TODO

================================================================================================

- spherical polar:

  exersize numbers now out of sequence.

- have some repeated curl = 0 references a few times in the mvpotential problems.  point those back to the theorem (chapter 2 -- vector identities section.)

================================================================================================

- have various boxedEquations in chapter 1.  Might be best to systematically convert those to lemmas.  Example

   eqn:generalizedWedge:640
- Figure 1.5 should have shading to show the difference in orientation.  Not clear -- even in color.

- Perhaps try again later: couldn't get hatch shading to look good for:

  Figure 3.12: Pillbox integration volume.

- Think that I had some review comments in dropbox to act on.

- Also have some review comments below.

- steal from ece1228?: parts of appendix F, G (cylindrical and spherical gradient operators and coordinates)

- review theorems and convert minor ones to lemmas (or use theorem for everything)

================================================================================================


- chapter 1:

   - incorporate subspaceDistance.tex

   - incorporate parts of lineAndPlane.tex into chapter 1 (summary portion as theorems, with extra details as problems).

   - Add table at the end summarizing the GA vs conventional forms.

- chapter 2.

volumeintegral.tex:82:(FIXME: Have I not spelled out the volume element and the vector derivative for three parameters before?  Check for this and remove

   Have d^3 \Bx image with the fundamental theorem for volume integrals.  It should be moved from
   volumeintegral.tex to volumeintegraldef.tex

   Do the same for surfaceintegral.tex.

- get rid of example block:

  Example 1.4: Multivector dot products.

- On the possibility of removing (1.67) : Mo:

   I think you should keep the definition of reverses and blade. In a book you do not have to use (later) every thing you define.
   Also a student may hear the term blade or reverse and may want to check the book to see what these refer to.
   So, in my opinion, removing these terms and definitions can do more harm than having it, which only will slightly increase the length of the book.
   Actually Eq. (1.67) is a very nice application of reverse, worth keeping.

- Manfred:

      For example, in Euclidean spaces we could use either of

      (1.69)
      i = (u ^ v)/sqrt(...)
      i = (u ^ v ^ w)/sqrt(...)

   => don't use i for both bivector and trivector pseudoscalar.

- didn't prove:

   Conversely, if the product of two vectors is a bivector, those vectors are orthogonal.

- Mo's marked up GA Chp1.pdf

   comments dealt with up to pg <= 36.

- define k-vector dot product.  multivector dot product will then just follow by superposition.  Same for the generalized wedge.

- mo:
   define the reverse of a k-vector, and then expand this to the reverse of a multivector.

- re: blades and reverse.
Mo suggested highlighting the fact that e_123^2 = -1, and e12^2 = -1 much earlier (presenting both as theorems).  Once that is done I probably won’t use that reverse formula, and will see if I even need it in the book.  I’m actually using blade for almost nothing now in the book (having switched to bivector since the distinction didn’t matter in many places).  Really the only place that it is left is chapter II where it was relevant for the generalized Stokes theorem.  Perhaps I can eliminate it entirely?

- switched to annotated tags:

   https://stackoverflow.com/a/7261049/189270

   rework the versioning to be based on that.

- busted link to fix for print version:

   ...:stressEnergyTensorValues.nb

- new msword document with followup required for Mo's read.

- followup for Mo's comments:

   4) audit all definitions, to ensure there are examples.

   6) make i = e1 e2, i^2 = -1 a theorem with proof to make stand out.

      handle sequencing issues.

   7) make I = e1 e2 e3, I^2 = -1 a theorem ...

   8) figure 1.7: change from e1,e2 to u, v, with u and v orthogonal, but different lengths.

   9) Figure 1.11: Rotation in a plane.

      x e^{i theta} = e^{-i \theta} x, provided x lies in the plane of the bivector i, so you can get a rotation with the opposite sense of x e^{i theta} with e^{i theta} x.  I'll add a note to try to clarify this, but think the label in the figure should be x e^{-i theta}.

   [figure adjusted, but not in this way (at least yet)].

   10) We can show that these unit vectors are orthogonal easily (eq 1.32)

      >> why grade selection.

         It looks like I haven't shown yet that the dot product is equivalent to grade 0 selection.

      "can find the velocity and acceleration by taking"
      add note:

   11)

> point of new 1.3 vs. 1.4 figures.

The proposed 1.3 and 1.4 illustrate the same areas, with the same orientation.  Part of the point was to illustrate orientation, but the other point was to show that there is no preferred geometric representation of a bivector.  Any closed figures with the same orientation and total area are equally valid representations of a given bivector.

12)

> On multi-vector vs. k-vector  (your comment on page 13), indeed this was one of my upcoming questions which caused me some confusion. It seems to me that when you say multi-vector you mean a superposition of k-vectors that can have mixed Grades. For example 1+e1+ e12, +e123  is a multi-vector. But when you say k-vector even though this can also be a superposition, but all constituents have the same grade, for example e123 + 3e231 -e321. Am I correct? If so please make the distinction clear in the text and the best way to do so is to give examples.

> I asked a version if this question in my email thus morning. Here is a more specific version of it.
For example when you write bivecotor you can have superposition of bivectors but grade is 2 and only 2 (no scalar, trivector,... k-vector)
when you write trivecotor you can have superposition of trivectors but grade is 3 and only 3.
Similarly for k-vector.
However when you write multivector you mean superpositions of scalar, vector, bivector, ...k-vector. That is to say that the grade depends on which term you look at.

A: Your comment on multivector vs. k-vector is exactly correct.  A k-vector, say, k=2, has only grade 2 (examples: e1 e2, or e1 e3 + e2 e3), whereas a multivector may have any grade.  Any k-vector is also a multivector.  I'll add more examples to make this clearer.

13)

      When you say vector almost always you mean 1-vector correct?
      Here is my jargons, see if I am correct.
      0-vector = scalar
      1-vector = vector (traditional vector)
      2-vector = bivector
      3-vector = trivector
      ..
      So for example your entire discussion in section 1.8 is about the product of two 1-vectors, leading to a dot product and wedge product of two 1-vector, correct?

      Dot product can be defined for the product of two multivector (and hence two 1-vector), correct? However, wedge product is always between two 1-vector and has the grade 2?

      in product ab, (no dot or wedge between a and b) can a and b be two multi-vectors?


Clarify.  My answers:

         Yes, a 1-vector a traditional vector.  I tend not to write 1-vector, since I don't think that adds anything.


         > So for example your entire discussion in section 1.8 is about the product of two 1-vectors, leading to a dot product and wedge product of two 1-vector, correct?

         Yes.

         > Dot product can be defined for the product of two multivector (and hence two 1-vector), correct?

         Correct.

         > However, wedge product is always between two 1-vector and has the grade 2?

         Yes.  The wedge product of two vectors (1-vectors) a, b is defined as the grade 2 selection of the product a b (which can only have grade 2).

         > in product ab, (no dot or wedge between a and b) can a and b be two multi-vectors?


         Yes.  Maxwell's equation is an example of such a product.  It has the form

         (gradient + (1/c)partial_t)(E + I eta H) = J

         The gradient is a vector, and the time partial is a scalar, so that sum is a multivector with grades 0 and 1.
         The electric field is a vector, but I H is a bivector, so E + I eta H is a multivector with grades 1 and 2.

14) Is use of * for duality opertaion common?  In my opinion it only can cause confusion.

Yes, especially in differential forms.  I don’t use it in the book, but thought it worth mentioning for the reader who had seen duality in another context.  I’ll remove this from the definition, but may still mention it in the comments or a footnote following the definition.

15) Definition 1.23: Dual

"follow this with an example. In fact, you already have the example in the form of I e_3 e_(1= -) e_2. You just need to state it as an example. "

- re-implement \makeexample using \newtcolorbox for consistency.  They have different border widths and thicknesses.

- In the report I made some notes about the limitations of my geometric calculus treatment.  These should be
  incorporated into the text.  There is also an interesting point in Spivak's manifolds about triangularization
  which is of interest given MacDonald's paper on the same for blades.

-
   \maketheorem{Far field representation.}{thm:potentialSection_farfield:1}{
   ...
   F &= -j \omega \lr{ 1 + \rcap } \lr{ \rcap \wedge \BA} \\
   F &= -j \omega \eta I \lr{ 1 + \rcap } \lr{ \rcap \wedge \BF }.

   dimensions of [eta \BF] should equal [c \BA], but do not here.

- I was also thinking of adding various symbols to the index, and probably putting a table of symbols somewhere for reference (perhaps with page references to the points of first use).

- refer to pdfs too in mathematica notebooks index.

- Pillbox figure.  Perhaps make a first figure that doesn't have the integration volume, but does have the
  regions, and an illustration of each of fields (vector+bivector) in each region.

- "employ the mean value theorem" :

   statement about the mean value theorem is vague, and notation is not defined.

- Have places that it would make sense to use label boxes.  I'd forgotten all about those.

- before a \makedefinition, theorem, ... should I end any sentance?  I think so, and don't in many places.

- colors: manually colored equations.  used:
      DarkOliveGreen,Maroon

      (general color scheme used elsewhere.)  Something that looks darker in equations might be nice.

- Is {eqn:isotropicMaxwells:520} the first use of \grad f = grad . f + I grad cross f?

(probably not: expect it's also in the Helmholtz content)

   This probably deserves some standalone discussion.

- Theorem 2.6: Line integral of a scalar function

   In proof of this theorem, the grade 2 component is ignored.  This probably deserves some thought and comment.
   Relation to vector curl in \R{3}?

- have now added a more complete proof of gagc up front.  Now need to streamline the surface and volume integral content.

- reciprocal frame: Alan:

   Better stated as a theorem. How do you know that there is a reciprocal frame? See my LAGA p. 94.

- Stated the fundamental theorm and stokes expicitly.  Under line, surface (and volume when done), go back and reference
  those theorem statements, covering just the differences from the general statement.

  Perhaps try a general proof of Stokes from the fundamental theorem, and get rid of the proofs for the special cases.

  May be able to get rid of the references to the distribution theorems used in the original Stokes' proof.  Will there
  be value keeping those?

- had used bigwedge notation in Stokes but don't think I defined it.  removed it ... search for other instances of this.

- add symbols to the index.

- Mo review:

The only thing missing is an explicit enumeration of the basis elements for the complete R^3 geometric algebra.  i.e.

1

e1, e2, e3

e_12, e_23, e_31

e_123


- Mo suggested 'is' instead of 'can be' in the bivector definition.  I use 'can be' many times, and
  something more definitive might be better.

- We don’t usually try to represent of a quantity that has the
apparent dimensions of a point graphically.

   >> not clear to mo.

-
      If you are inclined to skip this, please at least examine the
      stated dot product definition, since the conventional positive definite property is not assumed.

   >> perhaps eliminate this, since no non-Euclidean metric use has been introduced.

-

   There is no (concrete or CAS) worked example for _Integration with GA_ yet.
   I think you will think about it ;)
   Maybe there is a typical problem from a course for electro-engineers ?

- I've made right handed assumptions in my area and volume integral formulations.  Need to clarify those, or make a note to
  ensure that we pick a parameterization for which the hypervolume element is positive with respect to the standard basis.

- In general want to show connection between the area/volume element and jacobean.

   Did that for the specific case of spherical polar parameterization in R{3}.

- Wolfgang.  email: e&m draft - Greens with MM: Suggestion:

   Idea: you could collect your formulas in 2.5 in a mathematica notebook, to give the reader the possibility to run/test the solutions. Maybe you could demonstate it using some code of clifford?

   => produce an Appendix with links to all the Mathematica notebooks.  Will need to add metadata references for them.

- review: email attachments: e&m draft 348 - some hints

- review: Attachments-e&m draft version 321 - hints, screenshots, Eigenmath code

- review: Wolfgang/353-26.txt

- review: 348-greensIntro.pdf

   embedded pdf comments in this doc.

- review: Wolfgang/346-chapter25.txt

   Hints 345 §2.5:
   ---------------
   (As before I write  MY COMMENTS BIG to contrast from main text)

   - p.96: 'The fundamental theorem of geometric calculus is a generalization of Stokes’
   theorem __theorem__ 2.4 to multivector integrals'

   WHERE:
      # grep -in theorem.*cref.thm *tex
      fundamentalTheoremOfCalculus.tex:28:The fundamental theorem of geometric calculus is a generalization of Stokes' theorem \cref{thm:stokesTheoremGeometricAlgebra:1740} to multivector integrals.
      fundamentalTheoremOfCalculus.tex:125:For Stokes' theorem \cref{thm:stokesTheoremGeometricAlgebra:1740}, after the work of stating exactly what is meant by this theorem, most of the proof follows from the fact that for \( s < k \) the volume curl dot product can be expanded as

   - p.97: 'This theorem and Stokes’ theorem, both single formulas, are loaded and abstract statement ..'
     I don't understand the last 5 words.

   - p.97: 'A dual basis {x^i} reciprocal to the tangent vector basis xi ..'
     IS the dual basis the reciprocal basis? DUAL=RECIPROCAL?

   - p.100: Definition 2.2: Multivector Fourier transform pairs
     PLEASE give a small example following Def. Check with MM (=MatheMatica)?

   - p.100: Definition 2.3: Multivector phasor representation.
     PLEASE give a small example following Def. Check with MM (=MatheMatica)?
     It may be very new for your readers ..
     They need to get a 'working' comprehension ;)

   - I found a nice notation in Prof. Speck's lecture notes.
     He uses for clarity using = (Def vs. fact or proposition):
         def
     ..  =  ..
     Maybe it could be used in your text (construct a LaTeX \def ?)?

   - p.104:  'a gradient \/' acting bidirectionally on functions of x' '
     Please remember or explain what is meant by "bidirectionally".

   - p.105: appendix C.1 is a interesting computation!

   WHERE: c.1 helmholtz operator.

     Is it worth doing steps of it using/showing MM
     to compactify/verify the computation?
     I will try it with EigenMath or Maxima (wL.ToDo ;)

   - p.106: ' Theorem 2.8: Green’s function for the first order Helmholtz operator.
     The Green’s function satisfying ? ..'
                                     ?
     What is meant by del_arrow/back_arrow?
     I suspect: advancing (causal), and the receding (acausal)?
     One can rediscover the meaning following the proof in (2.152).

   - p.106: after  'This is relavant for bounded superposition states, which we will discuss next now that the
   proof of theorem 2.8 is complete.'  you should open a new line \lf? paragraph?

   -----------------------------------------------------------------------
   GENERAL REMARK:
   This part of your text is very formal e.g. algebraic.
   This could be problemeatic for the comprhension of amny readers.
   Is it possible to use 1-3 simple examples/physical situations
   througout this chap. 2.6 to visualize and concretize your results?
   I think of the example/s in

   Green's Function -- from Wolfram MathWorld, I send you the picture, or
   https://brilliant.org/wiki/greens-functions-in-physics/

   These done with MM and the new GA calculus ..
   -----------------------------------------------------------------------

- Wolfgang/346-chapter22-24.txt

   - p83:
     the torus can be stated directly
   Q:  (2.62)-(2.64)  I COULD NOT VERIFIY WITH CAS !?

   WHERE:
      torusCenterOfMassParameterization.tex:74:FIXME: show this with Mathematica too.

   RESOLUTION:

      Mathematica calculation of the basis and volume element.

   - p87: (Green's theorem) you should spend an concrete example for (2.78) using GA and using MM (me: EigenMath ;)

   WHERE:
      surfaceintegral.tex:362:FIXME: Add an example

   RESOLUTION:
      Add an example (lots to pick from in any 3rd term calc text)
      Also CAS example.

   - p88: you should give an concrete (also CAS) example for (2.85).

   WHERE: (line integral theorem)
      grep -n 'ine integral of a scalar functi' *tex

   RESOLUTION:
      As suggested.

   - General:
      ----------------------------------------------------------
      I think your impetus and your main motivation is to do Calculus using GA.
      IMO , therefore you should show some typical 'classical' vector analysis
      calculations using in contrast GA methods (plus MM).
      This is new, unusal and unkown and a big merit of your work/research -
      hence there should be in your ongoing work in the selection of contrasting
      examples and exercises in the spirit of
      [Mac]=[17] Macdonald (I like his style)
      or
      [Din] S. Dineen: "Multivariate Calculus and Geometry."
            Springer. ISBN 1-85233-472-X.
      (IMO a very good classical book with clear examples,
      struggeling for good understanding of concepts by the reader. No CAS used).
      I recommend to usesome examples/exercises form this books and solve them
      using GA ..
      -----------------------------------------------------------

   - p89: 2.4.2 the 'example' is a 'theoretical' implication.
     To staffold the reader you should give an concrete (also CAS) example for (2.84),
     perhaps [17, Ex.7.17 p.96 orEx.7.20] or [Din, p.53 Ex 6.1, page attached].

     WHERE:
      This referred to:
            2.4.2 One parameter specialization of Stokes’ theorem.

     RESOLUTION:
      This has now moved, and is now stated as a consequence of GAGC.  The concrete example
      comment still applies.  There should be multivector examples as well as scalar
      and vector examples throughout the line integral section (as well as the same for
      area and volume integrals)

   - p90: besides the 'abstract' fig. 2.7 one could take the fine
     Fig. 2.3 (Torus M), pick a concrete point p, paint the TpM and
     calculate the vec.deriv. there .. and show it in the Fig.
     You have done all prerequsites earlier .. ;)

     WHERE:
         ../figures/GAelectrodynamics/toroidFig1
         gaToroid.nb

     RESOLUTION:

         Nice idea.  Plot the tangent plane at a point as suggested.

   - p91: at the end of 2.4.3 a concrete example using GA methods like attached
     ex. like [din.Ex.9.3]?

     WHERE:

         2.4.3 Two parameter specialization of Stokes’ theorem.

     RESOLUTION:

         Do this (as suggested for the line integral case).

- Helmholtz:

   I'm not sure if there's really a good reason to include the Helmholtz theorem section at the end of the chapter. I happen to like it, since it's a nice compact derivation, but it's somewhat out of place since we don't have a good reason to work with separate divergence and curl operations in a GA context. Better would be a more general first order statement showing that the gradient (not its divergence and curl) uniquely determines the solution if the function or the Green's function vanishes on the boundary

- footnote about the CliffordBasic package now follows first use.  Add some more sample calculations, and move footnote earlier

   (perhaps generalizing to include other CAS engines like Maxima.)

-
   cylinderField.tex

      What was this?  It's now orphaned.

- far field application of Green's function for grad+jk.  GAFP does this nicely, but
it's a bit hard to follow, and their notation for the Green's function is not spelled
out for the uninitiated.  Perhaps try to do something like that, but in a more accessible
fashion (also not assuming mu = epsilon = c = 1 and using j \omega instead of -i \omega)

- placement of the primes in \lspacegrad' \rspacegrad' or their squares looks bad.

- differentiate: have two different theorem's labelled Green's theorem (one for area, and other for bounded Green's fucntions)

- mmacells.sty dependency isn't properly in the make sequence.

- example for reflection.  Do this one in 3d.

- Dr Wolfgang Lindner:

   > The first chapter on linear GA is very "heavy/dry" mathematical stuff.
   Im my opinion it would be wise to spread some concerte calculations/examples
   through the text,
   perhaps every 2-3 pages


   I agree.  I started out with something very informal, and I think I've gone too far in the opposite direction now.  I'll see if I can work on finding a better balance.


   > -- also using some free CAS like REDUCE, MAXIMA or EIGENMATH.
   Give screenshots to let control/reproduce the results.
   This could activate the novice learner for keeping on the track ;)


   There are a number of GA CAS packages available in different languages.  Let me start with putting together an appendix listing some of the packages available, and giving examples of their use.
    
   MATHEMATICA is very good, but not free avaílable: my private version 4.0
   (1999)
   is very outdated. Maybe I can run your code, because you give txt-access to
   the *.m file.
    
   There are a few Mathematica GA packages available (mine, plus some others).  It's possible that one of the others will also work with Mathematica 4.  I like the integrated symbolic capability of Mathematica, and found that it justified the 'personal use licence' Mathematica purchase.  A full version for professional use is horrendously expensive.

- It should be clear how to generalize the reciprocal basis calculation formulas:

   -- statement is a cop-out.  fix.

   workaround: just deleted this for now.  Stick to \R{3} and below.

- Got to try plotting the ring fields.

- Lorentz force, boosts: gabookII chapter 35.  Is there a reasonable paravector implementation of this?

- gapauli

   - refresh .pdf's associated with the .nb's
   - have introduced Private symbols for GA13 (but not GA20, GA30)
   - StandardForm for GA13 now looks funny:

       {1},
       {GA13`Private`\[Gamma][1]},
       {GA13`Private`\[Gamma][2]},
       {GA13`Private`\[Gamma][3]},
       {GA13`Private`\[Gamma][2] GA13`Private`\[Gamma][3]}
       ...

    A bigger issue is that cut and paste of the \gamma_... symbols from the various forms can be used as grade[...] operations.

- figure out how to factor out a grad dot operator from the Poynting stress equation.

- two new fixmes:  Add (or move) figures describing oriented bivectors and trivectors to where those are defined (before multivector space).

   - Also add cooresponding figures for vectors when those for bivectors are defined.

- hand drawn figures:

   Infinitesimal loop integral.
   Sum of infinitesimal loops
   Three parameter volume element

- superfluous use of:

   also
   observe
   many

   (probably lots of adjectives in general)

- spelling: unsure about these

   whos argument
   distributivity
   nondimensionalizing vs.  Non-dimensionalizing
   duals
   constitutivity
   subspaces
   reexpress
   factorizations

- copyright prologue check:

   for i in `cat spellcheckem.txt ` ; do grep -i copyright $i >/dev/null || echo $i ; done

- makeexample breaks in strange places sometimes.

- number line vs. 1d vector space plots show with different sizes.

- have boilerplate copyright.

- dedication.tex got cleaned out on make clean.  Rule that was in makefile doesn't appear to be right.  Revisit.

- exercise headers are bigger than the section headers (partially fixed)

- problem on <abc>_3 relation to wedge.  generalize that to _k.

- have some direct ref's that could use cref helper macros (forget how to do those though).

- Green's:
   Reference this derivation in Stokes' and make note of the fact that Stokes's derivation should properly consider the
   surface triangularization.

- Stokes'

   refer to the helper theorem \cref{} for in the problem.

- Fundamental theorem:

   Fixme's to deal with.
   Strip the definitions section out, since it should be covered previously.
   Bold vectors for Green's function.  Perhaps rework to be less general (assuming Euclidean).

- summation convention: any left?

- Review the whole doc for the use of \Abs instead of \Norm (believe that all the .tex files in this directory are now converted)

- fixmes:

   grep -ni fixme `cat spellcheckem.txt`

- table (2d multiplication) uses different color scheme, doesn't have a label or table number, and is too wide.

- figure for projection and rejection.  move from later material into SimpleProducts2