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Finite Element Methods for Maxwell's Equations$

Peter Monk

Print publication date: 2003

Print ISBN-13: 9780198508885

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198508885.001.0001

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(p.427) Appendix B Vector and Differential Identities

(p.427) Appendix B Vector and Differential Identities

Source:
Finite Element Methods for Maxwell's Equations
Publisher:
Oxford University Press

B.1 Vector identities

  1. (1) a x b = -b x a.

  2. (2) a · (b x c = (a x b) · c = (c x a) · b.

B.2 Differential identities

These differential identities are valid for smooth functions/vector functions:

(B.1)
×(p)=0,
(B.2)
·(×v)=0,
(B.3)
·(φv)=φ·v+φ·v,
(B.4)
×(φv)=φ×v+(φ)×v,
(B.5)
×(u×v)=u(·v)(u·)v+(v·)uv(.u),
(B.6)
×(×u)=(·u)Δu,
(B.7)
·(u×v)=v·×uu·×v,
(B.8)
××{xu(x)}=xΔu(x)+{u(x)+ρuρ(x)}.
In the (B.6) and (B.8), ▵u = (▵u1, ▵u2, ▵u3) in Cartesian coordinates only.

B.3 Differential identities on a surface

Let S be a smooth surface with unit normal υ‎ and let v and p be smooth functions defined a neighborhood of S. The following identities hold:

Sp=(ν×p|s)×ν,¯S×p=ν×Sp,S×v=S·(ν×v),S·v=S×(ν×v),S·(ν×v)=ν·(×v)|S.
The differntial equalities in this and the previous subsection can be extended to less smooth functions as discussed in the text.