If you are interested in seeing what I have on this website on the solutions
to selected problems
from the textbook New Foundations to Classical Mechanics (NFCM) by
David Hestenes (the
textbook uses Geometric Algebra), click on the link
New Foundations for Classical Mechanics: Selected Problems and Answers.
On this page, I present various notes I have prepared on quantum mechanics,
that use geometric algebra as the main computational tool. These notes are
still a work in progress, but I think they are sufficiently robust to offer
the reader some help in understanding the material.
Quantum Mechanics and Geometric Algebra.
On this page, I present various proofs about projective and conformal geometry,
using geometric algebra as the main computational tool. I also present my personal
notes to various papers on projective and conformal geometry. click on the link
projective and conformal geometry.
If you are interested in seeing what I have on this website on selected problems/computations
from the textbook Clifford Algebra to Geometric Calculus (CAGC): A Unified Language for
Mathematics and Physics by David Hestenes and Garret Sobczyk (the textbook uses
Geometric Algebra), click on the link
Clifford Algebra to Geometric Calculus: Selected Computations.
Vector calculus that uses Geometric Algebra/Calculus Common vector calculus identities, with proofs.
More vector calculus that uses (mostly) Geometric Algebra/Calculus More Vector Calculus Identities.
In this paper I adapt a problem solved by Michael Penn on his YouTube channel
concerning finding the intersection point of two tangents to a circle, using complex
numbers. I show how to convert the problem to a vector problem with solution
and then how to transform between complex numbers and vectors using geometric
algebra.
Transforming between
complex numbers and vectors using geometric algebra.
This link takes one to the famous Helmholtz Decomposition Theorem, which proves
that a vector field of suitable qualities can be computed if the divergence and curl of
the vector field are known. The main part of the proof does not use Geometric Calculus,
though some of the lemmas do.
The Helmholtz Decomposition Theorem.
Download PDF file Geometric algebra is used (in linear
algebra) to show that a
linear transformation from a vector space to itself has a two-sided inverse.
A couple significant results from the theory of nth-order linear DEQs
are produced here by the convenience of geometric algebra, such as Abel's Identity
and the Variation of Parameters. In the end we find some interesting geometry that
someone should figure out what its importance is.
nth-order linear Differential Equations
bAr = b . Ar +b /\ Ar .
However, this paper provides a proof to the similar identity
Arb = Ar . b + Ar /\ b.
Basic Products on the Right PDF file
A . B = (-1) s(r-s)B . A.
Reversion of the Inner Product PDF file
This paper is a redo of an article that first appeared in the Arizona Journal of Natural Philosophy, July, 1995. Some scientists seem to believe that the unreasonable effectiveness of Clifford algebras in physics is too good to be true. They resist using it or allowing themselves a chance to understand it because of psychological and/or philosophical prejudices. Some of them even admit all this. This short essay attempts to alleviate those doubts. Explaining the Mystery of Clifford Algebra PDF file
The following paper is a redo of an article that first appeared in the Arizona Journal of Natural Philosophy, July 1992. What's special about the solution here is that it uses geometric algebra.
Rotational Dynamics Problem Using Geometric Algebra PDF file
The following HTML file/PDF paper shows how relate the familiar area of a rectangle (ab sin θ) in geometric algebra in two different ways.
Area of a Parallelogram by Geometric Algebra
The following HTML file/PDF paper shows how to Determinant Form of the Area of a Triangle by Geometric Algebra. It may be helpful first to read the article above: "Area of a Parallelogram by Geometric Algebra".
Determinant Form of the Area of a Triangle by Geometric Algebra
The following HTML file/PDF paper shows how to prove that a triangle inscribed in a rectangle has half the area of the rectangle by using vectors and geometric algebra.
Area of Inscribed Triangle by Geometric Algebra
Old theorems about triangles proved.
Proofs about cones and planes.
Download PDF file Constructive proof that there exists a polynomial curve that focuses all incoming rays to a single point.