Appendix 2 - Formulary
Published online by Cambridge University Press: 13 October 2016
Summary
Vector Identities
• Multiple products:
(A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C)
A · (B × C) = B · (C × A) = C · (A × B)
A × (B × C) = B(A · C) − C(A · B)
• Rules for products with derivatives:
∇(fg) = f (∇g) + g(∇f)
∇(A · B) = A × (∇ ×B) + B × (∇ ×A) + (A · ∇)B + (B · ∇)A
∇ · (fA) = f (∇ · A) + A · (∇f)
∇ · (A × B) = B · (∇ ×A) − A · (∇ ×B)
∇ ×(fA) = f (∇ ×A) − A × (∇f)
∇ ×(A × B) = (B · ∇)A − (A · ∇)B + A(∇ · B) − B(∇ · A)
• Rules for second-order derivatives:
∇ · (∇ ×A) = 0
∇ ×(∇f) = 0
∇ ×(∇ ×A) = ∇(∇ · A) – A
Vectorial derivatives
We list here the vectorial derivatives in the three usual coordinate systems (see figure A2.1).
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- Introduction to Modern Magnetohydrodynamics , pp. 256 - 258Publisher: Cambridge University PressPrint publication year: 2016