aa r X i v : . [ m a t h . G M ] D ec Nested Coordinate Systems in Geometric Algebra
Garret SobczykUniversidad de las Am´ericas-PueblaDepartamento de Actuar´ıa F´ısica y Matem´aticas72820 Puebla, Pue., M´exicoJanuary 5, 2021
Abstract
A nested coordinate system is a reassigning of independent variablesto take advantage of geometric or symmetry properties of a particular ap-plication. Polar, cylindrical and spherical coordinate systems are primaryexamples of such a regrouping that have proved their importance in theseparation of variables method for solving partial differential equations.Geometric algebra offers powerful complimentary algebraic tools that areunavailable in other treatments.
AMS Subject Classification MSC-2020:
Keywords:
Clifford algebra, coordinate systems, geometric algebra, sepa-ration of variables.
Geometric algebra G is the natural generalization of Gibbs Heaviside vectoralgebra, but unlike the latter, it can be immediately generalized to higher dimen-sional geometric algebras G p,q of a quadratic form. On the other hand, Cliffordanalysis, the generalization of Hamilton’s quaternions, is also expressed in Clif-ford’s geometric algebras [1]. The main purpose of this article is to formulate theconcept of a nested coordinate system , a generalization of the well-known meth-ods of orthogonal coordinate systems to apply to any coordinate system. Werestrict ourselves to the geometric algebra G because of its close relationship tothe Gibbs-Heaviside vector calculus [3]. This restriction also draws attention tothe clear advantages of geometric algebra over the later, because of its powerfulassociative algebraic structure.The idea of a nested rectangular coordinate system arises naturally whenstudying properties of polar coordinates in the 2 and 3-dimensional Euclideanvector spaces R and R . We begin by discussing the relationship between or-dinary polar coordinates and the nested rectangular coordinate system N , ,before going on to the higher dimensional nested coordinate system N , , uti-lized in the reformulation of cylindrical and spherical coordinates. A detailed1iscussion of the geometric algebra G is not given here, but results are oftenexpressed in the closely related well-known Gibbs-Heaviside vector analysis forthe benefit of the reader. Let G := G ( R ) be the geometric algebra of 2-dimensional Euclidean space R .An introductory treatment of the geometric algebras G , G and G is given in[4, 5, 6]. Most important in studying the geometry of the Euclidean plane is theposition vector x := x [ x, ˆ x ] = x ˆ x (1)expressed here as a product of its Euclidean magnitude x and its unit direction ,the unit vector ˆ x . In terms of rectangular coordinates ( x , x ) ∈ R , x = x [ x , x ] = x e + x e , (2)for the orthogonal unit vectors e , e along the x and x axis, respectively. Theadvantage of our notation is that it immediately generalizes to 3 and higherdimensional spaces of arbitrary signature ( p, q ) in any of the definite geometricalgebras G p,q := G ( R p,q ) of a quadratic form.The vector derivative , or gradient in the Euclidean plane is defined by ∇ := e ∂ + e ∂ (3)where ∂ := ∂∂x and ∂ := ∂∂x are partial derivatives [3, p.105]. Clearly, e = ∂ x = e · ∇ x , e = ∂ x = e · ∇ x . Since ∇ is the usual 2-dimensional gradient, it has the well-known properties ∇ x = 2 , and ∇ x = ˆ x. With the help of the product rule for differentiation,2 = ∇ x = ( ∇ x )ˆ x + x ( ∇ ˆ x ) = ˆ x + x ( ∇ ˆ x ) . (4)Since in geometric algebra x = x , it follows that ˆ x = 1, so that for x ∈ R , ∇ ˆ x = 1 x and e · ∇ x = e · ˆ x = x x , e · ∇ x = e · ˆ x = x x . (5)Similarly, ∇ ˆ x = n − x for x ∈ R n . This is the first of many demonstrations ofthe power of geometric algebra over standard vector algebra.By a nested rectangular coordinate system N , ( x , x [ x , x ]), we mean x = x ˆ x = x [ x , x ] = x (cid:2) x , x [ x , x ] (cid:3) . Note in geometric algebra, unlike in standard vector analysis, we need not write ∇· x = 2.This has many important consequences in the development of the subject. x and x := p x + x to be independent. The partial derivatives with respect to these independentvariables is denoted by ˆ ∂ := ˆ ∂∂x and ˆ ∂ x := ˆ ∂∂x , the hat on the partial derivativesindicating the new choice of independent variables.For polar coordinates ( x, θ ) ∈ R , for x := p x + x ≥
0, 0 ≤ θ < π , and x := x [ x, θ ], x = x ˆ x [ θ ] = x ( e x x + e x x ) = x ( e cos θ + e sin θ ) , (6)where cos θ := x x and sin θ := x x . Using (5), ∇ ˆ x = ∇ ˆ x [ θ ] = ( ∇ θ ) ∂ ˆ x∂θ = 1 x ⇐⇒ ∇ θ = 1 x ∂ ˆ x∂θ , ∇ θ = 0 , (7)since ∇ ˆ x = ( ∇ θ ) ∂ θ ( e cos θ + e sin θ ) = ( ∇ θ )( − e sin θ + e cos θ ) , and ∇ θ = − ˆ xx ∂ θ ˆ x + 1 x ( ∇ θ ) ∂ θ ˆ x = − (cid:16) ˆ xx · ( ∂ θ ˆ x ) (cid:17) = 0 . The ⇐⇒ follows by multipling both sides of the first equation by the unitvector ∂ θ ˆ x , which is allowable in geometric algebra. Note also the use of thefamous geometric algebra identity 2 a · b = ( ab + ba ) for vectors a and b , [4,p.26].The 2-dimensional gradient ∇ , ∇ = e ∂∂x + e ∂∂x = e ∂ + e ∂ (8)already defined in (3), and the Laplacian ∇ is given by ∇ = ∂ ∂x + ∂ ∂x = ∂ + ∂ . (9)In polar coordinates,ˆ ∇ = ( ∇ x ) ˆ ∂ x + ( ∇ θ ) ˆ ∂ θ = ˆ x ˆ ∂ x + 1 x ( ˆ ∂ θ x ) ˆ ∂ θ (10)for the gradient where ˆ ∂ θ := ˆ ∂∂θ , and since ˆ ∇ θ = 0,ˆ ∇ = ˆ ∇ (cid:0) ˆ x ˆ ∂ x + ( ˆ ∇ θ ) ˆ ∂ θ (cid:1) = (cid:16) ˆ ∇ ˆ x + ˆ x · ˆ ∇ (cid:17) ˆ ∂ x + (cid:16) ˆ ∇ θ + ( ˆ ∇ θ ) · ˆ ∇ (cid:17) ˆ ∂ θ = ˆ ∂ x + 1 x ˆ ∂ x + 1 x ˆ ∂ θ . (11)for the Laplacian. The decomposition of the Laplacian (11), directly impliesthat Laplace’s differential equation is separable in polar coordinates.3hen expressed in nested rectangular coordinates N , ( x , x ), the gradient ∇ ≡ ˆ ∇ takes the formˆ ∇ := ( ∇ x ) ˆ ∂∂x + ( ∇ x ) ˆ ∂∂x = e ˆ ∂ + ˆ x ˆ ∂ x . (12)Dotting equations (8) and (12) on the left by e and ˆ x gives the transformationrules ∂ = ˆ ∂ + x x ˆ ∂ x , ˆ x · ˆ ∇ = x x ˆ ∂ + ˆ ∂ x = cos θ ∂ + sin θ ∂ . Using these formulas the nested Laplacian takes the formˆ ∇ = ˆ ∂ + 2 x x ˆ ∂ x ˆ ∂ + 1 x ˆ ∂ x + ˆ ∂ x = − ˆ ∂ + 2 ∂ ˆ ∂ + 1 x ˆ ∂ x + ˆ ∂ x . (13)The unusual feature of the nested Laplacian is that it is defined in terms of boththe ordinary partial derivative ∂ and the nested partial derivative ˆ ∂ . Whereaspartial derivatives generally commute, partial derivatives of different types donot. For example, it is easily verified that ∂ ˆ ∂ x x = 2 x , whereas ˆ ∂ ∂ x x = 4 x . Because the mixed partial derivatives ˆ ∂ x ˆ ∂ occurs in (13), Laplace’s differ-ential equation in the real rectangular coordinate system N , ( x , x ) is not, ingeneral, separable. Indeed, suppose that a harmonic function F is separable, sothat F = X X for X = X [ x ] , X = X [ x ]. Using (13),ˆ ∇ FX X = ∂ X X + (cid:16) ∂ x X + x ∂ x (cid:17) XX + 2 (cid:16) x ∂ X X (cid:17)(cid:16) ∂ x XxX (cid:17) = 0 . (14)The last term on the prevents F in general from being separable. However,it is easily checked that F = k x x is harmonic and a solution of (13). When X [ x ] = kx , it is easily checked that x ∂ X X = 1. Letting F = kx X [ x ], andrequiring ˆ ∇ F = 0, leads to the differential equation for X [ x ],3 ∂ x X + x∂ x X = 0 , with the solution X [ x ] = c x + c . The simplest example of a harmonic function F = X X is when X = x and X = x . A graph of this function is shown inFigure 1. Consider the real nested rectangular coordinate system ( x , x p , x ), defined by N , , := { ( x , x p , x ) | x = x ˆ x = x e + x p ˆ x p + x ˆ x } , Figure 1: The harmonic 2-dimensional function F = x x + x is shown.where x p = p x + x ≥ , x = p x + x + x ≥
0. In nested coordinates, thegradient ∇ = e ∂ + e ∂ + e ∂ takes the formˆ ∇ = ( ˆ ∇ x ) ˆ ∂ + ( ˆ ∇ x p ) ˆ ∂ p + ( ˆ ∇ x ) ˆ ∂ x = e ˆ ∂ + ˆ x p ˆ ∂ p + ˆ x ˆ ∂ x , (15)where ˆ ∂ p := ˆ ∂∂x p . Formulas relating the gradients ∇ and ˆ ∇ easily follow: ∂ = ˆ ∂ + e · ˆ x p ˆ ∂ p + e · ˆ x ˆ ∂ x = ˆ ∂ + x x p ˆ ∂ p + x x ˆ ∂ x (16) ∂ = e · ˆ x p ˆ ∂ p + e · ˆ x ˆ ∂ x = x x p ˆ ∂ p + x x ˆ ∂ x (17)and ∂ = e · ˆ x ˆ ∂ x = x x ˆ ∂ x . (18)For the Laplacian ∇ in nested coordinates, with the help of (15),ˆ ∇ = ˆ ∇ ( e ˆ ∂ + ˆ x p ∂ x p + ˆ x ˆ ∂ x ) = e · ˆ ∇ ˆ ∂ + ˆ ∇ · ˆ x p ˆ ∂ p + ˆ ∇ · ˆ x ˆ ∂ x e · ˆ ∇ ˆ ∂ + ˆ ∇ · ˆ x p ˆ ∂ p + ˆ ∇ · ˆ x ˆ ∂ x = (cid:16) ˆ ∂ + x x p ˆ ∂ p + x x ˆ ∂ x (cid:17) ˆ ∂ + (cid:16) x p + x x p ˆ ∂ + ˆ ∂ p + x p x ˆ ∂ p (cid:17) ˆ ∂ p + (cid:16) x + x x ˆ ∂ + x p x ˆ ∂ p + ˆ ∂ x (cid:17) ˆ ∂ x = ˆ ∂ + ˆ ∂ p + ˆ ∂ x + 2 (cid:18) x x p ˆ ∂ ˆ ∂ p + x x ˆ ∂ ˆ ∂ x + x p x ˆ ∂ p ˆ ∂ x (cid:19) + 1 x p ˆ ∂ p + 2 x ˆ ∂ x . (19)Another expression for the Laplacian in mixed coordinates is obtained with thehelp of (16),ˆ ∇ = − ˆ ∂ + ˆ ∂ p + ˆ ∂ x + 2 (cid:18) ∂ ˆ ∂ + x p x ˆ ∂ p ˆ ∂ x (cid:19) + 1 x p ˆ ∂ p + 2 x ˆ ∂ x . (20)Suppose F = F [ x , x p , x ]. In order for F to be harmonic, ˆ ∇ F = 0. Assum-ing that F is separable, F = X [ x ] X p [ x p ] X x [ x ], and applying the Laplacian(20) to F givesˆ ∇ F = ( ˆ ∂ X ) X p X x + X (cid:16)(cid:0) ˆ ∂ p + 1 x p ˆ ∂ p (cid:1) X p (cid:17) X x + X X p (cid:16) x ˆ ∂ x X x (cid:17) +2 (cid:18)(cid:0) x p ∂ p X p (cid:1)(cid:0) x ˆ ∂ x X x (cid:1) X + (cid:0) ∂ X p (cid:1) X x + X p (cid:0) ∂ X x (cid:1)(cid:19) . (21)We now calculate the interesting expression (cid:0) x p ∂ p X p (cid:1)(cid:0) x ˆ ∂ x X x (cid:1) X + (cid:0) ∂ X p (cid:1) X x + X p (cid:0) ∂ X x (cid:1) X X p X x = (cid:16) x p (cid:0) ∂ p log X p (cid:1)(cid:17)(cid:16)(cid:0) x ∂ x log X x (cid:1)(cid:17) + ∂ log( X p X x ) X . In general, because of the last term in (21), a function F = X X p X x willnot be separable. However, just as in the two dimensional case, there are 3-dimensional harmonic solutions of the form F = x k x mp x n . Taking the Laplacian(19) of F , with the help of [7], givesˆ ∇ F = (2 km + m ) x n x k x m − p + ( − k + k ) x n x k − x mp +(2 kn + 2 mn + n (1 + n )) x n − x k x mp = 0 . This last expression vanishes when the system of three equations, { km + m = 0 , − k + k = 0 , and 2 kn + 2 mn + n (1 + n ) = 0 } . All of the distinct non-trivial harmonic solutions F = x k x mp x n are listed in thefollowing Table 6 m n1 0 00 0 -11 -2 01 0 -31 -2 1 (22) Cylindrical and spherical coordinates are examples of nested coordinates N , ( R ),and N , ( R ), respectively. For the first, x = x [ x p , θ, x ] = x p [ x p , θ ] + x [ x ] , (23)where x p = x p ˆ x p [ θ ], x p = p x + x , and x = x e . Cylindrical coordinates( x p , θ, x ) ∈ R = R ∪ R are a decomposition of R into the polar coordi-nates ( x p , θ ) ∈ R , already studied in Section 1, and x ∈ R . For sphericalcoordinates, x p = x p ˆ x p [ θ ] the same as in cylindrical and polar coordinates, and x = x [ x, θ, ϕ ] = x ˆ x [ θ, ϕ ]) = x (cid:16) e cos ϕ + ˆ x p [ θ ] sin ϕ (cid:17) , (24)where x = q x + x + x , ˆ x [ θ, ϕ ] = e cos ϕ + ˆ x p [ θ ] sin ϕ, ˆ x p [ θ ] = e cos θ + e sin θ. The basic quantities that define both cylindrical and spherical coordinates areshown in Figure 2.The gradient ˆ ∇ and Laplacian ˆ ∇ for cylindrical coordinates are easily cal-culated. With the help of (7), (10), and (11),ˆ ∇ = ( ˆ ∇ x p ) ˆ ∂ p + ( ˆ ∇ θ ) ˆ ∂ θ + ( ˆ ∇ x ) ˆ ∂ for the cylindrical gradient, andˆ ∇ = ˆ ∇ (cid:16) ˆ x p ˆ ∂ p + ( ˆ ∇ θ ) ˆ ∂ θ + e ˆ ∂ (cid:17) = ˆ ∂ p + 1 x p ˆ ∂ p + 1 x p ˆ ∂ θ + ˆ ∂ (25)for the cylindrical Laplacian. Letting F [ x ] = X p [ x ] X θ [ θ ] X [ x ], the resultingequation is easily separated and solved by standard methods, resulting in threesecond order differential equations with solutions, X p [ x p ] = k J n [ βx p ] + k Y n [ βx p , ] X θ [ θ ] = k cos nθ + k sin nθX [ x ] = k cosh( α ( m − x )) + k sinh( α ( m − x )) , xx xx O x3 Figure 2: For cylindrical coordinates, x = x p ˆ x p [ θ ] + x e . For spherical coordi-nates, x = x ( e cos ϕ + ˆ x p [ θ ] sin ϕ ).where J n and Y n are Bessel functions of the first and second kind . The constantsare determined by the various boundary conditions that must be satisfied indifferent applications [8, p.254].Turning to spherical coordinates ( x, θ, ϕ ) ∈ R , the spherical gradientˆ ∇ = ( ˆ ∇ x ) ˆ ∂ x + ( ˆ ∇ θ ) ˆ ∂ θ + ( ˆ ∇ ϕ ) ˆ ∂ ϕ = ˆ x ˆ ∂ x + 1 x p ( ˆ ∂ θ ˆ x p ) ˆ ∂ θ + 1 x ( ˆ ∂ ϕ ˆ x ) ˆ ∂ ϕ , (26)where from previous calculations for polar and cylindrical coordinates,ˆ ∇ θ = 1 x p ( ˆ ∂ p ˆ x p ) , ( ˆ ∇ θ ) = 1 x p , ˆ ∇ θ = 0 , ˆ ∇ ϕ = 1 x ˆ ∂ ϕ ˆ x, ( ˆ ∇ ϕ ) = 1 x . (27)Furthermore, since ˆ x = ˆ x [ θ, ϕ ] = e cos ϕ + ˆ x p [ θ ] sin ϕ x = ˆ ∇ ˆ x = ( ˆ ∇ θ )( ˆ ∂ θ ˆ x ) + ( ˆ ∇ ϕ )( ˆ ∂ ϕ ˆ x ) = 1 x p ( ˆ ∂ θ ˆ x )( ˆ ∂ ϕ ˆ x ) + 1 x , it follows that ( ˆ ∂ θ ˆ x )( ˆ ∂ ϕ ˆ x ) = x p x = sin ϕ, and ˆ ∇ ϕ = x x x p . That ˆ ∇ ϕ = x x x p follows using (26) and (27),ˆ ∇ ϕ = ˆ ∇ (cid:0) x ˆ ∂ ϕ ˆ x (cid:1) = (cid:16) − ˆ xx + 1 x ˆ ∇ (cid:17) ˆ ∂ ϕ ˆ x − ˆ xx ˆ ∂ ϕ ˆ x − x (cid:16) ( ˆ ∇ x ) ˆ ∂ x ˆ ∂ ϕ ˆ x + ( ˆ ∇ θ ) ˆ ∂ θ ˆ ∂ ϕ ˆ x + ( ˆ ∇ ϕ ) ˆ ∂ ϕ ˆ x (cid:17) = − (cid:16) ˆ xx ˆ ∂ ϕ ˆ x + 1 x ( ˆ ∂ ϕ ˆ x )ˆ x (cid:17) + 1 xx p ( ˆ ∂ θ ˆ x p )( ˆ ∂ ϕ ˆ ∂ θ x ) = x x x p , since partial derivatives commute, ˆ ∂ x ˆ x = 0, and ˆ ∂ ϕ ˆ x = − ˆ x .For the spherical Laplacian, using (26) and (27),ˆ ∇ = ˆ ∇ (cid:16) ˆ x ˆ ∂ x + ( ˆ ∇ θ ) ˆ ∂ θ + ( ˆ ∇ ϕ ) ˆ ∂ ϕ (cid:17) = (cid:16) x + ˆ ∂ x (cid:17) ˆ ∂ x + ( ˆ ∇ θ ) · ˆ ∇ ˆ ∂ θ + (cid:16) ˆ ∇ ϕ + ( ˆ ∇ ϕ ) · ˆ ∇ (cid:17) ˆ ∂ ϕ = (cid:16) ˆ ∂ x + 2 x (cid:17) ˆ ∂ x + 1 x p ˆ ∂ θ + (cid:16) x x x p + 1 x ˆ ∂ ϕ (cid:17) ˆ ∂ ϕ , equivalent to the usual expression for the Laplacian in spherical coordinates [8,p.256].Just as in cylindrical coordinates, the solution of Laplace’s equation in spher-ical coordinates is separable, F = X x [ x ] X θ [ θ ] X ϕ [ ϕ ], resulting in three secondorder differential equations with solutions X x [ x ] = k x β + k x − ( β +1) ,X θ [ θ ] = k cos nθ + k sin nθ,X ϕ [ ϕ ] = k P mn (cos ϕ ) + k Q mn (cos ϕ ) , where P mn and Q mn are the Legendre functions of the first and second kind,respectively [8, p.258]. Acknowledgment
This work was largely inspired by a current project that author has with Pro-fessor Joao Morais of Instituto Tecnol´ogico Aut´onomo de M´exico, utilizingspheroidal coordinate systems. The struggle with this orthogonal coordinatesystem [2], led the author to re-examine the foundations of general coordinatesystems in geometric algebra [5, p.63].
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