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APPENDIX A
If ω and ϑ are two vectors defined as follow: ˆ ˆ ˆ , ˆ ˆ ˆ i j k i j k i j k i j k ω ω ω ω ϑ ϑ ϑ ϑ = + + = + + (A.1)
their dyadic product can be written as:
ˆ ˆˆ ˆˆ ˆ ˆ ˆˆ ˆˆ ˆ , ˆˆ ˆˆ ˆ ˆ ii ij ik ji jj jk ki kj kk ii ij ik ji jj jk ki kj kk ω ϑ Φ = ⋅ = Φ + Φ + Φ + Φ +Φ +Φ + Φ +Φ +Φ (A.2) where: . ii i i ij i j ik i k ji j i jj j j jk j k ki k i kj k j kk k k ω ϑ ω ϑ ω ϑ ω ϑ ω ϑ ω ϑ ω ϑ ω ϑ ω ϑ Φ = Φ = Φ = Φ = Φ = Φ = Φ = Φ = Φ = (A.3)
The dyadic product can be express in a matrix form as:
. ii ij ik ji jj jk ki kj kk Φ Φ Φ Φ = Φ Φ Φ Φ Φ Φ (A.4)
It is important to remark the dyadic product is not commutative:
1 ω ϑ 2 ϑ ω .
