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 LLCHelmholtztheorem 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)