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 LLCAppendix 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)