Geometric Algebra

Did you know that the inner product space Rn can be embedded in a vector space of dimension 2n which is also an associative algebra with unit, the geometric algebra?

Some members of the geometric algebra represent geometric objects in Rn. Other members represent geometric operations on the geometric objects.

Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas, including linear algebra, multivariable calculus, real analysis, complex analysis, and euclidean, noneuclidean, and projective geometry. They provide a unified mathematical language for physics (classical and quantum mechanics, electrodynamics, relativity), the geometrical aspects of computer science (e.g., graphics, robotics, computer vision), and engineering.

Textbook: Linear and Geometric Algebra

This textbook for the first undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering a majority of the usual linear algebra topics. The link is to the book's web page.


I have created a six video YouTube playlist Geometric Algebra, about 72 minutes in all, taken from the book. Some knowledge of linear algebra is a prerequisite for the videos, but no knowledge of geometric algebra is assumed. The book assumes no previous knowledge of linear algebra.

Textbook: Vector and Geometric Calculus

This textbook for the first undergraduate vector calculus course presents a unified treatment of vector calculus and geometric calculus, while covering a majority of the usual vector calculus topics. The link is to the book's web page.


I have created a five video YouTube playlist Geometric Calculus, about 53 minutes in all, taken from the book. It is a sequel to my Geometric Algebra playlist. Some knowledge of vector calculus is a prerequisite for the videos, but no knowledge of geometric calculus is assumed. The book assumes no knowledge of vector calculus.

A version of the videos was presented at Applied Geometric Algebra in Computer Science and Engineering, Barcelona, Spain, July 2015.


Professor Ahmad Eid, publisher of Geometric Algebra Explorer, asked me

for my thoughts about geometric algebra and its place in undergraduate education.

A Survey of Geometric Algebra and Geometric Calculus

Adv. Appl. Cliff. Alg. 27, 853–891 (2017). Invited paper for the proceedings of the conference Applied Geometric Algebra in Computer Science and Engineering, Barcelona, Spain, July 29-31, 2015.

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

  1. Published version improved.
  2. Fixed discussion of Pauli equation.
  3. Several changes in Geometric Calculus section.
  4. Fixed 1.5.4h.

An Elementary Construction of the Geometric Algebra

Adv. Appl. Cliff. Alg. 12, 1-6 (). (Somewhat improved.)

Presented at The Fifth International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa, Mexico, -

Abstract: We give a simple, elementary, direct, and motivated construction of the geometric algebra over Rn.

The Fundamental Theorem of Geometric Calculus via a Generalized Riemann Integral

Adv. Appl. Cliff. Alg. 8, 5-16 ().

Presented at Octonions and Clifford Algebras, Corvallis, -

Abstract: Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the vector derivative exists and is Lebesgue integrable. We also give sufficient conditions that the vector derivative exists.

Before tackling this paper you might want to read my paper Stokes' Theorem. See below.

Potentials, Fields, and Sources with Geometric Calculus

Not published.

Abstract: Geometric calculus offers significant advantages over other formalisms in the treatment of potentials, fields, and sources. We show this for 3D Euclidean space and 4D Minkowski space. As a corollary we show that charge conservation implies the existence of a field satisfying Maxwell's equations.


General Relativity in a Nutshell

Not published.

A 100 page book on general relativity (75 pages in the main text).

  1. Small changes.
  2. Small updates with new observations.
  3. Small updates with new observations.
  4. Used recently released cosmological parameters from the Planck spacecraft.
  5. Added description of binary white dwarf system J0651.
  6. Version 3.5. Added new evidence for dark matter from the bullet cluster and new results from the double pulsar.

From the preface:

"My purpose here is to provide, with a minimum of mathematical machinery and in the fewest possible pages, a clear and careful explanation of the physical principles and applications of classical general relativity. The prerequisites are single variable calculus, a few basic facts about partial derivatives and line integrals, a little matrix algebra, and some basic physics. Only a bit of the algebra of tensors is used; it is developed in about a page of the text. The book is for those seeking a conceptual understanding of the theory, not computational prowess. Despite it's brevity and modest prerequisites, it is a serious introduction to the physics and mathematics of general relativity which demands careful study. The book can stand alone as an introduction to general relativity or it can be used as an adjunct to standard texts."

Universal one-way light speed from a universal light speed over closed paths

Found. Phys. Lett. 16, 593-604 (). With Ettore Minguzzi.

Abstract: This paper gives two complete and elementary proofs that if the speed of light over closed paths has a universal value c, then it is possible to synchronize clocks in such a way that the one-way speed of light is c. The first proof is an elementary version of a recent proof. The second provides high precision experimental evidence that it is possible to synchronize clocks in such a way that the one-way speed of light has a universal value. We also discuss an old incomplete proof by Weyl which is important from an historical perspective.

Einstein's Hole Argument

Much improved version of Am. J. Phys. 69, 223-225 ().

Abstract: In general relativity, spacetime is inseparable from a gravitational field: no field, no spacetime. This is a lesson of Einstein's hole argument. We use a simple transformation in a Schwartzschild spacetime to illustrate this.

  1. Much improved from the published version.

On the Marzke-Wheeler and Desloge Constructions

Found. Phys. Lett. 3, 493 ().

Abstract: There is no indication of time dilation of clocks or of length contraction of rods in Marzke and Wheeler's clock or in Desloge's metrosphere.

Clock synchronization, a universal light speed, and the terrestrial redshift experiment

Am. J. Phys. 51, 795-797 (1983).
Abstract: This paper (i) gives necessary and sufficient conditions that clocks in an inertial lattice can be synchronized, (ii) shows that these conditions do not imply a universal light speed, and (iii) shows that the terrestrial redshift experiment provides evidence that clocks in a small inertial lattice in a gravitational field can be synchronized.

Special and General Relativity Based on the Physical Meaning of the Spacetime Interval

Not published.

Abstract: We outline a simple development of special and general relativity based on the physical meaning of the spacetime interval. The Lorentz transformation is not used. The approach is suitable for beginning students.

John Wheeler wrote to me (): "It makes sense to me. After all one is dealing with a local problem, that means a local inertial Lorentz frame, that means just what you say: 'The metric postulate of general relativity rests on (1)-(3), and not the strong equivalence principle.'"

Quantum Theory

Entanglement, joint measurement, and state reduction

Slightly improved version of Int. J. Theor. Phys. 42, 943-953 ().

Presented at Quantum Composite Systems: theory, experiment and applications, Ustron, Poland, -

Abstract: Entanglement has been called the most important new feature of the quantum world. It is expressed in the quantum formalism by the joint measurement formula. We prove the formula for projection valued observables from a plausible assumption, which for spacelike separated measurements is an expression of relativistic causality. The state reduction formula is simply a way to express the joint measurement formula after one measurement has been made, and its result known.

  1. New footnote citing a result of Gisin.
  2. New footnote clarifying the term "nonlocal".

Quantum theory without measurement or state reduction problems

Much improved version of invited paper presented at Quantum Theory without Collapse, Rome, -

Abstract: There is a consistent and simple interpretation of the quantum theory of isolated systems. The interpretation suffers no measurement problem and provides a quantum explanation of state reduction, which is usually postulated. Quantum entanglement plays an essential role in the construction of the interpretation.

  1. Improved readability. No new content.

Thermal Physics

The Form of Magnetic Work in a Fundamental Thermodynamic Equation for a Paramagnet

Am. J. Phys. 67, 613-615 (). With Martin Barrett.

Abstract: Magnetic work takes two forms in the thermodynamics of a paramagnet as developed in many textbooks. We observe that in the case when the lattice energy is excluded, the form dW = BdM cannot be used in a fundamental thermodynamic equation. This shows that there are thermodynamic systems with no fundamental thermodynamic equation.

A New Statement of the Second Law of Thermodynamics

Am. J. Phys. 63, 1122-1127 ().

Abstract: A new statement of the second law of thermodynamics is given. The law leads almost effortlessly, for very general closed systems, to a definition of absolute entropy S, a demonstration that ΔS ≥   0 in adiabatic processes, a definition of temperature, and a demonstration that dS ≥ δQ/T  along quasistatic processes. Entropy is given a clear physical meaning.

  1. Fixed a gap in the discussion of temperature.


Stokes' Theorem

An abridged version appeared in Real Analysis Exchange 27, 739-747 ().

Abstract: We give a simple proof of Stokes' theorem on a manifold assuming only that the exterior derivative is Lebesgue integrable. The proof uses the integral definition of the exterior derivative and a generalized Riemann integral.

Old Mathematics Papers


The Price Equation

Not published.

I give concise derivations of Price's equation and the criteria for kin and group selection, prove that kin and group selection are equivalent, and discuss the controversies about altruism.