Differential Forms in Geometric Calculus

  title={Differential Forms in Geometric Calculus},
  author={David Hestenes},
Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant application of the system is to relativistic physics where it is referred to as Spacetime Calculus. The… Expand
The Shape of Differential Geometry in Geometric Calculus
  • D. Hestenes
  • Mathematics, Computer Science
  • Guide to Geometric Algebra in Practice
  • 2011
We review the foundations for coordinate-free differential geometry in Geometric Calculus. In particular, we see how both extrinsic and intrinsic geometry of a manifold can be characterized by aExpand
Spacetime Geometry with Geometric Calculus
Geometric Calculus is developed for curved-space treatments of General Relativity and comparison is made with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle ofExpand
The Fundamental Theorem of Geometric Calculus via a Generalized
Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative ∇V F exists and is Lebesgue integrable. WeExpand
A new gauge theory of gravitation on flat spacetime has recently been developed by Lasenby, Doran, and Gull in the language of Geometric Calculus. This paper provides a systematic account of theExpand
The fundamental theorem of geometric calculus via a generalized riemann integral
Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative ∇VF exists and is Lebesgue integrable. WeExpand
Gauge Theory Gravity with Geometric Calculus
A new gauge theory of gravity on flat spacetime has recently been developed by Lasenby, Doran, and Gull. Einstein’s principles of equivalence and general relativity are replaced by gauge principlesExpand
The Intersection of Rays with Algebraic Surfaces
Although a well known result in the traditional language of multi-variable calculus, in this paper it is shown, using the language of geometric algebra, that for any real-valued polynomial definedExpand
A multivector data structure for differential forms and equations
We use tools from algebraic topology to show that a class of structural differential equations may be represented combinatorially, and thus, by a computer data structure. In particular, everyExpand
Geometry of complex data
  • K. J. Sangston
  • Computer Science
  • IEEE Aerospace and Electronic Systems Magazine
  • 2016
This tutorial provides a basic introduction to geometric algebra and presents formulations of known electrical engineering and signal processing concepts to illustrate some inherent advantages of geometric algebra for formulating and solving problems involving vectors. Expand
This book provides a synopsis of spacetime calculus with applications to classical electrodynamics, quantum theory and gravitation. The calculus is a coordinate-free mathematical language enabling aExpand


Simplicial calculus with Geometric Algebra
We construct geometric calculus on an oriented k-surface embedded in Euclidean space by utilizing the notion of an oriented k-surface as the limit set of a sequence of k-chains. This method providesExpand
Hamiltonian Mechanics with Geometric Calculus
Hamiltonian mechanics is given an invariant formulation in terms of Geometric Calculus, a general differential and integral calculus with the structure of Clifford algebra. Advantages overExpand
The claim that Clifiord algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry. ClifiordExpand
A Unified Language for Mathematics and Physics
Clifford Algebra provides the key to a unified Geometric Calculus for expressing, developing, integrating and applying the large body of geometrical ideas running through mathematics and physics. Expand
Lie-groups as Spin groups.
It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is availableExpand
Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5.Expand
Proper particle mechanics
Spacetime algebra is employed to formulate classical relativistic mechanics without coordinates. Observers are treated on the same footing as other physical systems. The kinematics of a rigid bodyExpand
Clifford algebras and their applications in mathematical physics
General Surveys.- A Unified Language for Mathematics and Physics.- Clifford Algebras and Spinors.- Classification of Clifford Algebras.- Pseudo-Euclidean Hurwitz Pairs and Generalized FueterExpand
Space-time algebra
Preface to the Second Edition.- Introduction.- Part I:Geometric Algebra.- 1.Intrepretation of Clifford Algebra.- 2.Definition of Clifford Algebra.- 3.Inner and Outer Products.- 4.Structure ofExpand
The discovery of Mathematical Viruses is announced here for the first time. Such viruses are a serious threat to the general mental health of the mathematical community. Several viruses inimical toExpand