This is an archival listing of my notes from 2007 and earlier.
October 22, 2007 Radial components of vector derivitives.
This and the other October docs below are all based on very ad-hoc notes I dumped into the Wikipedia Geometric algebra page , which had very little actual content in it. As I tried to learn GA from the much too advanced treatments I found on the web, I put my notes in the wiki as I puzzled things out myself. Eventually I bought some books and if I had done so earlier, I wouldn’t have needed to figure out a lot of this on my own. It was however time well spent as I learned things more thoroughly trying to figure it out from the basic properties. I wanted to add details to the wiki pages that didn’t really fit with a wiki page, or were hard to do with wiki markup compared to latex, so eventually I stopped trying to make my notes in wiki and just typed them up directly here. I never did go back and see if there was anything extra that I added to these that would be worth re-adding to the wiki pages. What I have here is all based on my wiki page additions (and I was the only contributor at the time), and excludes the minimal bits that were there previously.
October 22, 2007 Rotational dynamics. Angular velocity.
Angular velocity as radial component of velocity. Kepler’s law from Salas/Hille reworked with GA. Acceleration, and circular motion special case expressed in GA form, noting how close that ends up being to the scalar equivalent, but vector product and inversion encodes the direction too.
October 13, 2007 Maxwell’s equations expressed with Geometric Algebra.
Taking the hint that Maxwell’s equations could be expressed naturally using GA. I didn’t get it as far as the GAFP book since I didn’t see how to also formulate the spacetime gradient in the STA basis. This is probably a treatment that can be understood without too much GA prep.
October 13, 2007 My wikipedia Geometric Algebra notes.
Basic identities, and comparisions to non GA formulations. Wedge vs. Cross. Norm using GA square compared equivalent dot product method. Lagrange identity in cross and wedge product form. Determinant bivector and cross product expansions. Cross and wedge product formula for planes. Projection and rejection. Parallelogram area, and parallopiped volume. Vector angle. Vector inversion. Symetric and antisymetric product representations of dot and wedge products. Reversion. Complex numbers. Rotation in arbitrary plane. Cross product as wedge dual.
October 16, 2007 Cramer’s rule using wedge products.
An example showing how Cramer’s rule is just a special case of linear equation solution using the wedge product. Hestenes uses examples like this in his book. Without using GA, the grassman algebra book also has a worked example of this which is nice. I find the GA approach is easier to understand than the Grassman book approach. The grassman book avoids the dot product to use the metric free regressive product approach. I find the explicit use of the metric and associated dot product in GA more naturally when trying to bridge from traditional vector identities to a wedge product treatment.
October 13, 2007 Torque expressed with geometric algebra.
This contains scalar and bivector formulations of torque based on rate of change of work with respect to angle.
October 16, 2007 Derivatives of a unit vector.
This is a derivation of the unit (radially expressed) vector time derivative in terms of the GA wedge product.
May 10, 2007 The cross product in three and more dimensions
Revisiting my cross product treatment, but after discovering differential forms and the wedge product (Harley Flanders. “Differential forms and Applications to the Physical Sciences”). This was curtosy of Tor who was smart enough to search for it.
I had trouble with relating the wedge product on differential quantities to vectors. I could see this was related to my old attempt to generalize the vector dot product. I went back to my old notes and tried to lead from those ideas to the wedge product in a natural way.
This doc contains a few interesting bits.
A coordinate description of vector rejection (difference from projection) in terms of determinants is developed. This provides a way to calculate the normal component of a vector with respect to a spanning set.
This is used to calculate the area and volume of an N dimensional hyper-parallelogram, or hyper-parallopiped.
Also included is a calculation of a general normal to a set of vectors based on Null space calculation. Of particular interest is the description of the case when such a normal cannot necessarily be unambiguously defined.
The ideas here are then used to try to intuitively introduce the wedge product and a dot product on pairs of wedge products, without also introducing differentials. The wedge product and the normal and rejection concepts are all related by the determinant (in fact the determinant really ought to be viewed as generated by the wedge product).
I end with an poor attempt to introduce differential forms for geometric area and volume elements. When attempting to find a treatment of differnential forms that I could understand I ended up discovering Geometric Algebra. That subject has exactly what I was looking for as the “answer” for all this generalized vector algebra. Due to the time required to study that algebraic toolbox I have not yet gotten back to the differential forms that started me down that path. Soon I hope.
March 25, 2000 Various formulations of Maxwell’s equations.
Integral and differential forms of Maxwell’s equations in differential, and integral form, as well as explicit normal and tangential form (as in my Dad’s old 1960s Encyclopedia Britanica). Derivation of the wave equation in free space and in presence of matter using the normal triple cross product vector relation identity method. Followup with a four vector complex number treatment that puts the equations in a much more symmetrical form, that highlights the complex number factor that is naturally associated with the magnetic field.
I later followed up on this idea of complex representation in more detail in an email to Tor (at which point I got excited about it and wondered why this wasn’t in all the E&M books since it was so simple and natural and appropriate seeming). I should dig up and write that up since it would be a nice bridge to the GA ideas.
January 17, 2000. Triangulating distance to a star from orbital angle measurements. Scan a picture and add to this old note from Feynman's lectures so I can toss the old beat up paper sitting in the back of the book.
Circa ~1999 Old cross product generalization musings.
The three dimensionality (and even non-2 dimensionality) of the cross product always bugged me. I never saw any hint (or if I did I didn’t recognize it) about existing generalizations or degeneralizations of the cross product, even with four years of Engineering classes. So, one day I felt like “mathing” as my wife and kids call now it, and played with generalizing it.
Starting point was looking at torque in a plane (aka: Feynman Lectures, Vol I), and considering the work done by an incremental rotation, then doing the same for three dimensions. You end up with the cross product as an operator, and it takes the form of a completely antisymmetric matrix. I now know that this naturally expresses the antisymmetry of the wedge product (ie: cross product as the dual of a bivector).
In these notes I attempted to take this cross product operator form and extract some of the structure. In particular I factored this antisymmetric matrix into the product of diagonal and permutation product matrixes, and tried to use that as a way to define a generalized cross product. I was able to create a 4D cross product generalization in matrix form that had at least some of the 3D cross product properties (and properties of the 2D de-generalization of the cross product). However, the core problem here is that one doesn’t know how to define the normal that is intrinsic to the cross product, so the whole approach is kind of busted.
After starting with generalizing the cross product using incremental rotation as a starting point, I went back to the other starting point in the cross product definition, where the cross product is defined in terms of Euclidean normal vector properties (where one such orientation of the general normal between two vectors is picked).
There is a final attempt to generalize this further to 5D, but I don’t think I did a particularily good job at it. The most notable issue is that the 4D version wasn’t really well defined, and worse, I invented a 4D cross product generalization that I didn’t know any applications for.
Perhaps even worse than inventing a cross product generalization for was that I spent a bunch of time working out stuff that others had already figured out much more completely than I, and I didn’t go looking to see what others had done first to ensure I wasn’t (badly) reinventing the wheel. My friend Tor Aamodt (Prof at UBC), says this is one of the biggest reasons that grad students have thesis advisors. If I had stayed in school and done this in an academic context I would have had the good fortune to have somebody help point out that I was wasting time.
The final bits in this (long) set of notes was as far as I went with these ideas for quite a while and ended up dropping math as a hobby again for a number of years.