This is an archival listing of my notes from 2008, a full year of math and physics learning! I learned a lot of electrodynamics, some special relativity, a lot of Geometric Algebra, some Lagrangian theory, and a touch of quantum mechanics.
December 30, 2008. Some Poynting vector notes. Doran/Lasenby once again too cryptic for me. Do the calculation of the electromagnetic energy density flux through surface for the charge and current free case in enough detail that it makes sense to me. Followup with a complete energy conservation treatment for the field including currents, and an expansion of the energy momentum tensor for the gamma_0 parameter (a four vector containing both energy density and Poynting vector "momentum density" components.). Also want to figure out the attempt to get the field energy from the Hamiltonian field congugate momentum. Tried that here and didn't succeed.
December 27, 2008. Some E&M notes from reading of Rayleigh-Jeans law discussion in Bohm's Quantum Theory. Notes on gauge transformations, wave equations for the vector potential and Poisson solution of Laplacian.
December 26, 2008. Walk through the details of the relativistic velocity addition from Pauli's book. Did the entire messy calculation of the observed velocity as well as confirming the x component from Pauli's book. Would like to later compare this to the STA form. Is the simplicity of that wedge/dot quotient really the equivalent of this mess?
December 23, 2008. Some very rough notes on Susskind's QM lectures from itunes U. Bra and Ket notations, projections, dual space, Hermitian operators, position and momentum eigenfunctions and states, QM postulates.
December 21, 2008. An attempt to decode the Dirac equation Lagrangians found in wikipedia. Calculate the field equations from the Lagrangians once all the terms were understood. Includes a translation between the matrix and Doran/Lasenby notations for dagger and Dirac adjoint.
December 19, 2008. Start of some notes trying to develop some intuition on Green's functions with eventual goal of tackling retarded time potentials. Paper notes only partially written up.
December 18, 2008. A few small notes on rapidity angles.
December 15, 2008. Calculate the integral for the time for two particles to converge in space from an initial position. Just for fun (recall trying this in school and making a mess of it). Updated Jan/4. This approach was totally busted as numeric evaluation proved. Have updated this to say so, but haven't yet revisited this to do it right.
December 14, 2008. A small program using boost::numeric::ublas to print out the Dirac gamma matrix multiplication table. A perl script used to generate part of the C++ program which generates part of the latex output, which generates the pdf output;) Once given the generated output in its completeness, a few further observations were made. Was looking for the matrix operations that correspond to grade selection. Hermitian operations don't do the job like they do with the Pauli matrices, instead providing a split into relative spacetime vectors and the spacelike vectors.
December 13, 2008. Split out the QM equation motivation from the wave Lagrangian notes of Dec/2 below. That previously included an ad-hoc way of arriving at the Dirac QM equation by
taking (vector) roots of the (scalar) Klien-Gordon equation. This is
just an algebraic maniupulation, and I'm sure Dirac did it with a lot
more physical understanding, but it is still cool. Also added for completeness is the typical non-relativisitic motivation for the wave equation.
December 12, 2008. Derivation and notes on the Bohr model. Do the derivation myself to understand some of the ideas involved. Eventually would like to revisit this and see how one would incorporate not just the Coulomb subset of Maxwell's equations (as well as the rest of the Lorentz force) in an argument of this nature.
December 7, 2008. Figuring out how Pauli algebra works. Find the equivalents of all the clifford algebra constructs in the Pauli algebra, and figure out how to work with the the commutator and anticommutator notation that is a de-facto standard in physics. Having done this I expect that some of the math in the QM texts will no longer be so scary. Also have to learn the "pedestrian" lingo as Avodyne puts it in the following interesting thread http://www.physicsforums.com/showthread.php?t=277054 , on the Dirac equation. Still have to think through how to work with the Dirac matrix algebra, but now that I've got a handle on the (anti)commutator and Pauli algebra this shouldn't be too hard.
December 2, 2008. Comparing the Lagrangian and field equations for vibration, electromagnetism, and quantum mechanics. Collect up some assorted bits and pieces to see similarities.
December 1, 2008. Expressing wave equation exponential solutions in four vector form. Take std wave equation for wave in 3D with specified velocity, and put exponential solution in a tidy covariant form, expressing the wave-number/velocity parameter in terms the proper velocity for the position four vector. Don't know yet if this is good for anything but the tidyness is nice.
November 30, 2008. Calculating an interpolating rotor.
November 27, 2008. Start of some notes on Hestenes' New Foundations for Classical Mechanics.
November 25, 2008. Examining equivalance of a GA polynomial function with matrix form. Turns out the function considered has a null determant (at least for R^2). Also has some thoughts on how to (if inclined) persue the idea of how would an arbitrary matrix be represented as a GA polynomial function.
November 24, 2008 Problem 1.5 from Purcell (field due to semicirclular charged arc). Generalized to arbitrary subtended angle, and point not necessarily at the origin. GA exponentials used to express the rotations involved. The integral for the most general case wasn't fully evaluated since I didn't figure out how to do so (is a numerical solution required?)
November 23, 2008 Electric field due to a non-infinite length wire. Do the infinite length calculation in vector form instead of coordinates, then calculate the variation for finite length.
November 22, 2008 Flux equality through spherical surface and tilted surface.
November 22, 2008 Some notes on spherical polar rotations and gradient
I'd intended to do a more complete treatment for myself than what is in these notes, including:
- analytic derivation of the gradient.
- divergence curl, and Laplacian in spherical polar coordinates.
I did get the transpose issues with transformation of basis vs. transformation of coordinates figured out (was confusing me for a bit). Revisit this later for the remaining bits.
November 8, 2008
Wrote gasym.cpp, a C++ poor man's mathematica.
This is source code for a simple multivector symbolic calculator.
package since I have used
these classes as a backend for the multivector parts. The symbolic
calculations were done the quick and dirty lazy way using a stl::map and
expression tree. If you are interested in trying
(or extending this) for your own calculation please let me know that you are doing so (I'd like to keep the core bits single sourced, with enhancements and bugfixes in one place).
As coded this "calculator" is applied towards the
expansion of a quaternion (or scalar plus bivector multivector in the GA lingo) representation of Euler angle exponentials to calculate the rotation matrix in terms of the half angles, and to also compute the rotation matrix in terms of
November 2, 2008 Some notes on Euler angles. Compare the rotor form to the matrix form. Tried the derivation in matrix form too and found almost the same result. Resolution of the mistake and discussion ends up being somewhat interesting by itself.
Update: Nov 10. After a digression to write some symbolic GA calculator software, the Cayley-Klein parameterization of the rotation matrix has been revisited. Found the parameterization associated with problem 2.9 in GAFP. Identifying the last differences from the book problem was done with a scripted search and replace to generate the 120 different matrixes that are possible with permutation of the parameters, and toggling of their signs (bivector terms also have a degree of freedom in the sign). This was probably a silly exercise to take to absolute conclusion since the root idea ... that various half angle sine and cosine parameterizations can be used to generate that form of rotation matrix ... is what was important. Very OCD of me to actually solve this beastie!
October 30, 2008 Derive (non-multivector) form of Noether's equations for a field Lagragian. Example calculations include the conserved current calculations for the Schroedinger phase invariance with the relativistic Lagragian, and the conservation quantity for a boosted/rotated Maxwell field Lagrangian.
October 26, 2008
Some rough notes (mostly questions) about GravitoElectroMagnetism.
Revisit after/when learning some GR.
October 22, 2008 Application of Noether's to Lorentz transformed interaction Lagrangian Boost and or rotation leaves the Lorentz force Lagrangian unchanged implying a conserved quantity. The conserved quantity is computed (a set of wedge product results relating four vectors A, x, v). This is also expressed in purely vector form for a specific observer frame, producing a pair of vector equations, one of which supplies the torque on the charged particle supplied by the field. Finally the result is expressed in tensor and (four-vector) coordinate form, which like the wedge product representation is also nicely symmetrical.
October 19, 2008 Demonstrate Lorentz invariance of Maxwell field and force Lagrangians. Invariance result using clifford algebra. This is followed up with the exercise of seeing how to express the Lorentz transform of a vector in tensor form, as well as a derivation of the transformational rules for the transformation of a bivector expressed in tensor form. This is used to derive the same result as done in multivector form.
October 13, 2008 Derive Euler Lagrange equations using Feynmans variational procedure. Step one towards getting a feel for the multivector Lagrangian treatment: understand the scalar case. Have considered the following in addition to the Euler-Lagrange derivation itself
October 12, 2008 Derive the covariant Lorentz force equation entirely in tensor form. Done as an exersize in tensor manipulation (previously did the same using STA algebra).
October 11, 2008 Derive the field form of the Euler Lagrange equations. To confirm the derivation this is used to calculate Maxwell's equation from its Lagrangian density, and also Schroedinger's equation from the Lagrangian for it Goldstein. Updated Oct/29. Added relativistic Schroedinger's calculation to examples.
October 9, 2008 Revisit Lagrangian derivation of Lorentz force equation. Take more care with the metric signature this time. Also relates this to the traditional observer dependent E + v x B form of the equations. For the first time the use of the wedge with the time basis vector as a means to select rest frame observer dependent quantities makes sense to me. Perhaps just because this had nothing to do with the Lorentz velocity addition where I had trouble seeing how the wedge and the traditional means were related.
October 5, 2008 Euler-Lagrange calculation for the Schwartzchild metric. A set of notes mostly seeing how generalized SR metrics lead to the metric forms of GR, plus a small calculation at the end using the Schwartzchild metric.
September 28, 2008 Revisit Stokes equation derivation in full generality. Algebraically describe the boundary surface instead of pictorially. The pictorial description of the surface and loop orientation obviously doesn't work well for 4D and greater volumes. Having formulated this boundary description, this leads naturally to a proof of the general case (the specific cases of Stokes theorem for the vector and bivector cases were proved below.) There are two bits of handwaving in the proof, but relatively minor ones. Enough that my Engineering bias considers the proof complete enough for practical purposes.
September 27, 2008 Apply bivector form of Stokes equation to Maxwell's equation. Updated to include all of the following
September 20, 2008 Prove the clifford algebra formulation of some Stokes/Green's theorem equations.
Like so many things it seems I can't find treatments that I can understand easily. In particular I've found it hard to understand in an intuitive fashion the generalized Stokes law or Green's theorem treatment in both GA and differential forms texts.
So I've calculated this myself.
Covered in this set of notes is:
Everything treated here is done in a fashion that applies to any metric, as well as any number of dimensions.
Left untackled are the four-volume and even higher dimensional varients of Stokes' law. This is probably enough of the general multivector integral transformation equations that I can go back and comfortably tackle the integral equivalents of the electrodynamics field equations.
Also not done are specific examples of volumes and surfaces to get a feel for this. Try the hypersphere volume and area calculations from the Flanders that were obvious to him but not me. I also suspect that working some examples of this will allow me to figure out how to represent the boundary surfaces in the general case (suspect that this can be expressed as an alternating form, much like the reciprocal basis calculation ... that intuitively makes sense since we have normal elements constrained to the subspace spanned by the volume).
September 13, 2008 Lagrangian field density used to derive Maxwells equation.
I couldn't find a derivation that used the electrodynamic field density to derive Maxwell's equations that I could understand, but puzzled it out myself, using Feynmans ad-hoc method.
Interestingly, which differs from other writeups that I have found, I had to use both the E dot E - B dot B part (to generate the vector half of the equation), and used the E dot B part to generate the trivector part of Maxwells equation.
The wikipedia article on the electrodynamic tensor has a rather unexplained description of a Maxwells equation derivation with some sort of field varient of the Euler Lagrange equations where they apparently do this derivation using only the scalar part of the squared field.
Regardless of whether the E dot B term can be implied in some way, including it results in a very neat result. We can form a complex valued Lagrangian density from the curl of the vector potential, and use that directly to find the STA formulation of Maxwells equation without having to reconstruct this from the tensor results. This may actually be something that I'm not just rediscovering!
September 10, 2008 Extracting traditional vector forms of Maxwell's equations using projective operations on bivector form.
The end results here are nice, but the messy algebra in between is yucky. Shows exactly which projective type operations are required to extract Maxwell's equations in the differential vector form that we see in school.
September 7, 2008 Extracting tensor equations from bivector Maxwell equation.
While examining metric issues below I also looked at how the tensor and bivector forms were related more closely. I ended up with a fairly complete comparison of the tensor and bivector field equations, and have split my notes for that into a separate less meandering writeup.
Also calculated is the Lagrangian density in terms of potential directly from the spacetime curl of the potential. There is no attempt here to try to recover the field equations from the density (followup separately).
September 6, 2008 Examination of metric dependencies in STA and relationships to tensor expression.
John Denker has an excellent STA exposition for the em bivector field. This however uses a different metric than Doran/Lasenby, so I went back and reexamined a number of the metric dependencies to avoid getting sign mixups translating between the two notations.
An interesting (to me) observation about the gradient as a vector derivative is also made.
September 2, 2008 Solutions to Goldstein Mechanics problems from chapter I and II.
Some of the Goldstein problems in chapter I were also in the Tong problem set. This is some remaining ones and a start at chapter II problems.
September 2, 2008 Question on GAFP Example 4.1 (outermorphism example).
Type up question for asking on Google Geometric Algebra group.
This was originally a question on PF, in this thread Geometric Algebra for Physicists has an example of a linear operator. , but it was too obscure for PF, and nobody ever answered. Eckhard who runs http://gaupdate.wordpress.com/ suggested the creation of a GA forum in response to my asking if he knew of one. I created the google group above and used this old unanswered question to seed it.
I think that my final conclusion is that "it follows" in the GAFP text really means that "it can be shown".
September 1, 2008 Looking at the meaning of the term Canonican Momentum.
Put canonical momentum in vector form to see what this looks like when expressed in a coordinate free fashion.
August 30, 2008 Quicky explaination of metric tensor with respect to PF question on Lorentz diagonality.
Really amounts to an explaination for my own benifit to see if I understand things. Hopefully also helpful to the OP.
August 28, 2008 proper Lagrangian with mass variation.
In the Force-free motion from a busted Lagrangian PF thread, Mentz posted a question about the equations of motion resulting from a Lagrangian where the Kinetic energy term was allowed to vary. His question used observer dependent time and one dimension, and I had a bit of fun generalizing this result. I wrote this up specifying enough detail that all conventions and assumptions that somebody that wasn't me could also tell what I was talking about.
August 25, 2008 My solutions to David Tong's Lagrangian problem set.
Solutions to the Lagrangian problems, associated with the online text by Dr. David Tong on classical mechanics.
Written up for myself to throw out my messy notebooks, and jot down extra notes that I wanted to potentially refer to later (I also liked my approach in some cases). Problems include:
August 22, 2008 Derivation of covariant Lorentz force law from proper time Lagrangian equations.
Take a guess at the covariant form of the Lagrangian equations for relativistic correctness. Use this first derive the proper time force from the kinetic term, and followup with a derivation of the Lorentz force law (in covariant form) after looking at the result of applying this form of Lagrange equations to a velocity plus field dot product. Although this doesn't start at the most fundamental point (ie: it assumes the form of the Lagrange equations instead of demonstrating that they actually solve the minimization problem), it is still a very cool result, and what I expect may be a high point in my current understanding of pre-quantum level physics for quite a while.
August 19, 2008 Expanded notes on covarient Lorentz force law.
I now understand enough of the GAFP derivation (section 5.5.3, The Lorentz force law) that I can follow it. However, I found that it still needed a lot of margin notes to make it clear for myself. Here I try the derivation on my own. Without the GAFP text, this is likely not a good standalone explaination.
Once I got as far as understanding this covariant form, apply it to expressing the Lorentz force in potential form. Having seen the Lagrangian for electrodynamics using the electrostatic potential field, and the vector magnetic potential field, the obvious question is how one would express that Lagrangian in a more symmetric four vector form. I get as far as expressing the proper Lorentz force as a spacetime gradient. However, having done so I'm left having to go back to relativistic dynamics to see how one expresses a work done integral in four vector form (where just summing over the spatial force and spatial displacement is not likely enough).
Another followup bit was added, which is an explicit validation that the spacetime curl produces the electric and magnetic field components expected in terms of electrostatic and vector potentials.
August 15, 2008 Take the four vector expression of the separate electric and magnetic fields and put into STA form.
This was a messy algebraic exercise, but shows the structure behind the normal formulation of this. Also show how the Lorentz guage is a natural condition to impose in order for the electromagnetic field to be specified as the gradient of the four potential, instead of having to express it as the spacetime curl of this same potential.
August 14, 2008 Explicit expansion of Lorentz boost applied to rest frame event vector.
Dumbly plug through the algebra to see why GAFP can casually write their four velocities as: v R = R gamma_0. Use this result to show how the projective split that utilizes the wedge product and dot product ratios of proper velocities works. That's yet another result from GAFP that I didn't understand reading their text. Also transform both proper velocities to switch which worldline is observed in its rest frame. This demonstrates that the relative velocity is independant of such a common transformation which matches intuition. That said, intuition with relativitistic issues is a hard to come by and I don't yet trust what I do have of it. What I did not demonstrate yet is whether this projective split is independent of any boost. I'll probably return to that later (and assume for now.)
August 13, 2008 Cauchy Riemann equations expressed as a gradient.
With some manipulation, it is shown that the pair of Cauchy-Riemann equations that describe the conditions that the function is analytic (has a derivative), may also be expressed as a single gradient equation. Specifically, that the gradient of a complex function is zero in all analytic regions. The gradient here is used in multivector form, where the complex imaginary is expressed as a plane unit spatial pseudoscalar (unit bivector). We are used to seeing that real and imaginary components of complex functions are harmonic (sum of second partials equal zero). We see that this follows directly from the fact that the gradient (a vector square root of the scalar Laplace operator) is zero. I thought this was a kind of neat result, and suspect that it can be related to residue calculus in a plane.
UPDATE: August 26 2008. Last page was all wrong (attempt to generalize to reciprocal frame vector form of gradient. Deleted.)
August 9, 2008 Derivation of gradient in non-orthonormal frame using Lagrangian.
Using a derivation of Newton's Law from the Lagrangian action minimization for a purely kinetic point particle in a position dependent (conservative) field, the normal orthonormal basis definition of the gradient is observed. Then the same calculation is done utilizing reciprocal frame pairs in order to find the correct form for the gradient when expressed using an arbitrary not neccessarily orthonormal basis (ie: get the upper and lower indexes all in the right places).
August 3, 2008 Rotor form of the Lorentz boost, and invariance of four vector dot product.
Examine the invariance of the Minkowski dot product with respect to Lorentz transformation. This can be done trivially with the exponential rotor form of the Lorentz boost, but it is not so trivial to demonstrate that this rotor form works. Also examine rotation with a Minkowski basis (a sign change is required).
July 28, 2008 Potential and Kinetic Energy.
A demonstration of how much I've forgotten after being out of school for 10 years. Derivation of potential energy, it's relationship to the gradient, and total energy for conservative forces. After doing this I'm now able to get the signs right in my Lagrangians.
July 20, 2008 Magnetic field due to N parallel lines.
One of the problems from the excellent book: A Student's Guide to Maxwell's Equations.. Here complex numbers (as can be expressed at whim using the geometric product) are used to evaluate this at any point outside of the wires instead of just between them.
July 16, 2008 Lorentz transformation of spacetime gradient
Having expressed the spacetime gradient with a (STA) Minkowski basis, and knowing that Maxwell’s equation written using the spacetime gradient is Lorentz invariant, we therefore expect that the square root of the wave equation (4-Laplacian) operator is also Lorentz invarient. Here this idea is explored, and we look at how the spacetime gradient behaves under Lorentz transformation.
July 12, 2008 Geometric Algebra: Back to Maxwell’s equations
A step back and look at Maxwell’s equations in more detail. In particular looking at how we get from integral to differential to GA form. I had a version of this earlier (below) that I was unsatisfied with, but that one is perhaps more natural if just getting the feel for the algebra.
July 8, 2008 Radial decomposition of angular velocity and angular velocity.
This is a derivation of the radial velocity decomposition for a physics forum poster who couldn’t understand my GA approach. My intention was just to provide a hint, but he didn’t have enough math background to use this hint for his own derivation.
Updated July 18, 2008 with acceleration derivation.
June 30, 2008 Covariant derivative and tensor notes.
My notes on tensors and index notation, to try to clarify the references to this in “Geometric Algebra for Physicists” (GAFP). That text uses the upper and lower index notation but assumes in many ways that it is already known to the reader. This is an attempt to clarify for myself some of the notation and details missing in that text, where the focus is on the coordinate free approach.
June 25, 2008 A derivation of the Lorentz transformation from wave equation.
There are some subtleties to understand what it means for the speed of light to be constant. I cannot say definitively that I still have an satisfactory or intuitive feel for such this constancy. I think it is easier to to conceptualize the idea that light (ie: an electrodynamic field) is wavelike regardless of the motion of the observer. Specifically, that light must satisfy the wave equation (ie: Maxwell’s equations) regardless of the space and time parameterization.
This little doc (which I am pleased with) presents a derivation of the Lorentz transformation using the wave equation. This approach does requires a bit more math than the typical expanding light shell method, but is possibly more intuitive. I’m curious if somebody else also new to the subject of relativity would agree?
June 10, 2008 Summarizing for myself the various four-vectors of mechanics.
Used to preview a question made in a PF post . Not a useful doc by itself, but kept because I want to followup on many of the relativistic ideas based on invarients.
May 30, 2008 Angular Velocity and Acceleration.
A derivation of angular velocity and acceleration, as related to radial components of velocity and acceleration. There is another such attempt of this below from when I was first learning geometric algebra, but this one is more coherent.
May 25, 2008 Geometric Algebra: Signs of electromagnetic field tensor components?
Another PF question (resolved in the thread) . I couldn’t understand the signs of the electrodynamic tensor as expressed as a bivector dotted with the sets of bivector reciprocal frames. Note that I never went back and corrected the original question writeup, so this document has all my original errors.
May 17, 2008 Oblique projection and reciprocal frame vectors.
Calculate the directed projection along a line or hypervolume onto a target hypervolume. I called this oblique projection but I’m not sure that is the right term.
May 16, 2008 Projection and Moore-Penrose vector inverse.
This is a look at the similarity of the Moore-Penrose vector inverse and the geometric product vector inverse.
May 16, 2008 Projection with generalized dot product.
Calculate the projection matrix for a non-Euclidian dot product.
May 15, 2008 Matrix of k-vector linear transformations.
Computing the matrix of a linear transformation expressed as a geometric product.
May 13, 2008 Satellite location by measuring direction from two points.
There is a mistake in the rotor formulas in this doc that I have identified (and have a correct expression for on paper). Have to go back and correct the latex.
May 9, 2008 Questioning equation 8.8 from GASC.
Confused about a bit of the book. Questions sent to the authors and my own response trying to reconcile things for myself while waiting for an answer.
May 7, 2008 Solving Lorentz Force Differential Equation.
My attempt to answer a PF thread about how to solve the Lorentz force equation. Used the classical mechanics ideas from Tong’s rigid body treatment. I don’t think that I really followed this to its complete conclusion, but would like to first understand the relativistic treatment of the same problem in both the GAFP and NFCM books (but that will take considerably more study before I’m capable of understanding that stuff).
April 30, 2008 Kinetic Energy in rotational frame.
Fill in the missing details of the rotational KE derivation in Dr. David Tong’s excellect classical dynamics lecture notes . Contrast matrix and GA approach.
April 11, 2008 Projection expressed in matrix form
Line, plane, and hypervolume projection matrixes are calculated with respect to general and orthonormal basis. Developed the matrix form in detail and compared to GA projection.
April 11, 2008 Rough notes on Moore Penrose Inverse and SVT.
Some notes on application of projection as left pseudoinverse (ie: linear fitting). Removed from projection matrix doc source for later rewrite in coherent fashion (not yet done).
April 7, 2008 Notes on shear transformation.
GAFP and GASC both use similar examples for determining inverses of what they call the shear transformation. Some notes on this for myself.
April 1, 2008 Orthogaonal decompostion take II.
A treatment of the vector blade product and components of the vector within and dual to that blade is made. A small standalone treatment of this that I liked (also done in other ways in some of the earlier notes).
March 31, 2008 Exterior derivative and chain rule components of the gradient
Components of the gradient tangential and dual to a curve is calculated. I think this is a cool little result, as it appears to show the chain rule and exterior derivative both hiding in the gradient. If I actually could follow the differential forms books that I have it probably wouldn’t seem like much of a big deal since I expect that subject shows how all this sort of stuff fits together in its natural context.
March 25, 2008 Blade grade reduction.
Perform the proof of the identity to reduce certain forms of triple wedge products to dot product form. This follows the hint on how to do this in NFCM (not really detailed fully there).
March 21, 2008 Reciprocal Frame Vectors
The reciprocal frame vector concept is extremely powerful, and the way too dense and short treatment of this in the GAFP book isn’t sufficient to learn from in my opinion. Here are my notes on this which are more thorough and dumb than the GAFP book. Despite the treatment in that book being so short it has a rather fundamental place in much of the notation used later in the book (like all the index upper and lower notation). Perhaps this isn’t well covered in their book because they assume familiarity by virtue of having seen this in the context of tensor algebra. There appear to be enough people thoroughly confused by the tensor notation and covariant and contravarient nomenclature that I think this formulation in terms of simple vector operations well justified. It also makes it easier to understand the more complex tensors. An example is the electromagnetic field tensor which we later see to be a coordinate description of a bivector.
March 17, 2008 Angle between line and plane.
Like the line and line angle, this is developed for a line and plane. A little note on the measurement orientation of the cosine between is also included.
March 17, 2008 An attempt to intuitively introduce the dot, wedge, cross, and geometric products
This (and the Pythagorus note below that I intended to incorporate into this) was related to this attempt by me to write up a good justification and introduction of the GA products from scratch in a natual way. Turns out that the GASC book ended up covering this so well I didn’t bother with this little project after purchasing that.
March 17, 2008 Pythagoras law.
2D proof illustration, and justification of the 3D/ND coordinate length rule/definition.
March 13, 2008 Expanding the grade 0 part of a multivector product.
Dumb proof of the fact that the order of the terms in a scalar selection can be commuted (compared to the slick but harder to understand proof of the same in GAFP).
March 12, 2008 K-vector exponential
Here I play with the use of exponentials with multivector arguments. Split this out from the laplace notes below for clarity.
March 8, 2008 Trivector geometry.
Some of the bivector treatment below expanded to trivectors.
March 5, 2008 Bivector Geometry.
Looking at multivector exponential values in the context of the Laplace equation generated some bivector products that warranted understanding the general properties of how these multiply. In particular vector vector products are well covered in the texts (GAFP, NFCM), but other higher order products are covered in much less detail. Some of the questions that I had when writing up these notes led to the purchase of the book “Geometric Algebra for Computer Science” (GASC). That book, which has by far the best introduction to the subject to be had, ended up covering much of the material I felt missing in the other texts. Much of that missing detail I had worked myself by the time I had received that book, but it was still a very worthwhile read. Later I found the PhD thesis by one of the authors which contains much of that text, and is just as readable as the book. If you want a good intro to the subject before purchasing any texts, give that thesis a read. I find a book much easier to read than a computer screen, but this particular book uses a different notation than all the others, which is distracting, so you can get an idea of some of the content before a purchase by looking at the thesis. An additional great resource for learning GA is their interactive graphical GA calculator GAViewer , and their Game Developer tutorial which uses it.
Covered in these particular notes are:
All types of products of bivectors (scalar, bivector, and quadvector), how these are related to symmetric and antisymetric decompositions when possible.
Intersection of planes, and relation to duals.
Action of the grade two term of a bivector product as a rotation. Projection and rejection.
Angle between intersecting planes.
Rotation of a plane.
Bivector products compared to dot product of cross product expressions from traditional R^3 vector algebra.
February 28, 2008 Exponential Solutions to Laplace Equation in RN
Expressing planar Laplace equation solutions using multivector exponentials.
February 19, 2008 Rotor Notes.
Again, reading one of the GAFP or NFCM books I found the treatment of rotors lacking many details that somebody dumb like me needed to understand them from just reading. Here was my notes on the use of Rotors. Rotors are two sided rotation operators that generalize complex exponentials and quaternions as rotation operators, and act on vectors or multivectors.
February 15, 2008 Inertial tensor.
The Geometric Algebra for Physicists book has a lot of excellent content, but is much too dense (at least for me). Here I expand on their inertial tensor section, adding details that helped me understand it better.
February 8, 2008 Quaternions.
Notes on quaternions which like complex numbers may naturally be represented using a scalar plus bivector pair.
February 3, 2008 NFCM Exercise 8.4. Legendre Polynomials.
This is one of the GAFP problems. Notes that clarify the notation used in the book for that problem (not obvious notation).
February 2, 2008 Taylor’s Theorem
A more dumbed down derivation of Taylors Theorem than the one provided in New Foundations for Classical Mechanics.
January 30, 2008 Wedge product formula for hyperplanes.
Having purchased “New Foundations for Classical Mechanics” by Hestenes, I made more a more detailed exposition of the formula for a plane and hyperplanes than covered in the book (which covered the same stuff for a line in much more detail). These notes include planar and hyperplanar rejection, and components within and dual to that space.
Jan 2008 Wedge product norm and GA bivector norm comparision.
Here I meant a comparison with the norm for a wedge product as I thought it ought to be based on area, and volume, and the Geometric algebra norm, which differs from that by a sign factor.