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E. Hitzer, Basic Calculus, Proceedings of FAN 2008, Hiroshima, Japan, 23+24 October 2008.

Basic Multivector Calculus

Eckhard Hitzer, Department of Applied Physics, Faculty of Engineering, University of Fukui

Abstract: We begin with introducing the generalization of real, complex, and numbers to hypercomplex numbers, also known as Clifford numbers, or of . Multivectors encode everything from vectors, rotations, transformations, improper transformations (reflections, inversions), geometric objects (like lines and spheres), spinors, and , and the like. Multivector calculus allows to define functions mapping multivectors to multivectors, differentiation, integration, norms, multivector Fourier transformations and wavelet transformations, filtering, windowing, etc. We give a basic introduction into this general mathematical language, which has fascinating applications in physics, engineering, and computer science.

more didactic approach for newcomers to geometric algebra. 1. Introduction G(I) is the full geometric algebra over all vectors in the “Now faith is being sure of what we hope for and certain of what we do not see. This is what the ancients were    n-dimensional unit pseudoscalar I  e1  e2  ...  en . commended for. By faith we understand that the universe was formed at God’s command, so that what is seen was not made 1 A  G (I) is the n-dimensional vector sub-space of out of what was visible.” [7] n The German 19th century mathematician H. Grassmann had    grade-1 elements in G(I) spanned by e1,e2 ,...,en . For the clear vision, that his “extension theory (now developed to geometric calculus) … forms the keystone of the entire    1 vectors a,b,c  An  G (I) and scalars , ,, ; structure of .”[6] The algebraic “grammar” of this universal form of calculus is geometric algebra (or Clifford G(I) has the fundamental properties of algebra). That geometric calculus is a truly unifying approach  associativity to all of calculus will be demonstrated here by developing     some basics of the vector differential calculus part of a(bc)  (ab)c, a  (b  c)  (a  b)  c, (1) geometric calculus.   The basic geometric algebra[1,2] necessary for this is  commutativity a  a, a  b  b  a, (2) compiled in section 2. Then section 3 develops vector  distributivity differential calculus with the help of few simple definitions.            This approach is generically coordinate free, and fully shows a(b  c)  ab  ac, (b  c)a  ba  ca, (3) both the concrete and abstract geometric and algebraic beauty       of the “keystone” of mathematics. The full generalization to  linearity (a  b)  a b  (a  b), (4) multivector calculus is shown in [2,10], applications in [11].  square (vector length  ) a 2. Basic Geometric Algebra 2     2 This section is a basic summary of important relationships in a  aa  a  a  a . (5) geometric algebra. This summary mainly serves as a reference   section for the vector differential calculus to be developed in The geometric ab is related to the (scalar) inner the following section. Most of the relationships listed here are   to be found in the synopsis of geometric algebra and in product a b and to the ( or 2-vector) chapters 1 and 2 of [1], as well as in chapter 1 of [2], together   with relevant proofs. Beyond that [1] and [2] follow a much a  b by       ab  a b  a  b , (6) a(A  B)  aA  aB . (19) with The inner and outer products of homogeneous multivectors

 1      (13)  A and B are defined ([2], p. 6, (1.21), (1.22)) as a  b  (ab  ba)  b  a  ab  a  b  ab , (7) r s 2 0

(13) A  B  A B for r,s  0 , (20)  1       r s r s r s a  b  (ab  ba)  b  a  ab  a  b  ab (8) 2 2 A  B  0 for r  0 or s  0 , (21) The inner and the outer product are both linear and distributive r s        A  B  A B , (22) a  (b  c)  a b  a  c , (9) r s r s r  s        a  (b  c)  a  b  a  c . (10) Ar      Ar  Ar for scalar  . (23) A unit vector aˆ in the direction of a is The inner (and outer) product is again linear and distributive  a 2 (A )  B  A  (B )  (A  B )  A  B (24) aˆ  , with aˆ  aˆaˆ  1, a  aˆ a . (11) r s r s r s r s a A  (B  C )  A  B  A C , (25) The inverse of a vector is r s t r s r t   1 1 a a aˆ . (26) a     . (12) (Bs  Ct )  Bs  Ct a a2 a 2 a The reverse of a multivector is A multivector A can be uniquely decomposed into its ~ n A  (1)k (k1) / 2 A . (27) homogeneous grade k parts ( grade k selector):  k k k1

A  A  A  A  ...  A  ...  A (13) [2] uses a dagger instead of the tilde. 0 1 2 k n Special examples are scalar vector bivector k vector pseudo scalar ~ ~  ~      , a  a , (a  b )  b  a  a  b , …(28) If A is homogeneous of grade k one often simply writes

A  A  A . (14) The scalar magnitude A of a multivector A is k k

Grade selection is invariant under scalar n 2 ~ ~ A  A  A  A 2  A  A , (29)  A  A . (15)  0  r r k k scalar product r1

The consistent definition of inner and outer products of 2 where the separate term A is in particular due to the 0 vectors a and r-vectors A is r definition of the inner product in [2], p. 6, (1.21). The magnitude allows to define the inverse for simple k-blade   1  r  a  A  aA  (aA  (1) A a), (16) vectors r r r 1 2 r r ~ 1 A 1 1   1  r  , with . (30) a  A  aA  (aA  (1) A a) (17) A  2 A A  AA 1 r r r 1 2 r r A By linearity the full geometric product of a vector and a Alternative ways to express a A  G1(I) are multivector A is then n aA  a  A  a  A . (18) I  a  0 or Ia  I  a . (31) This extends to the distributive multiplication with arbitrary The projection of a into A  G1(I) is multivectors A, B n     n n (a  b)  (c  d)  a  (b  (c  d))  P (a)  P(a)  ak a  a  a ak  a, (32) (42) I  k  k         k 1 k 1 (a  d)(b  c)  (a  c)(b  d ),      where ak is the reciprocal frame defined by (a  b)2  (a  b)  (a  b)  (a b)2  a2b 2 

      2 if  (b  a)  (a  b)   a  b , (43) k  k 1 j  k a  a j   j  Kronecker delta   . (33) 0 if j  k  and the Jacobi identity           A general convention is that inner products a b and a  (b  c)  b  (c  a)  c  (a  b)  0. (44)  The commutator product of multivectors A,B is outer products a  b have priority over geometric products 1  A B  (AB  BA). (45) ab , e.g. 2     One useful identity using it is a bc  de  (a b)(c  d )e . (34)      (a  b)  A  ab  A  A  ab  ab  A  A  ab.(46) The The projection of a multivector B on a subspace described by a commutator product is to be distinguished from the cross simple m-vector (m-blade) A  a  a  ...  a ,m  n m 1 2 m product, which is strictly limited to the three-dimensional is Euclidean case with unit pseudoscalar I : sm 3 P (B)  (B  A)  A1  A1  (A B ), (35) A   (s)       general   a  b  (b  a)I  (a  b)I . (47) degree dependent 3 3 For more on basic geometric algebra I refer to [1,2,9] and to P ( B )  B , P ( B )  B  AA1, (36) A 0 0 A n n section 3 of [3]. the exceptions for scalars B and pseudoscalars 0 3. Vector Differential Calculus This section shows how to differentiate functions on linear B being again due to the definition of the inner product in n subspaces of the universal geometric algebra G by vectors. [2], p. 6, (1.21). A projection of one factor of an inner product has the effect It has wide applications particularly to mechanics and physics    in general [1]. Separate concepts of gradient, divergence and a  P(b)  P(a)  P(b)  P(a) b. (37) curl merge into a single concept of vector derivative, united by the geometric product. For a multivector B G(A ) , with A  A we have m m The relationship of differential and derivative is clarified. The Taylor expansion (P. 12) is applied to important examples, (a  B)  A  (a  B)A  a  (BA) if a  A  0 . (38) yielding e.g. the Legendre polynomials (P. 36). The Reordering rules for products of homogeneous multivector integrability (P. 42, etc.) of multivector functions are defined are and discussed. Throughout this section a number of basic differentials and derivations are performed explicitly A  B  (1)r(sr) B  A for r  s , (39) r s s r illustrating ease and power of the calculus. ([1]-[5]). As for the notation: P. 7 refers to proposition 7 of this A  B  (1)rs B  A . (40) r s s r section. Def. 13 refers to definition 13 of this section. (6) Elementary combinations that occur often are refers to equation number (6) in the previous section on basic      geometric algebra. a  (b  c)  (a  b)c  (a  c)b  a  bc  a  cb, (41) Standard definitions of continuity and scalar differentiability apply to multivector-valued functions, F  F((x)),   (x) scalar valued function. because the scalar product determines a unique “distance”  A  B between two elements A, B G(I) . Proposition 8 (identity) a  x  a . Definition 1 (directional derivative) Proposition 9 (constant function) A independent of x :   F  F(x) multivector-valued function of a vector variable a  A  0 . Proposition 10 (vector length) x defined on an n-dimensional A  G1(I) , I n      a  x   a   x    a  xˆ . unit pseudoscalar. a  An . x         dF(x  a ) dF(x  a )  dF(x) x  a  F   lim xˆ   unit vector in the direction of x (11). d  0  x Nomenclature: derivative of F in the direction a , Proposition 11 (direction function)     a  a  xˆxˆ xˆxˆ  a a -derivative of F. ([1] uses    , [2] uses    .) a  xˆ   . x x Proposition 2 (distributivity w.r.t. vector argument)       Proposition 12 (Taylor expansion) (a  b)  F  a  F  b  F , a,b  An    (a  )k Proposition 3 For scalar  F(x  a)  exp(a  )F(x)  F(x).  k!     k 0 (a )  F   (a  F ) Definition 13 (continuously differentiable, differential) Proposition 4 (distributivity w.r.t. multivector-valued function) F is continuously differentiable at x if for each fixed a     a  (F  G)  a  F  a  G a  F(y) exists and is a continuous function of y for

F  F(x) , G  G(x) multivector-valued functions of a each y in a neighborhood of x . vector variable x . In the notation of Def. 13: If F is defined and continuously differentiable at x , then, for  F  G  F  G . fixed x , a  F(x) is a of a , the (first)

Proposition 5 () differential of F.             a  (FG)  (a  F)G  F(a  G) F(a, x)  Fa (x)  a  F(x) .

In the notation of Def. 13: FG  FG  FG . ([1], p. 107 uses F  F .)

Suppressing x , or for fixed x : Proposition 6 (grade invariance)    F  F(a)  F  a  F .   a a   F  a  F k k Proposition 14 (linearity)      a   is therefore said to be a scalar differential operator. F(a  b)  F(a)  F(b) Proposition 7 (scalar chain rule)  scalar: F(a)  F(a)   dF a  F  (a  ) Proposition 15 (linear approximation) d    Proposition 23 For r  x  x0 sufficiently small: For x independent of x and r  r  x  x :       F(x)  F(x0 )  F(x  x0 )  F(x)  F(x0 ) .  r r Proposition 16 (chain rule) a  r  a   a  rˆ , where rˆ  . r r dF   d     x(t)   x(t)  F(x)    dt dt x  x(t)  rˆrˆ  a   Proposition 24 a  rˆ  . Definition 17 (vector derivative) r      2 Differentiation of F by its argument x :  x F(x)  F ,  rˆ  a Proposition 25 a   (rˆ  a )  .  r with the differential operator   , assumed to x     rˆ  aa  rˆ 1 Proposition 26 a  (rˆ  a)  . (i) have the algebraic properties of a vector in An  G (I) , I r unit pseudoscalar; and      rˆ  a rˆ  a      Proposition 27 a   rˆ  a   . (ii) that a   x with a An is a   x F as in Def. 1. r  Proposition 18 (algebraic properties of   )  1 1 1 x Proposition 28 a     a . r r r  (31)  (31)  I     0 , I   I    x x x    1 a  rˆ (32) n Proposition 29 a   2  2 3 .    k   r r  x  PI ( x )  a ak  x , k1  ˆ 2 ˆ  2 1   2 1 3(a  r)  r  a k Proposition 30 (a  )  . where the a express the algebraic vector properties and the 2 r 2 r 4   Proposition 31 ak   x the scalar differential properties. 2 1  1  4(a  rˆ)3  4 rˆ  a a  rˆ Definition 19 (gradient) (a  )3  . 6 r 2 r5       The vector f  f (x)   (x)   for a scalar x     a  r  Proposition 32 a  logr  . function   (x) is called the gradient of . r 2 Propostion 20 (3-dimensional ) Proposition 33  For integer k and r  0 if k<0 : For b independent of x  A  G1(I ) : 3 3  a  r2k  2ka  rr2(k 1).    a  (x  b)  a  b . Proposition 34 Proposition 21 For integer k and r  0 if 2k+1<0 :   a   x  A  a  A ,  2k 1 2k   r r a  r  r (a  2ka  rˆrˆ). A independent of x . 1 Proposition 22 Proposition 35 (Taylor expansion of ) x  a     a  [x  (x  b)]  a  (x  b)  x  (a  b) The differential of composite functions is the composite of 1 1 1 1 1 1 1   a  a a  ... . differentials. x  a x x x x x x F(x,a)  G( f (x), f (x,a)) (explicit) Proposition 36 (Legendre Polynomials)

The Legendre Polynomials Pn are defined by: Definition 41 (second differential) 1  P (xˆa)  P (xa)        n  n . F  (x)  b  a  F (x) .     n1   2n1 ab x  a n0 x n0 x     Suppressing x : F   b  a  F . The explicit first four polynomials are: ab P (xa)  1 0 Proposition 42 (integrability condition) F   F .    ab ba P1(xa)  x  a The second differential is a symmetric bilinear function of its  1   2 2 2 P2 (xa)  3(x  a)  a x  2 differential arguments a, b. 1  (x  a)2  (x  a)2 Remark: Please see [2,9,10] for more results and proofs. 2  1   3 2 2   Acknowledgements: “Soli deo gloria.” (Latin: To God alone P3 (xa)  5(x  a)  3a x x  a  2 be the glory.) [8] I thank my family for their patient support. 3 (x  a)3  x  a(x  a)2 2 References n n n [1] D. Hestenes, New Foundations for Classical Mechanics, P (xa)  x P (xˆa)  x a P (xˆaˆ). n n n 2nd ed., Kluwer Dordrecht, 1999. Definition 37 (redefinition of differential, over-dots) [2] D. Hestenes, G. Sobczyk, Clifford Algebra to Geometric Calculus, Kluwer, Dordrecht, 1999.  (7) 1   F(a)  a  F  aF  aF  , [3] E. Hitzer, Antisymmetric Matrices Are Real ,   2 Mem. Fac. Eng. Fukui Univ. 49(2), 283 (2001). where the over-dots indicate, that only F is to be differentiated [4] L. Dorst, The inner products of geometric algebra, in C. and not a . Doran, et al, eds. Applied Geometrical Alegbras in computer Science and Engineering, AGACSE 2001, Proposition 38 For a  A  G1(I) , P  P : n I Birkhauser, 2001.        [5] C. Doran, Geometric Algebra and its Appl. to Mat. Physics, a     a  P(  )  P(a)    , P(a) . x x x An Ph.D. thesis, University of Cambridge, 1994. Proposition 39 [6] H. Grassmann, (F. Engel, editor) Die Ausdehnungslehre  von 1844 und die Geometrische Analyse, 1, part 1, Teubner, F(a)  F(P(a))  P(a)  F . Leipzig, 1894. [7] Hebrews, chapter 11, verses 1 to 3, Holy Bible, New F(a)  0 , if P(a)  0 . International Version. Proposition 40 (differential of composite functions) [8] Johann Sebastian Bach. [9] E. Hitzer, Vector Differential Calculus, Mem. Fac. Eng. For F(x)  G( f (x)) and Fukui Univ. Vol. 50, No. 1 (2002). [10] E. Hitzer, Multivector Differential Calculus, Advances in f : x  A  G 1(I)  f (x)  A  G 1 (I) n n Applied Clifford Algebras 12(2) pp. 135-182 (2002).   [11] E. Hitzer, B. Mawardi, Clifford Fourier Transf. on a  F  f (a)  G Multivector Fields and Uncertainty Principles for Dims n = 2 (mod 4) and n = 3 (mod 4), Adv. in Applied Cliff. F(a)  G( f (a)) (Def. 13) Alg., DOI 10.1007/s00006-008-0098-3, Online First.