Geometric Calculus Research Articles

Clifford Algebra Implementations in Maxima

This tutorial focuses on the packages $\texttt{clifford}$ and $\texttt{cliffordan}$ for the computer algebra system Maxima. Maxima is the open source descendant of the first computer algebra system and features a rich functionality from a large number of shared packages. The Maxima language is based on the ideas of functional programming, which is particularly well suited for transformations of formal mathematical expressions. While $\texttt{clifford}$ implements Clifford algebras $C\ell_{p,q,r}$ of arbitrary signatures and order based on the elementary construction of Macdonald, $\texttt{cliffordan}$ features geometric calculus functionality. Using $\texttt{clifford}$ expressions containing geometric, outer and inner products can be simplified. Applications of $\texttt{clifford}$ and $\texttt{cliffordan}$ in linear algebra and calculus are demonstrated.

Application of geometric calculus in numerical analysis and difference sequence spaces

The main purpose of this paper is to introduce the geometric difference sequence space ℓ∞G(ΔG) and prove that ℓ∞G(ΔG) is a Banach space with respect to the norm ‖.‖ΔGG. Also we compute the α-dual, β-dual and γ-dual spaces. Finally we obtain the Geometric Newton–Gregory interpolation formulae.

Geometric Calculus of the Gauss Map

In this paper we connect classical differential geometry with the concepts from geometric calculus. Moreover, we introduce and analyze a more general Laplacian for multivector-valued functions on manifolds. This allows us to formulate a higher codimensional analog of Jacobi`s field equation.

Analytic Study Regarding Yarn Geometry of Two Synthetic Materials Used in Furniture and Automotive Industry

In this study the knitted fabric side of two leather substitute materials was analysed. The two synthetic leather material used are usually fabricated for the use as upholstery in the furniture and automotive industry. The first material has a polyurethane (PU) matrix and the second has a polyvininylchloride (PVC) matrix.The main analysis done in this study was the geometrical calculus of the yarn eye using different geometric models and the evaluation of the relative deformation in surface for a biaxial tensile loading (along the rows and the columns of the knitted fabric). Results show a relative deformation of 6.29% for the PU matrix material and a 9.41% relative deformation for the PVC matrix material.The main goal of this study was to obtain a better characterization of the materials studied in order to have indications based on loading conditions when using these materials in their different industrial application.

Sparse Representations of Clifford and Tensor Algebras in Maxima

Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G3 = Cl(3,0) as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symbolic tensor calculations supported by Maxima, based on the itensor package. The Clifford algebra functionality of Maxima is distributed as two new packages called clifford - for basic simplification of Clifford products, outer products, scalar products and inverses; and cliffordan - for applications of geometric calculus.

A Survey of Geometric Algebra and Geometric Calculus

This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper.

Coordinate Free Integrals in Geometric Calculus

We introduce a method for evaluating integrals in geometric calculus without introducing coordinates, based on using the fundamental theorem of calculus repeatedly and cutting the resulting manifolds so as to create a boundary and allow for the existence of an antiderivative at each step. The method is a direct generalization of the usual method of integration on function of a real variable. It may lead to both practical applications and help unveil new connections to various fields of mathematics.

Geometric-Algebra LMS Adaptive Filter and Its Application to Rotation Estimation

This letter exploits geometric (Clifford) algebra (GA) theory to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from geometric calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3-D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3-D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.

Geometric-Algebra Adaptive Filters

This paper reformulates adaptive filters (AFs) in the framework of geometric algebra (GA), developing a complete study of the resulting geometric-algebra adaptive filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of GA, a comprehensive mathematical language well suited for the description of geometric transformations. Also, differently from standard adaptive-filtering theory, geometric calculus (the extension of GA to differential calculus) allows for applying the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex numbers, quaternions, etc. Relying on those characteristics (among others), a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular AFs to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least mean squares (LMS) AF variants via algebraic isomorphisms: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing very good agreement. Real-data experiments highlight the good tracking performance of the GAAFs in a joint linear prediction of different signals.

Natural Boundary Conditions in Geometric Calculus of Variations

Abstract In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y - the manifold of dependent and independent variables underlying a given problem - as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in ℝn+1, and the Lagrangian is first-order (in particular, the hypersurface area).

Geometric Calculus-based Postulates for the Derivation and Extension of the Maxwell Equations

Clifford analysis, particularly application of the geometric algebra of three-dimensional physical space and its associated geometric calculus, enables a compact formulation of Maxwell’s electromagnetic (EM) equations from a set of physically relevant and mathematically pleasing postulates. This formulation results in a natural extension of the Maxwell equations yielding wave solutions in addition to the usual EM waves. These additional solutions do not contradict experiment and have three properties in common with the apparent properties of dark energy. These three properties are that the wave solutions 1) propagate at the speed of light, 2) do not interact with ordinary electric charges or currents, and 3) possess retrograde momentum. By retrograde momentum, we mean that the momentum carried by such a wave is directed oppositely to the direction of energy transport. A “gas” of such waves generates negative pressure.

Looking for a Mathematical Solution for a Self-Centering Device

Proper centering problem is common to processing by cutting the metal or nonmetallic cylindrical parts. This paper proposes a solution useful in self-centering devices with mobile jaws, both in terms of cinematic jaw movement - rotation instead of translation - as well as the active profile of the jaw - curvilinear instead of straight. The method of problem solving is mathematical deduction, on the one hand by establishing the geometrical calculus schemes that describes the graphic curves for the envelope of a single-sided curvilinear profile for jaw, and on the other by the generalization of the solution for the case the active surfaces of the jaws are both curvilinear profile.

Tutorial on Geometric Calculus

The book Clifford Algebra to Geometric Calculus is the first and still the most complete exposition of Geometric Calculus (GC). But it is more of a reference book than a textbook, so can it be a difficult read for beginners. This tutorial is a guide for serious students who want to dig deeply into the subject. It presents helpful background and aims to clarify objectives, important results and methods in the book. It is supplemented by a hot-linked annotated bibliography of papers elaborating various aspects of Geometric Algebra and Calculus.

Double exponential map on symmetric spaces

We establish an asymptotic formula for the double exponential map operator on affine symmetric spaces. This operator plays an important role in the geometric calculus of symbols of (pseudo)differential operators on manifolds with connection, whose foundations were laid by Sharafutdinov. To obtain this result, we essentially use the structural theory of symmetric spaces and techniques of the Lie group theory. One of the key moments is an application of the Campbell-Hausdorff series in Dynkin form.

Kant’s 1768 attack on Leibniz’ conception of space

: This paper examines two features of Kant’s 1768 critique of Leibniz’ conception of space. Firstly, Leibniz’ proposed geometrical calculus called ‘analysis situs’; secondly, Leibniz’ relational conception of space. The main thesis of the paper is that Kant’s arguments are more powerful than generally recognized. With regard to the analysis situs, I will show that Kant was quite well informed about this proposed science and that his arguments severely undermine Leibniz’ claims to what it could perform. With regard to the relational theory of space, Kant’s argument would require Leibniz to present a complex story about the relation between God’s act of creation and our spatial experience to defend his relational view, rather than using the simple principle of sufficient reason.

Complex mapping of aerofoils – a different perspective

In this article an application of conformal mapping to aerofoil theory is studied from a geometric and calculus point of view. The problem is suitable for undergraduate teaching in terms of a project or extended piece of work, and brings together the concepts of geometric mapping, parametric equations, complex numbers and calculus. The Joukowski and Karman–Trefftz aerofoils are studied, and it is shown that the Karman–Trefftz aerofoil is an improvement over the Joukowski aerofoil from a practical point of view. For the most part only a spreadsheet program and pen and paper is required, only for the last portion of the study of the Karman–Trefftz aerofoils a symbolic computer package is employed. Ignoring the concept of a conformal mapping and instead viewing the problem from a parametric point of view, some interesting mappings are obtained. By considering the derivative of the mapped mapping via the chain rule, some new and interesting analytical results are obtained for the Joukowski aerofoil, and numerical results for the Karman–Trefftz aerofoil.

Some basic results on the sets of sequences with geometric calculus

As an alternative to the classical calculus, Grossman and Katz [Non-Newtonian Calculus, Lee Press, Pigeon Cove, Massachusetts, 1972] introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and bigeometric calculus. Following Grossman and Katz, we construct the field C(G) of geometric complex numbers and the concept of geometric metric. Also we give the triangle and Minkowski's inequalities in the sense of geometric calculus. Later we respectively define the sets w(G), l∞(G), c(G), c0(G) and lp(G) of all, bounded, convergent, null and p-absolutely summable sequences, in the sense of geometric calculus and show that each of the set forms a complete vector space on the field C(G).

Geometric calculus for second-order differential equations and generalizations of the inverse problem of Lagrangian mechanics

We review the main features of the geometric calculus which has been introduced over the past 15 years in the study of second-order ordinary differential equations and then explain how a recently introduced generalization of the inverse problem of Lagrangian mechanics can be very concisely dealt with by this calculus in an intrinsic way. This paper is an account of the lecture with the same title presented at the ICDVC-2010 Conference in Hangzhou, May 12–14 (2010).

Proposition 10, Book 2, in the Principia, revisited

In Proposition 10, Book 2 of the Principia, Newton applied his geometrical calculus and power series expansion to calculate motion in a resistive medium under the action of gravity. In the first edition of the Principia, however, he made an error in his treatment which lead to a faulty solution that was noticed by Johann Bernoulli and communicated to him while the second edition was already at the printer. This episode has been discussed in the past, and the source of Newton’s initial error, which Bernoulli was unable to find, has been clarified by Lagrange and is reviewed here. But there are also problems in Newton’s corrected version in the second edition of the Principia that have been ignored in the past, which are discussed in detail here.

The Homological Kähler‐De Rham Differential Mechanism part I: Application in General Theory of Relativity

The mechanism of differential geometric calculus is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalars defined together with the associated de Rham complex. In this communication, we demonstrate that the dynamical mechanism of physical fields can be formulated by purely algebraic means, in terms of the homological Kähler‐De Rham differential schema, constructed by connection inducing functors and their associated curvatures, independently of any background substratum. In this context, we show explicitly that the application of this mechanism in General Relativity, instantiating the case of gravitational dynamics, is related with the absolute representability of the theory in the field of real numbers, a byproduct of which is the fixed background manifold construct of this theory. Furthermore, the background independence of the homological differential mechanism is of particular importance for the formulation of dynamics in quantum theory, where the adherence to a fixed manifold substratum is problematic due to singularities or other topological defects.