# Differential Forms in Geometric Calculus

@inproceedings{Hestenes1993DifferentialFI, title={Differential Forms in Geometric Calculus}, author={David Hestenes}, year={1993} }

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

#### 34 Citations

The Shape of Differential Geometry in Geometric Calculus

- 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 a… Expand

Spacetime Geometry with Geometric Calculus

- Mathematics
- 2020

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 of… Expand

The Fundamental Theorem of Geometric Calculus via a Generalized

- Mathematics
- 2005

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. We… Expand

SPACETIME CALCULUS for GRAVITATION THEORY

- 1998

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 the… Expand

The fundamental theorem of geometric calculus via a generalized riemann integral

- Mathematics, Physics
- 1998

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. We… Expand

Gauge Theory Gravity with Geometric Calculus

- Mathematics
- 2005

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 principles… Expand

The Intersection of Rays with Algebraic Surfaces

- Mathematics
- 2014

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 defined… Expand

A multivector data structure for differential forms and equations

- Mathematics
- 2000

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, every… Expand

Geometry of complex data

- 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

SPACETIME CALCULUS

- 1997

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 a… Expand

#### References

SHOWING 1-10 OF 20 REFERENCES

Simplicial calculus with Geometric Algebra

- Mathematics
- 1992

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 provides… Expand

Hamiltonian Mechanics with Geometric Calculus

- Mathematics
- 1993

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 over… Expand

UNIVERSAL GEOMETRIC ALGEBRA

- Mathematics
- 1988

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. Clifiord… Expand

A Unified Language for Mathematics and Physics

- Computer Science
- 1986

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.

- Mathematics
- 1993

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 available… Expand

Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics

- Mathematics
- 1984

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

- Physics
- 1974

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 body… Expand

Clifford algebras and their applications in mathematical physics

- Mathematics
- 1986

General Surveys.- A Unified Language for Mathematics and Physics.- Clifford Algebras and Spinors.- Classification of Clifford Algebras.- Pseudo-Euclidean Hurwitz Pairs and Generalized Fueter… Expand

Space-time algebra

- Mathematics
- 1966

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 of… Expand

MATHEMATICAL VIRUSES

- 1991

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 to… Expand