APPENDIX D

COMMON EQUATIONS

605

T a b l e D.3: Linear transformation f applied to common products.

Product

Linear transformation

Reference

outer product scalar product left contraction

f[A ∧ B] = f[A] ∧ f[B] f[A ∗ B] = A ∗ B f[AB] = f −1 [A] f[B]

(4.3) (4.9) (4.13)

−1

f∗ [A∗ ] = det(f) f [A∗ ]

dual

f[a × b] f[In ] f[A ∩ B] f[A ∪ B]

cross product pseudoscalar meet join

(4.14)

−1

= det(f) f [a × b] = det(f) In = f[A] ∩ f[B] (or with det(f)) = f[A] ∪ f[B] (or with det(f))

(4.15) (4.7) Section 5.7 Section 5.7

T a b l e D.4: Basic equations in geometric algebra.

Operation symmetry of outer product symmetry of outer product symmetry of left contraction symmetry of right contraction contraction vector on blade contraction vector on versor contraction vector on blade reciprocal frame dual undual in Rn (and Rn+1,1 ) duality relationship duality relationship projection of X onto B rejection of vector x by B reflection of direct X in dual D applying an even versor V applying an odd versor V inverse of a versor V

Formula a∧B B∧a aB Ba

Reference = = = =

1 2 1 2 1 2 1 2

 a) (a B + B  (B a + a B)  (a B − B a)  (B a − a B)

a(b ∧ C) = (a · b) ∧ C − b ∧ (aC) a(b C) = (a · b) C − b (aC) x(a1 ∧ · · · ∧ ak ) = k i−1 a ∧ · · · (xa ) ∧ · · · a 1 i k i=1 (−1) i b = (−1)i−1 (b1 ∧ · · · b˘ i ∧ · · · ∧ bn )In−1 X∗ X−∗ (A ∧ B)∗ (AB)∗

= = = =

XIn−1 (−1)n(n−1)/2 X∗ A(B∗ ) A ∧ (B∗ ), for A ⊆ I

X → (XB)B −1 = (XB)/B wrong x → (x ∧ B)B −1 = (x ∧ B)/B X → (−1)xd D X D X → X → −1 V =

V X V −1  V −1 VX  V ) V/(V

(6.10) (6.11) (6.12) (6.13) from (3.10) (C.5) (3.16) (3.31) Section 3.5.3 Section 3.5.3 (3.24) (3.24) Section 3.6, 6.4.2 Section 3.6, 6.4.2 Table 7.1 (7.19) (7.20) (21.1)