1397电磁场理论

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 · (c¯ · B) = (A · c¯) · B (B.11) A × (c¯ × B) = (A × c¯) × B (B.12) C · (a¯ · ¯b) = (C · a¯) · ¯b (B.13) (a¯ · ¯b) · C = a¯ · ( ¯b · C) (B.14) A · (B × c¯) = −B · (A × c¯) = (A × B) · c¯ (B.15) A × (B × c¯) = B · (A × c¯) − c¯(A · B) (B.16) A · ¯I = ¯I · A = A (B.17) Integral theorems Note: S bounds V , � bounds S, nˆ is normal to S at r, ˆl and mˆ are tangential to S at r, ˆl is tangential to the contour �, mˆ × ˆl = nˆ, dl = ˆl dl, and dS = nˆ d S. Divergence theorem ∫ V ∇ · A dV = ∮ S A · dS (B.18) ∫ V ∇ · a¯ dV = ∮ S nˆ · a¯ d S (B.19) ∫ S ∇s · A d S = ∮ � mˆ · A dl (B.20) Gradient theorem ∫ V ∇a dV = ∮ S adS (B.21) ∫ V ∇A dV = ∮ S nˆA d S (B.22) ∫ V ∇sa d S = ∮ � mˆa dl (B.23) Curl theorem ∫ V (∇ × A) dV = − ∮ S A × dS (B.24) ∫ V (∇ × a¯) dV = ∮ S nˆ × a¯ d S (B.25) ∫ S ∇s × A d S = ∮ � mˆ × A dl (B.26) Stokes’s theorem ∫ S (∇ × A) · dS = ∮ � A · dl (B.27) ∫ S nˆ · (∇ × a¯) d S = ∮ � dl · a¯ (B.28) Green’s first identity for scalar fields∫ V (∇a · ∇b + a∇2b) dV = ∮ S a ∂b ∂n d S (B.29) Green’s second identity for scalar fields (Green’s theorem)∫ V (a∇2b − b∇2a) dV = ∮ S ( a ∂b ∂n − b ∂a ∂n ) d S (B.30) Green’s first identity for vector fields∫ V {(∇ × A) · (∇ × B) − A · [∇ × (∇ × B)]} dV =∫ V ∇ · [A × (∇ × B)] dV = ∮ S [A × (∇ × B)] · dS (B.31) Green’s second identity for vector fields∫ V {B · [∇ × (∇ × A)] − A · [∇ × (∇ × B)]} dV =∮ S [A × (∇ × B) − B × (∇ × A)] · dS (B.32) Helmholtz theorem A(r) = −∇ [∫ V ∇′ · A(r′) 4π |r − r′| dV ′ − ∮ S A(r′) · nˆ′ 4π |r − r′| d S ′ ] + + ∇ × [∫ V ∇′ × A(r′) 4π |r − r′| dV ′ + ∮ S A(r′) × nˆ′ 4π |r − r′| d S ′ ] (B.33) Miscellaneous identities ∮ S dS = 0 (B.34) ∫ S nˆ × (∇a) d S = ∮ � adl (B.35) ∫ S (∇a × ∇b) · dS = ∫ � a∇b · dl = − ∫ � b∇a · dl (B.36) ∮ dl A = ∫ S nˆ × (∇A) d S (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) − ∇2A (B.47) ∇ · (∇a) = ∇2a (B.48) ∇ · (∇ × A) = 0 (B.49) ∇ × (∇a) = 0 (B.50) ∇ × (a∇b) = ∇a × ∇b (B.51) ∇2(ab) = a∇2b + 2(∇a) · (∇b) + b∇2a (B.52) ∇2(aB) = a∇2B + B∇2a + 2(∇a · ∇)B (B.53) ∇2a¯ = ∇(∇ · a¯) − ∇ × (∇ × a¯) (B.54) ∇ · (AB) = (∇ · A)B + A · (∇B) = (∇ · A)B + (A · ∇)B (B.55) ∇ × (AB) = (∇ × A)B − A × (∇B) (B.56) ∇ · (∇ × a¯) = 0 (B.57) ∇ × (∇A) = 0 (B.58) ∇(A × B) = (∇A) × B − (∇B) × A (B.59) ∇(aB) = (∇a)B + a(∇B) (B.60) ∇ · (a ¯b) = (∇a) · ¯b + a(∇ · ¯b) (B.61) ∇ × (a ¯b) = (∇a) × ¯b + a(∇ × ¯b) (B.62) ∇ · (a¯I) = ∇a (B.63) ∇ × (a¯I) = ∇a × ¯I (B.64) Identities involving the displacement vector Note: R = r − r′, R = |R|, ˆR = R/R, f ′(x) = d f (x)/dx . ∇ f (R) = −∇′ f (R) = ˆR f ′(R) (B.65) ∇ R = ˆR (B.66) ∇ ( 1 R ) = − ˆR R2 (B.67) ∇ ( e− jk R R ) = − ˆR ( 1 R + jk ) e− jk R R (B.68) ∇ · [ f (R) ˆR] = −∇′ · [ f (R) ˆR] = 2 f (R) R + f ′(R) (B.69) ∇ · R = 3 (B.70) ∇ · ˆR = 2 R (B.71) ∇ · ( ˆR e− jk R R ) = ( 1 R − jk ) e− jk R R (B.72) ∇ × [ f (R) ˆR] = 0 (B.73) ∇2 ( 1 R ) = −4πδ(R) (B.74) (∇2 + k2)e − jk R R = −4πδ(R) (B.75) Identities involving the plane-wave function Note: E is a constant vector, k = |k|. ∇ (e− jk·r) = − jke− jk·r (B.76) ∇ · (Ee− jk·r) = − jk · Ee− jk·r (B.77) ∇ × (Ee− jk·r) = − jk × Ee− jk·r (B.78) ∇2 (Ee− jk·r) = −k2Ee− jk·r (B.79) Identities involving the transverse/longitudinal decomposition Note: uˆ is a constant unit vector, Au ≡ uˆ · A, ∂/∂u ≡ uˆ · ∇, At ≡ A − uˆAu , ∇t ≡ ∇ − uˆ∂/∂u. A = At + uˆAu (B.80) ∇ = ∇t + uˆ ∂ ∂u (B.81) uˆ · At = 0 (B.82) (uˆ · ∇t ) φ = 0 (B.83) ∇tφ = ∇φ − uˆ∂φ ∂u (B.84) uˆ · (∇φ) = (uˆ · ∇)φ = ∂φ ∂u (B.85) uˆ · (∇tφ) = 0 (B.86) ∇t · (uˆφ) = 0 (B.87) ∇t × (uˆφ) = −uˆ × ∇tφ (B.88) ∇t × (uˆ × A) = uˆ∇t · At (B.89) uˆ × (∇t × A) = ∇t Au (B.90) uˆ × (∇t × At ) = 0 (B.91) uˆ · (uˆ × A) = 0 (B.92) uˆ × (uˆ × A) = −At (B.93) ∇φ = ∇tφ + uˆ∂φ ∂u (B.94) ∇ · A = ∇t · At + ∂ Au ∂u (B.95) ∇ × A = ∇t × At + uˆ × [ ∂At ∂u − ∇t Au ] (B.96) ∇2φ = ∇2t φ + ∂2φ ∂u2 (B.97) ∇ × ∇ × A = [ ∇t × ∇t × At − ∂ 2At ∂u2 + ∇t ∂ Au ∂u ] + uˆ [ ∂ ∂u (∇t · At ) − ∇2t Au ] (B.98) ∇2A = [ ∇t (∇t · At ) + ∂ 2At ∂u2 − ∇t × ∇t × At ] + uˆ∇2 Au (B.99) ELECTROMAGNETICS Table of Contents Appendix B Useful identities Algebraic identities for vectors and dyadics Integral theorems Derivative identities Identities involving the displacemen vector Identities involving the plane-wave function Identities involving the transverse/longitudinal decomposition © 2001 by CRC Press LLC: © 2001 by CRC Press LLC