtztheorem A(r) =?? bracketleftbiggintegraldisplay V ? prime · A(r prime ) 4π|r ? r prime | dV prime ? contintegraldisplay S A(r prime ) · ?n prime 4π|r ? r prime | dS prime bracketrightbigg + +?× bracketleftbiggintegraldisplay V ? prime × A(r prime ) 4π|r ? r prime | dV prime + contintegraldisplay S A(r prime ) × ?n prime 4π|r ? r prime | dS prime bracketrightbigg (B.33) Miscellaneous identities contintegraldisplay S dS = 0 (B.34) integraldisplay S ?n × (?a) dS = contintegraldisplay Gamma1 adl (B.35) integraldisplay S (?a ×?b) · dS = integraldisplay Gamma1 a?b · dl =? integraldisplay Gamma1 b?a · dl (B.36) contintegraldisplay dl A = integraldisplay S ?n × (?A) dS (B.37) Derivative identities ? (a + b) =?a +?b (B.38) ?·(A + B) =?·A +?·B (B.39) ?×(A + B) =?×A +?×B (B.40) ?(ab) = a?b + b?a (B.41) ?·(aB) = a?·B + B ·?a (B.42) ?×(aB) = a?×B ? B ×?a (B.43) ?·(A × B) = B ·?×A ? A ·?×B (B.44) ?×(A × B) = A(?·B) ? B(?·A) + (B ·?)A ? (A ·?)B (B.45) ?(A · B) = A × (?×B) + B × (?×A) + (A ·?)B + (B ·?)A (B.46) ?×(?×A) =?(?·A) ?? 2 A (B.47) ?·(?a) =? 2 a (B.48) ?·(?×A) = 0 (B.49) ?×(?a) = 0 (B.50) ?×(a?b) =?a ×?b (B.51) ? 2 (ab) = a? 2 b + 2(?a) · (?b) + b? 2 a (B.52) ? 2 (aB) = a? 2 B + B? 2 a + 2(?a ·?)B (B.53) ? 2 ˉa =?(?·ˉa) ??×(?×ˉa) (B.54) ?·(AB) = (?·A)B + A · (?B) = (?·A)B + (A ·?)B (B.55) ?×(AB) = (?×A)B ? A × (?B) (B.56) ?·(?×ˉa) = 0 (B.57) ?×(?A) = 0 (B.58) ?(A × B) = (?A) × B ? (?B) × A (B.59) ?(aB) = (?a)B + a(?B) (B.60) ?·(a ˉ b) = (?a) · ˉ b + a(?· ˉ b) (B.61) ?×(a ˉ b) = (?a) × ˉ b + a(?× ˉ b) (B.62) ?·(a ˉ I) =?a (B.63) ?×(a ˉ I) =?a × ˉ I (B.64) Identities involving the displacement vector Note: R = r ? r prime , R =|R|, ? R = R/R, f prime (x) = df(x)/dx. ? f (R) =?? prime f (R) = ? R f prime (R) (B.65) ?R = ? R (B.66) ? parenleftbigg 1 R parenrightbigg =? ? R R 2 (B.67) ? parenleftbigg e ?jkR R parenrightbigg =? ? R parenleftbigg 1 R + jk parenrightbigg e ?jkR R (B.68) ?· bracketleftbig f (R) ? R bracketrightbig =?? prime · bracketleftbig f (R) ? R bracketrightbig = 2 f (R) R + f prime (R) (B.69) ?·R = 3 (B.70) ?· ? R = 2 R (B.71) ?· parenleftbigg ? R e ?jkR R parenrightbigg = parenleftbigg 1 R ? jk parenrightbigg e ?jkR R (B.72) ?× bracketleftbig f (R) ? R bracketrightbig = 0 (B.73) ? 2 parenleftbigg 1 R parenrightbigg =?4πδ(R) (B.74) (? 2 + k 2 ) e ?jkR R =?4πδ(R) (B.75) Identities involving the plane-wave function Note: E is a constant vector, k =|k|. ? parenleftbig e ?jk·r parenrightbig =?jke ?jk·r (B.76) ?· parenleftbig Ee ?jk·r parenrightbig =?jk · Ee ?jk·r (B.77) ?× parenleftbig Ee ?jk·r parenrightbig =?jk × Ee ?jk·r (B.78) ? 2 parenleftbig Ee ?jk·r parenrightbig =?k 2 Ee ?jk·r (B.79) Identities involving the transverse/longitudinal decomposition Note: ?u is a constant unit vector, A u ≡ ?u · A, ?/?u ≡ ?u ·?, A t ≡ A ? ?uA u , ? t ≡ ???u?/?u. A = A t + ?uA u (B.80) ?=? t + ?u ? ?u (B.81) ?u · A t = 0 (B.82) (?u ·? t )φ = 0 (B.83) ? t φ =?φ ? ?u ?φ ?u (B.84) ?u · (?φ) = (?u ·?)φ = ?φ ?u (B.85) ?u · (? t φ) = 0 (B.86) ? t · (?uφ) = 0 (B.87) ? t × (?uφ) =??u ×? t φ (B.88) ? t × (?u × A) = ?u? t · A t (B.89) ?u × (? t × A) =? t A u (B.90) ?u × (? t × A t ) = 0 (B.91) ?u · (?u × A) = 0 (B.92) ?u × (?u × A) =?A t (B.93) ?φ =? t φ + ?u ?φ ?u (B.94) ?·A =? t · A t + ? A u ?u (B.95) ?×A =? t × A t + ?u × bracketleftbigg ?A t ?u ?? t A u bracketrightbigg (B.96) ? 2 φ =? 2 t φ + ? 2 φ ?u 2 (B.97) ?×?×A = bracketleftbigg ? t ×? t × A t ? ? 2 A t ?u 2 +? t ? A u ?u bracketrightbigg + ?u bracketleftbigg ? ?u (? t · A t ) ?? 2 t A u bracketrightbigg (B.98) ? 2 A = bracketleftbigg ? t (? t · A t ) + ? 2 A t ?u 2 ?? t ×? t × A t bracketrightbigg + ?u? 2 A u (B.99)