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)