I’ve posted a refreshed version of my old fluid mechanics course notes (aka Continuum mechanics). Also included are instructions to clone the git repositories, and make the pdf from the latex sources (which would allow customization if desired).
Updated statistical mechanics notes.
February 13, 2019 math and physics play
I’ve posted a minor update to my old stat mech notes, plus instructions on how to clone the github repos and the latex, should somebody wish to attempt to fork these notes for their own purposes. Enjoy!
Mathematica notebooks updated, and a bivector addition visualization.
February 10, 2019 math and physics play bivector, Manipulate, Mathematica, Wolfram
This blog now has a copy of all my Mathematica notebooks (as of Feb 10, 2019), complete with a chronological index. I hadn’t updated that index since 2014, and it was quite stale.
I’ve also added an additional level of per-directory indexing. For example, you can now look at just the notebooks for my book, Geometric Algebra for Electrical Engineers. That was possible before, but you would have had to clone the entire git repository to be able to do so easily.
This update includes a new notebook written today, which has a Manipulate visualization of 3D bivector addition that is kind of fun.
Bivector addition, at least in 3D, can be done graphically almost like vector addition. Instead of trying to add the planes (which can be done, as in the neat illustration in Geometric Algebra for Computer Science), you can do the task more simply by connecting the normals head to tail, where each of the normals are scaled by the area of the bivector (i.e. it’s absolute magnitude). The resulting bivector has an area equal to the length of that sum of normals, and a “direction” perpendicular to that resulting normal. This fun little Manipulate lets you interactively visualize this process, by changing the radius of a set of summed bivectors, each oriented in a different direction, and observing the effects of doing so.
Of course, you can interpret this visualization as nothing more than a representation of addition of cross products, if you were to interpret the vector representing a cross product as an oriented area with a normal equal to that cross product (where the normal’s magnitude equals the area, as in this bivector addition visualization.) This works out nicely because of the duality relationship between the cross and wedge product, and the duality relationship between 3D bivectors and their normals.
Small update to old notes for phy450, Relativistic Electrodynamics
February 9, 2019 Uncategorized
I’ve updated the pdf for my old phy450 notes (Relativistic Electrodynamics) from the current latex sources. Also included on that page are a contents listing, and instructions for forking the git repos. That should allow for building the pdf from the latex, so if somebody had changes they’d like to make, either for themselves or as feedback, they should be able to do so.
Why to study electromagnetism with geometric algebra.
February 3, 2019 Geometric Algebra for Electrical Engineers
The current draft of my book really ought to have some motivation in the preface. This is what I was thinking of.
Why you want to read this book.
When you first learned vector algebra you learned how to add and subtract vectors, and probably asked your instructor if it was possible to multiply vectors. Had you done so, you would have been told either “No”, or a qualified “No, but we can do multiplication like operations, the dot and cross products.” This book is based on a different answer, “Yes.” A set of rules that define a coherent multiplication operation are provided.
Were you ever bothered by the fact that the cross product was only defined in three dimensions, or had a nagging intuition that the dot and cross products were related somehow? The dot product and cross product seem to be complimentary, with the dot product encoding a projection operation (how much of a vector lies in the direction of another), and the magnitude of the cross product providing a rejection operation (how much of a vector is perpendicular to the direction of another). These projection and rejection operations should be perfectly well defined in 2, 4, or N dimemsions, not just 3. In this book you will see how to generalize the cross product to N dimensions, and how this more general product (the wedge product) is useful even in the two and three dimensional problems that are of interest for physical problems (like electromagnetism.) You will also see how the dot, cross (and wedge) products are all related to the vector multiplication operation of geometric algebra.
When you studied vector calculus, did the collection of Stokes’s, Green’s and Divergence operations available seem too random, like there ought to be a higher level structure that described all these similar operations? It turns out that such structure is available in the both the language of differential forms, and that of tensor calculus. We’d like a toolbox that doesn’t require expressing vectors as differentials, or resorting to coordinates. Not only does geometric calculus provides such a toolbox, it also provides the tools required to operate on functions of vector products, which has profound applications to electromagnetism.
Were you offended by the crazy mix of signs, dots and cross products in Maxwell’s equations? The geometric algebra form of Maxwells’s equation resolves that crazy mix, expressing Maxwell’s equations as a single equation. The formalism of tensor algebra and differential forms also provide simpler ways of expressing Maxwell’s equations, but are arguably harder to relate to the vector algebra formalism so familiar to electric engineers and physics practitioners. In this book, you will see how to work with the geometric algebra form of Maxwell’s equation, and how to relate these new techniques to familiar methods.