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14 Oct 2015: b. a. a. b. c. ab. abc. bc. L1 S5. What is a vector? ... a. b. L1 S8. Quaternions. Generalises complex numbers, introduced the cross product and some notation still in use today.

14 Oct 2015: L1 S4. Vectors and Vector Spaces. a. b. ab. b. a a. ... product and an area term. a. b. L1 S8. Quaternions. Generalises complex.

22 Feb 2015: Any multiple of ray represents same point. x,y  a,b,c. ... U A A’B’B R. 3D Projective Geometry• Points represented as vectors in 4D• Form the 4D geometric algebra. •

20 Nov 2015: Careful with typographical. ordering. Blade product. L8 S23. bladeprod (a,n) (b,m) = (x,r). ... Blades] [Blades]. AB=simplify([bladeprod(a,b) | a <- A, b <- B]). Form every combination of product.

20 Nov 2015: Careful with typographical ordering. Blade product. L8 S23. bladeprod (a,n) (b,m) = (x,r). ... Blades]. [Blades]. AB=simplify([bladeprod(a,b) | a <- A, b <- B]). Form every combination of product from the two lists.

22 Feb 2015: a  b  I a  b. B  Iaa  IB. ... I(B) is a known linear function of these mapping bivectors to bivectors. •

19 Feb 2015: B = 12 (B BE), EB = B. (4.32). These give rise to the two separate instanton numbers, one for each of the SU(2)subgroups.

14 Feb 2015: Oziewicz, B. Jancewicz, and A. Borowiec,editors, Spinors, Twistors, Clifford Algebras and Quantum Deformations, page233. ... Academic Press Ltd.,London, 1966. 14. [10] B. de Witt. Supermanifolds. Cambridge University Press, 1984.

18 Feb 2015: R(ab) a(b) b(a) (a)(b) (22). and is a linear function mapping bivectors to bivectors. ... R = 2(a(a)) (43). and. G(a) = [(a) a(b(b))]. (44). 6.

19 Feb 2015: a'!)ao ¡ ¡! } m Å b b ¿c b h (Ì / Ê£ÍÏÎÐÉ¡ÊhÑ Ë ÎÐÏ. ... oj! ª M j mh u¤ ª "j b h£¤ b b_ b ¥ o £ ; _K Oo_ª mq £ ¤j J µ ¡ ¥5P ¡ h_º!

16 Feb 2015: R(ab) a(b) b(a) (a)(b). (2.14). From this we define the covariant Riemann tensor. ... R(B) = ḢBetet H2B 14κ2BSS 12κ(BD)S, (3.11). 9. for an arbitrary bivector B.

19 Feb 2015: b a 4.9. a abb aba. 4.10. The multivector analog of complex conjugation is defined by. ... To verify this, consider Holland’s otherchoice:. b b. 4.18. On substituting this into 4.13 we find that.

25 Mar 2015: 2. The bivector B gives us the plane of rotation (cf Lie groups andquaternions). ... where is the relative (3-d) vector derivativeMaxwell’s equations:. E = ρ B = 0.

18 Feb 2015: a) a(b(b)) = M(αann) M a((αn)n αnn) (57). which is also a total divergence and can be converted to a surface integral. ... M ρ3 2π. 0. dφ. dz sin2φ cosφ(iσ3)[i(βσ2n)]. ρ′ρ. d3x ρ′ cosφ(iσ3)[(γ2) γ2(b (b))]. =

18 Feb 2015: R(ab) a(b) b(a) (a)(b). (2.17). From this we define the covariant Riemann tensor. ... The Riemann tensor and its adjoint are related by. R(B) R̄(B) = b[B(aR(ab))] 1.

19 Feb 2015: B»>µ[><><>º9»>9p»>C¿p<Cºi,»CS,ÄáÕ ì>[5»>µ[>9p»>pÉÔÀ¿º|B<>[B>º»PC<>:>ÎC. "# ÿ X À> º|(,ÄpBÆ>º|pÏPµ[,»><p»G:5»>µ[ ... Ê»<>Ä>iº9pBº|µ[iµCB<>º9»>pwpÄà,µ[,Ä>|ºiµCB[ÉSÔ,» mÃp»>CwÄ[<>>5,q,Bĺ9»>àGÆ>ºi

2 Feb 2015: 7. product ab is scalar-valued, since ab ba = (a b)2 a2 b2. ... x b a b|a b| x. a b|a b|b (4). which represents the simple rotation in the ab plane.

19 Feb 2015: dMN>OHRA L]vw?%DBA>@MN>OHRA L ANV9M m<%hR>OHRQDGG>IA B. ( ... DGS $A BIv[]DGSHRFX7;v f v7}Ev B. %. v wxA MTETDXMBH L]W.

16 Feb 2015: Rh1(ab) a(b) b(a) (a)(b). (2.13). The Ricci tensor, Ricci scalar and Einstein tensor are formed from contractions ofthe Riemann tensor:. ... The Riemanntensor for this solution evaluates to. R(B) = M2ω3 (B 3σγBσγ), (4.18).

19 Feb 2015: 2. Euclidean Geometry and Spinors. Let R3 be a three-dimensional Euclidean vector space whose inner product isdenoted by (a,b) 7 ab. ... 2.11) (ba) x(ba)1. = baxa1b1 =baxab. ‖a‖2‖b‖2Let u a/‖a‖, v b/‖b‖ and R vu be the corresponding unit

5 Feb 2015: For vectors a = 〈a〉1 and b = 〈b〉1, Eq. (3.3) reduces to. ... K(B) = B. (4.55)From this we construct a generator basis for the invariance group of K.

18 Feb 2015: b)a. To linear order, the stress-energy tensor of the baryons is. ... T (b)ab = ρ(b)uaub 2(ρ(b) p(b))u(av(b)b) p(b)hab, (2.17). which shows that the baryon heat flux is (ρ(b)

5 Feb 2015: TA(a) = 12Ba, (4.35). so thatT(a)a = 12 (Ba)a = B. ... The appropriate projection operator is. Pb(Ar) = (Arb1)b = b(b1Ar), (5.2).

14 Feb 2015: We start by defining the field-strength via. R(ab)A [Da,Db]A, (1). = R(ab) = a(b) b(a) (a)(b). ... Covariant quantities such as R(B) and D are writtenwith calligraphic (‘curly’) symbols.

19 Feb 2015: B 9. K! vN J& U N. 0 0.005 0.0110. 0. ... w Z. R?wy y ze U U N? w L ) N| B_ y.

19 Feb 2015: La,Lb] = Lc, (69). 14. wherec = Lab ω(a)b Lba ω(b)a. ... Ricci Tensor: R(b) = γµR(γµb) (72)Ricci Scalar: R = γµR(γµ) (73).

19 Feb 2015: We take the circle to be defined by three unit vector points A, B and C. ... Ŷ e)(A e)(B e)(C e) = 0 (62). Expanding out (61) we obtain.

19 Feb 2015: ÊÔÕÙÍÙ{ÚEÊÄÅ ' -. $ " )B B " ÞHí&ÅÈ{ÆÕÓÅYÈ{ÎàXúÍHÙ cÄKÈÊMNMa&PÈ{ÍHÎN LXPÈÓÅñÊÄÅÛ{ÅGÙØÅGÊÓwÔÐÜÝHÓÙ"ÎHÒHÐ[ÊÒHÍHÔÊÅYÖ0ÊÄÅYÖÅÜÔÍÊÄHÅÖÔÍÛÆÕÅîÝHÓwÙÎHÒKÐ[ÊMN P.

14 Feb 2015: 1.7). It follows thatab ba = (a b)2 a2 b2 (1.8). ... c(ab) = 12 [c(ab) (ab)c]= 14 [cab cba abc bac]= 14 [2(ca)b acb abc 2b(ca) bca cba]= 12 [(ca)b b(ca) a(bc) (bc)a]=

19 Feb 2015: zIfXYZk@b<8Z f. –3. –2. –1. 0. 1. 2. 3. –4 –3 –2 –1 1 2 3 4. ... EzZ [BY[ CAsNPzJki IKZbWY[@OZ3FZbJ8>AIKBQ[B Z If[ ] I< [ ð.ªK¢£cj9y+=MCGyk£ñ} 9ôy{?=}Q4Cx9y} C 4DPZD?#xzy8Cy_Z? =¡

16 Feb 2015: DM = DM DM, (1.28). whereDM b (b DM)DM b (b DM). ... b) aR = 0.7. -10. -5. 0. 5. 10. -10 -5 5 10.

18 Feb 2015: b. a. Pð$þýõ K ë )íõÔö4ðùýõñ[ü(õöôýõô%!ÐøHý óÐðHõñ[ü(øHý8!F I=. ) íõïñôôý4ýøSö4þ4ñ[üAáôï(ø #?úø úôHï$üí4õ. ... Áø<üíÐôü dõNôýõáôHÿððfôýðùüíüíõNðfú4úõýWáïñôôý

18 Feb 2015: A The Dirac operator algebra 101. B Some results in multivector calculus 102. ... af(a) = ab〈af(b)〉 = f(b)b. (2.48). It follows that for symmetric functions.

15 Feb 2015: theexplicit appearance of γ0 in the formulae for E and B shows how this split isobserver-dependent. ... We use overdots forthis, so that in the expression ̇AḂ the operator acts only on B.