I just started teaching two courses on geometric algebra at the Physics Department of Ateneo de Manila University. The first is Ps 121: Vector Calculus and Complex Numbers. The second is Ps 195: Vector Calculus in Manifolds for Physicists.
Ps 121 is for 12 sophomore physics students. It is a good thing that vectors and complex numbers will be taught in the same course. So I told my students that I will simultaneously teach them vector and complex analysis, because imaginary numbers are eithe bivectors or trivectors. The first half of the course is on two-dimensions; the second half is on three-dimensions.
I began the course with . We squared the vector and derived the relations and . We derived the properties of the unit bivector . We showed that is an imaginary that anticommutes with vectors and also a rotation operator.
We factored out in a vector to transform it into a complex number form: . This makes explicit the relationship of vectors and complex numbers. The square of a vector then is equal to the product of a complex number and its conjugate: . For the product of two vectors, we showed that . I remarked that is an oriented area in Grassmann algebra. I promised that in the next meeting we shall discuss reflections and rotations.
Ps 195 should have been a special topics course. But since the teacher who designed it described it as Vector Calculus on Manifolds for Physicists, I adopted the name. This is a tutorial course for three senior students and two sit-ins. I cannot teach the couse using the framework of differential forms and tensor analysis, because I do not understand them much. It is difficult to teach an old dog new tricks. But a more apt metaphor would be the parable of the finding of a treasure in a field: you sell everything you have and buy the field. Or the parable of the pearl of great price: the man who finds it will sell everything he has and buys the pearl. In the same way, when one knows geometric algebra, you cannot help but give up all the other algebras that you know, or at least reinterpret them in the light of the new formalism.
I told my students that the study of manifolds and differential geometry is essentially the study of maps. There are several transformations in map making: translation, reflection, rotation, dilation, projection, and wrapping. What we want to know is if there are geometric properties in the original object that remains preserved by the mapping and how do other properties change. Some of the properties are angles, lengths, areas, and volumes. To analyze these properties, we shall use the methods of differential geometry and poisson brackets.
I followed the same introductory lecture as Ps 121. But since these students were under me before in their Ps 41 when they were still freshmen, they already know the Pauli identity for geometric products and the generalized vector rotation expression via half-angle exponentials. I made an introductory lecture in the complex vector algebra of the plane, so that I would not alienate my two sit-ins. They can follow the lecture and respond to my questions. This is a good sign. Normally, it is difficult to teach geometric algebra to older students and faculty because they already know too many mathematics. I really prefer to teach geometric algebra to freshmen and sophomore undergraduates because they have less mathematical baggage.
For this course I was able to reach reflections (actually only flips) before the hour ends. But the reflection is only about the x-axis: . I told them that traditional complex analysis can also do the same using complex conjugation. But there are reflections (flips) that complex analysis cannot do, such as flip about any arbitrary axis in the plane. I shall discuss the general reflection equation next meeting.