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 (B3) A.(B+C)=A·B+A.C (B.4) A×(B+C)=A×B+A×C (B5) A·(B×C)=B·(C×A)=C·(A×B) (B.6) Ax(B×C)=B(AC)一C(A·B)=B×(A×C)+C×(B×A) (B7) (A×B)·(C×D)=A·[B×(C×D)=(B·D)(AC)-(B·C(A·D)(B.8) (A×B)×(C×D)=CIA·(B×D)一D[A.(B×C A×[B×(C×D)=(B·D)(AxC)-(B·C)(A×D) A·(·B)=(Ac)·B (B.11) Ax(×B)=(Axc)×B (B12) C·(a.b)=(Ca).b (B.13) (ab)·C=a.(b·C) (B×=-B·(A×c)=(AxB) A×(B×c)=B·(A×己一c(A·B) A.I=I·A=A Integral theorems Note:S bounds v, r bounds S, n is normal to S at r, i and m are tangential to S at r, i is tangential to the contour r, m xi=A, dl=ldl, and ds=ndS Divergence theorem v. adv= da ds (B.18) ②2001 by CRC Press LLC
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)
B.20) Gradient theorem adv=fa B21) aDv=f dv (V×a)dv Vx×AdS a dl Stokes’ s theoren (V×A)·ds A. dI (B.27) 鱼·(V×a)dS=∮dla Greens first identity for scalar fields ds Greens second identity for scalar fields(Green's theorem) Green s first identity for vector fields LI(VxA).(VxB)-A[V x(V xB)dV= V·[Ax(V×B)]dv A×(V×B) (B.31) Green's second identity for vector fields [A×(×B)一B×(V×A)·dS (B32) ②2001 by CRC Press LLC
V ∇ · a¯ dV = S nˆ · a¯ d S (B.19) S ∇s · A d S = mˆ · A dl (B.20) Gradient theorem V ∇adV = S adS (B.21) V ∇A dV = S nAˆ d S (B.22) V ∇sadS = mˆ a 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∇2 b) dV = S a ∂b ∂n d S (B.29) Green’s second identity for scalar fields (Green’s theorem) V (a∇2 b − b∇2 a) 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) 4rr_rdv'- 4r-r' ⅴ×A+dAr)×n Miscellaneous identities (Va×Vb)ds=aVb·d bVa. dI BBBB dIA n×(VA)dS 37 Derivative identities V(a+b)=Va+vb V·(A+B)=V·A+V·B V×(A+B)=V×A+V×B V(ab=avb+bva V·(aB)=aV·B+B·Va V×(aB)=aV×B-Bxva BBBB V·(A×B)=B.V×A-AV×B V×(A×B)=A(V·B)-B(V·A)+(B.VA一(AV)B (B.45) V(A·B)=Ax(×B)+B×(×A)+(A.V)B+(B·V)A V×(×A)=V(VA)-VA v·(V BBBB V·(V×A)=0 V×(Va)=0 (B50) V×(aVb)= Vaxvb (B.51) b+2(Va)·(Vb)+b (B52) V(aB)=av-b+Bva+2(Va V)B (B53) Va=V(v·a)-×(vxa) (B54) V·(AB)=(VA)B+A·(VB)=(·A)B+(A·V)B (B55) V×(AB)=(V×A)B-Ax(VB) (B56) V·(V×a)=0 (B57) ②2001 by CRC Press LLC
Helmholtztheorem 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) − ∇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 a¯ = ∇(∇ · 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)
V×(VA)=0 V(A×B)=(VA)×B一(VB)×A (B59) V(ab)=(vaB+a(vB) v·(ab)=(Va)b+a(Vb) V×(ab)=(Va)×b+a(V×b) (B.62) (B.63) V×(a1)=VaxI (B.64) Identities involving the displacement vector Note: R=r-r,R=R,R=R/R, f(r)=df(x)/dx Vf(R)=-Vf(R)=Rf(r) R=R (B.67) R R jk [(R)k]=-V·[(RR=2fAR) +f'(R) (B70) R R I(R)R e-jkR R=-4x8(R) Identities involving the plane-wave function Note: E is a constant vector, k= k v(e-jk-r)=-jke-j V.Ee-jk-r)==ik. Ee-jkr ②2001 by CRC Press LLC
∇ × (∇A) = 0 (B.58) ∇(A × B) = (∇A) × B − (∇B) × A (B.59) ∇(aB) = (∇a)B + a(∇B) (B.60) ∇ · (ab¯) = (∇a) · b¯ + a(∇ · b¯) (B.61) ∇ × (ab¯) = (∇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|, Rˆ = R/R, f (x) = d f (x)/dx. ∇ f (R) = −∇ f (R) = Rˆ f (R) (B.65) ∇ R = Rˆ (B.66) ∇ 1 R = − Rˆ R2 (B.67) ∇ e− jkR R = −Rˆ 1 R + jk e− jkR R (B.68) ∇ · f (R)Rˆ = −∇ · f (R)Rˆ = 2 f (R) R + f (R) (B.69) ∇ · R = 3 (B.70) ∇ · Rˆ = 2 R (B.71) ∇ · Rˆ e− jkR R = 1 R − jk e− jkR R (B.72) ∇ × f (R)Rˆ = 0 (B.73) ∇2 1 R = −4πδ(R) (B.74) (∇2 + k2 ) e− jkR 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)
jk B.79 Identities involving the transverse/longitudinal decomposition Note: u is a constant unit vector,Aa≡0·A,a/u≡aV,A≡A-aAa,V V-ua/8 A=A,+uAu V=V;+ (B.81) B.82 (·V)φ=0 (B.83) (B.84) Vt·(φ)=0 (B87) V×(的)=-0×Vφ (B.88) V:×(×A)=avA2 89 ax(V×A)=VAn (B.90) 0×(V×A1)=0 (B.91) 0.(0×A)=0 (B.92) a×(×A) (B.93) (B.94) V·A=VA2+ (B.95) V×A=VxA+ax/4 vA a-o (B.97) V×V×A=|V,×V×A *EdA an|+a(V1·A)-V2A (B.98) vA=V(v·A,) v×V1×A|+av2A ②2001 by CRC Press LLC
∇ × Ee− jk·r = − jk × Ee− jk·r (B.78) ∇2 Ee− jk·r = −k2 Ee− jk·r (B.79) Identities involving the transverse/longitudinal decomposition Note: uˆ is a constant unit vector, Au ≡ uˆ · A, ∂/∂u ≡ uˆ · ∇, At ≡ A − uˆ Au, ∇t ≡ ∇ − uˆ∂/∂u. A = At + uˆ Au (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) ∇2 φ = ∇2 t φ + ∂2φ ∂u2 (B.97) ∇×∇× A = ∇t × ∇t × At − ∂2At ∂u2 + ∇t ∂ Au ∂u + uˆ ∂ ∂u (∇t · At) − ∇2 t Au (B.98) ∇2 A = ∇t(∇t · At) + ∂2At ∂u2 − ∇t × ∇t × At + uˆ∇2Au (B.99)
