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Geometric Calculus (Exponential Calculus)

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Non-Newtonian Calculus >

Geometric calculus (also called exponential calculus) is an extension of geometric algebra to include differentiation and integration [1].

Basic Problem of Geometric Calculus and Fundamental Concepts

The basic problem of geometric calculus is as follows:

“Suppose that the value of a positive function h is known at an argument r, and suppose that f, the geometric derivative of h, is continuous and known at each number in [r, s]. Find h(s).” [2]

The solution is:
geometric calculus

Where the number on the left
geometric integral

is the geometric integral.

The fundamental concepts of Geometric Calculus are [3]:

  • Coordinate-free differential geometry: Develops concepts on any form or manifold without reference to a particular coordinate system. Although they exist independently of any choice of basis.
  • Differentials and codifferentials (which go in the opposite direction from differentials) for mappings and fields.
  • Directed integrals (a generalization of the standard Riemannintegral) and differential forms.
  • Lie groups (groups that are also differentiable manifolds) as Spin groups. A differentiable manifold is a set on which derivatives and integrals can be calculated.
  • Linear and multilinear algebra (tensors, determinants).
  • Universal Geometric Algebra – arbitrary dimension and signature.
  • Vector derivative and the fundamental theorem of calculus.
  • Vector manifolds (for representing any manifold).

Uses of Geometric Calculus

Geometric calculus is challenging to learn and implement; Using it requires some basic knowledge of geometric algebra, an extension of vector algebra. Therefore, you’ll unlikely see it for any general applications. However, the field has brought about several innovations, including Chris Doran and colleagues improvement on general relativity called Gauge Theory Gravity [4, 5].


[1] Geometric Calculus. Retrieved July 27, 2021 from:
[2] Grossman, M. & Katz, R. Non-Newtonian Calculus
A Self-contained, Elementary Exposition of the Authors’ Investigations…. Lee Press.
[3] D. Hestenes. Tutorial on Geometric Calculus. Retrieved July 27, 2021 from:
[4] C. Doran and A. Lasenby. Geometric Algebra for Physicists. Cambridge: The University Press, 2003.
[5] D. Hestenes, Gauge Theory Gravity with Geometric Calculus, Foundations of Physics 36,
903-970 (2005).

Stephanie Glen. "Geometric Calculus (Exponential Calculus)" From Calculus for the rest of us!

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