Page 515 - Electromagnetics

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
ˆ
Note: S bounds V , bounds S, ˆ n is normal to S at r, l and ˆ m are tangential to S at
ˆ
ˆ
ˆ
r, l is tangential to the contour , ˆ m × l = ˆ n, dl = l dl, and dS = ˆ n dS.
Divergence theorem

∇· A dV = A · dS (B.18)
V S

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