diff --git a/figures b/figures index cdd5db7a4..d5387a5be 160000 --- a/figures +++ b/figures @@ -1 +1 @@ -Subproject commit cdd5db7a4a4db3055e66cabbb4f3dd7ede9a779b +Subproject commit d5387a5beb0c29619fd906328b38c31812ab9a5e diff --git a/notes/.GAelectrodynamics/.junk/gradeselectionProblems.tex b/notes/.GAelectrodynamics/.junk/gradeselectionProblems.tex deleted file mode 100644 index 50cc6eb84..000000000 --- a/notes/.GAelectrodynamics/.junk/gradeselectionProblems.tex +++ /dev/null @@ -1,26 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - - - - - - - - - - - - - - -%\makeanswer{problem:vectorproduct:cyclicpermutationII}{ -%FIXME: todo. -%} % answer -% - - - - diff --git a/notes/.GAelectrodynamics/.junk/introGAproblems.tex b/notes/.GAelectrodynamics/.junk/introGAproblems.tex deleted file mode 100644 index 360ed010f..000000000 --- a/notes/.GAelectrodynamics/.junk/introGAproblems.tex +++ /dev/null @@ -1,41 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%%{ -%\input{../blogpost.tex} -%\renewcommand{\basename}{introGAproblems.tex} -%%\renewcommand{\dirname}{notes/phy1520/} -%\renewcommand{\dirname}{notes/ece1228-electromagnetic-theory/} -%%\newcommand{\dateintitle}{} -%%\newcommand{\keywords}{} -% -%\input{../peeter_prologue_print2.tex} -% -%\usepackage{peeters_layout_exercise} -%\usepackage{peeters_braket} -%\usepackage{peeters_figures} -%\usepackage{siunitx} -%%\usepackage{mhchem} % \ce{} -%%\usepackage{macros_bm} % \bcM -%%\usepackage{txfonts} % \ointclockwise -% -%\beginArtNoToc -% -%\generatetitle{XXX} -%%\chapter{XXX} -%%\label{chap:introGAproblems.tex} -%% \citep{sakurai2014modern} pr X.Y -%% \citep{pozar2009microwave} -%% \citep{qftLectureNotes} -%% \citep{doran2003gap} -%% \citep{jackson1975cew} -%% \citep{griffiths1999introduction} -% - - - - -%%} -%\EndArticle -%%\EndNoBibArticle diff --git a/notes/.GAelectrodynamics/.junk/vectorproductProblems.tex b/notes/.GAelectrodynamics/.junk/vectorproductProblems.tex deleted file mode 100644 index 293033ace..000000000 --- a/notes/.GAelectrodynamics/.junk/vectorproductProblems.tex +++ /dev/null @@ -1,11 +0,0 @@ - - -%\makeproblem{Cyclic permutation}{problem:vectorproduct:cyclicpermutationII}{ -%Any cyclic permutation of the vectors within a grade zero selection leaves the result unchanged, such as -% -%\begin{dmath}\label{eqn:vectorproduct:280} -%\gpgradezero{ \Bx \By \Bz } = \gpgradezero{ \Bz \Bx \By }. -%\end{dmath} -% -%Show that this is the case. -%} % problem diff --git a/notes/.GAelectrodynamics/2dreciprocalMatrixCalculation.tex b/notes/.GAelectrodynamics/2dreciprocalMatrixCalculation.tex deleted file mode 100644 index d66ea5179..000000000 --- a/notes/.GAelectrodynamics/2dreciprocalMatrixCalculation.tex +++ /dev/null @@ -1,72 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{Reciprocal frame for two dimensional subspace.}{problem:reciprocal:2dsubspaceRecip}{ -Prove \cref{eqn:reciprocal:120}. -} % problem - -\makeanswer{problem:reciprocal:2dsubspaceRecip}{ - -Assuming the representation of \cref{eqn:reciprocal:100}, the dot products are - -\begin{dmath}\label{eqn:2dreciprocalMatrixCalculation:200} -\begin{aligned} -1 &= \Bx_1 \cdot \Bx^1 = a \Bx_1^2 + b \Bx_1 \cdot \Bx_2 \\ -0 &= \Bx_2 \cdot \Bx^1 = a \Bx_2 \cdot \Bx_1 + b \Bx_2^2 \\ -0 &= \Bx_1 \cdot \Bx^2 = c \Bx_1^2 + d \Bx_1 \cdot \Bx_2 \\ -1 &= \Bx_2 \cdot \Bx^2 = c \Bx_2 \cdot \Bx_1 + d \Bx_2^2 -\end{aligned} -\end{dmath} - -This can be written out as a pair of matrix equations - -\begin{dmath}\label{eqn:2dreciprocalMatrixCalculation:220} -\begin{aligned} -\begin{bmatrix} -1 \\ -0 -\end{bmatrix} -&= -\begin{bmatrix} -\Bx_1^2 & \Bx_1 \cdot \Bx_2 \\ -\Bx_2 \cdot \Bx_1 & \Bx_2^2 \\ -\end{bmatrix} -\begin{bmatrix} -a \\ -b -\end{bmatrix} \\ -\begin{bmatrix} -0 \\ -1 -\end{bmatrix} -&= -\begin{bmatrix} -\Bx_1^2 & \Bx_1 \cdot \Bx_2 \\ -\Bx_2 \cdot \Bx_1 & \Bx_2^2 \\ -\end{bmatrix} -\begin{bmatrix} -c \\ -d -\end{bmatrix}. -\end{aligned} -\end{dmath} - -The matrix inverse is -\begin{dmath}\label{eqn:2dreciprocalMatrixCalculation:240} -{ -\begin{bmatrix} -\Bx_1^2 & \Bx_1 \cdot \Bx_2 \\ -\Bx_2 \cdot \Bx_1 & \Bx_2^2 \\ -\end{bmatrix} -}^{-1} -= -\inv{ \Bx_1^2 \Bx_2^2 - \lr{\Bx_1 \cdot \Bx_2}^2 } -\begin{bmatrix} -\Bx_2^2 & -\Bx_1 \cdot \Bx_2 \\ --\Bx_2 \cdot \Bx_1 & \Bx_1^2 \\ -\end{bmatrix} -\end{dmath} - -Multiplying by the \( (1,0) \), and \( (0,1) \) vectors picks out the respective columns, and gives \cref{eqn:reciprocal:120}. -} % answer diff --git a/notes/.GAelectrodynamics/2dvectorsquare.tex b/notes/.GAelectrodynamics/2dvectorsquare.tex deleted file mode 100644 index 11604b909..000000000 --- a/notes/.GAelectrodynamics/2dvectorsquare.tex +++ /dev/null @@ -1,71 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{}{problem:multiplication:2dvectorsquare}{ -Generalize the calculation of \cref{eqn:gaTutorial:80} to calculate the square of an \R{n} vector. - -\begin{dmath}\label{eqn:multiplication:100} -\Bx = \sum_i x_i \Be_i -\end{dmath} -} % problem - -\makeanswer{problem:multiplication:2dvectorsquare}{ -Consider the 2D case to start with - -\begin{dmath}\label{eqn:multiplication:120} -\Bx^2 -= -\lr{ x \Be_1 + y \Be_2} -\lr{ x \Be_1 + y \Be_2} -= -\lr{ x \Be_1 } \lr{ x \Be_1 } -+ -\lr{ y \Be_2 } \lr{ y \Be_2 } -+ -\lr{ x \Be_1 } \lr{ y \Be_2 } -+ -\lr{ y \Be_2 } \lr{ x \Be_1 } -= -x^2 \Be_1^2 -+ -y^2 \Be_2^2 -+ -x y \lr{ \Be_1 \Be_2 + \Be_2 \Be_1 } -= -x^2 + y^2 -+ -x y \lr{ \Be_1 \Be_2 + \Be_2 \Be_1 }. -\end{dmath} - -The contraction axiom requires the bivector terms to sum to zero, as also demonstrated previously for the specific example \( \Bx = \Be_1 + \Be_2 \). - -More generally for \R{N} - -\begin{dmath}\label{eqn:multiplication:121} -\Bx^2 -= -\lr{ \sum_i x_i \Be_i } -\lr{ \sum_j x_j \Be_j } -= -\sum_{ij} x_i x_j \Be_i \Be_j -= -\sum_{i = j} x_i x_j \Be_i \Be_j -+ -\sum_{i \ne j} x_i x_j \Be_i \Be_j -= -\sum_{i} x_i^2 -+ -\sum_{i \ne j} x_i x_j \Be_i \Be_j -= -\sum_{i} x_i^2 -+ -\sum_{i < j} x_i x_j (\Be_i \Be_j + \Be_j \Be_i). -\end{dmath} - -The contraction axiom requires all the bivector pairs to sum to zero. That is, for each \( i \ne j \) - -\begin{dmath}\label{eqn:introGAproblems:140} -\Be_i \Be_j = -\Be_j \Be_i. -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/2subspaceR3reciprocalExample.tex b/notes/.GAelectrodynamics/2subspaceR3reciprocalExample.tex deleted file mode 100644 index 105104123..000000000 --- a/notes/.GAelectrodynamics/2subspaceR3reciprocalExample.tex +++ /dev/null @@ -1,75 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Two vector reciprocal frame}{problem:2subspaceR3reciprocalExample:2subspaceR3reciprocalExample}{ -Calculate the reciprocal frame for the \R{3} subspace spanned by \( \setlr{ \Bx_1, \Bx_2 } \) where - -\begin{dmath}\label{eqn:2subspaceR3reciprocalExample:20} -\begin{aligned} -\Bx_1 &= \Be_1 + 2 \Be_2 \\ -\Bx_2 &= \Be_2 - \Be_3. -\end{aligned} -\end{dmath} -} % problem - -\makeanswer{problem:2subspaceR3reciprocalExample:2subspaceR3reciprocalExample}{ -The bivector for the plane spanned by this basis is - -\begin{dmath}\label{eqn:2subspaceR3reciprocalExample:40} -\Bx_1 \wedge \Bx_2 -= -\lr{ \Be_1 + 2 \Be_2 } \wedge -\lr{ \Be_2 - \Be_3 } -= -\Be_{12} - \Be_{13} - 2 \Be_{23} -= -\Be_{12} + \Be_{31} + 2 \Be_{32}. -\end{dmath} - -This has the square -\begin{dmath}\label{eqn:2subspaceR3reciprocalExample:60} -\lr{ \Bx_1 \wedge \Bx_2 }^2 -= -\lr{ \Be_{12} + \Be_{31} + 2 \Be_{32} } -\cdot -\lr{ \Be_{12} + \Be_{31} + 2 \Be_{32} } -= --1 -1 -4 -= --6. -\end{dmath} - -Dotting \( -\Bx_1 \) with the bivector is -\begin{dmath}\label{eqn:2subspaceR3reciprocalExample:80} -\Bx_1 \cdot \lr{ \Bx_2 \wedge \Bx_1 } -= --\lr{ \Be_1 + 2 \Be_2 } \cdot \lr{\Be_{12} + \Be_{31} + 2 \Be_{32} } -= --\lr{ \Be_2 - \Be_3 - 2 \Be_1 - 4 \Be_3 } -= 2 \Be_1 - \Be_2 + 5 \Be_3, -\end{dmath} - -For \( \Bx_2 \) the dot product with the bivector is - -\begin{dmath}\label{eqn:2subspaceR3reciprocalExample:100} -\Bx_2 \cdot \lr{ \Bx_1 \wedge \Bx_2 } -= -\lr{ \Be_2 - \Be_3 } \cdot \lr{\Be_{12} + \Be_{31} + 2 \Be_{32} } -= -- \Be_1 - 2 \Be_3 - \Be_1 - 2 \Be_2 -= -- 2 \Be_1 - 2 \Be_2 - 2 \Be_3, -\end{dmath} - -so -\begin{dmath}\label{eqn:2subspaceR3reciprocalExample:120} -\begin{aligned} -\Bx^1 &= \inv{3} \lr{ \Be_1 + \Be_2 + \Be_3 } \\ -\Bx^2 &= \inv{6} \lr{ -2 \Be_1 + \Be_2 - 5 \Be_3 }. -\end{aligned} -\end{dmath} - -It is easy to verify that this has the desired semantics. -} % answer diff --git a/notes/.GAelectrodynamics/ComplexInnerProductVsDotAndCrossProduct.tex b/notes/.GAelectrodynamics/ComplexInnerProductVsDotAndCrossProduct.tex deleted file mode 100644 index 65f6b864e..000000000 --- a/notes/.GAelectrodynamics/ComplexInnerProductVsDotAndCrossProduct.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Complex inner product vs. dot and cross product.}{problem:introGAproblems:ComplexInnerProductVsDotAndCrossProduct}{ -Given two 2D vectors \( (a,b) \) and \( (a', b') \), and a complex number representation of these vectors \( z = a + ib, w = a' + i b' \), show that the components of the complex inner product have the representation -given by \cref{eqn:GAmotivation:220}. -} % problem - -\makeanswer{problem:introGAproblems:ComplexInnerProductVsDotAndCrossProduct}{ -\begin{dmath}\label{eqn:introGAproblems:20} -z w^\conj -= -(a + ib)(a' - ib') -= -a a' + b b' -+ i \lr{ a' b - a b' } -\leftrightarrow -( a a' + b b', a' b - a b' ). -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/FrontBackmatter/preface.tex b/notes/.GAelectrodynamics/FrontBackmatter/preface.tex deleted file mode 100644 index be92698cc..000000000 --- a/notes/.GAelectrodynamics/FrontBackmatter/preface.tex +++ /dev/null @@ -1,35 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -% -%\chapter{Preface} -% this suppresses an explicit chapter number for the preface. -\chapter*{Preface}%\normalsize - \addcontentsline{toc}{chapter}{Preface} - -A book on Geometric Algebra applications to electromagnetism. - -The target audience are undergraduate student with sufficient background in electromagnetism that knowledge of Maxwell's equations can be presumed. -Alternatives to the usual 3D vector or coordinate based derivations of various problems will be presented that highlight some of the ways that Geometric Algebra can be used to tackle electromagnetic problems. -%the students that come from physics but are still undergraduate. I can possibly introduce some material also to antenna students (just to derive the classical Hertz dipole and magnetic dipole radiation in a (possibly) simpler way). - -%\withproblemsetsMessage{ -%\textcolor{Maroon}{ -%\textit{THIS DOCUMENT IS REDACTED. THE PROBLEM SET SOLUTIONS AND ASSOCIATED MATHEMATICA CODE IS NOT VISIBLE. PLEASE EMAIL ME FOR THE FULL VERSION IF YOU ARE NOT TAKING THIS COURSE.} -%} -%} - -\paragraph{These notes contain:} - -\begin{itemize} -\item An introduction to Geometric Algebra (GA). -\item Application of Geometric Algebra to electromagnetism, with a focus on engineering applications. -\end{itemize} - -There are two potential audiences for these notes. The first is for a student new to GA faced with the learning curve of both GA itself and the notational changes needed to apply it to electromagnetism. The second audience is the individual who already has some knowledge of GA, and may want to skim yet another boilerplate ``Introduction to Geometric Algebra'' to get an idea of the notation in use, and move on to the electromagnetism applications. To serve both potential audiences, much of the substance of the introductory GA material has been deferred to the problems. Students new to GA should attempt all these problems before falling back to just reading the solutions. - -Peeter Joot \quad peeterjoot@protonmail.com - -Prof. Mauro Mongiardo \quad mauro.mongiardo@gmail.com diff --git a/notes/.GAelectrodynamics/GAelectrodynamics.sty b/notes/.GAelectrodynamics/GAelectrodynamics.sty deleted file mode 100644 index 36f042b2d..000000000 --- a/notes/.GAelectrodynamics/GAelectrodynamics.sty +++ /dev/null @@ -1,55 +0,0 @@ -%\usepackage{siunitx} -%\usepackage{esint} % \oiint - -%\newcommand{\timeaverage}[1]{\left[#1\right]} - -\newcommand{\nbref}[1]{% -\itemRef{GAelectrodynamics}{#1}% -\index{Mathematica}% -} - -% with an alternate label for the link. -% {nb}{text} -% nb of the form: ps2b:countItersAndPlot.m -\newcommand{\nbcite}[2]{% -\itemCite{GAelectrodynamics}{#1}{#2}% -} - -\usepackage{peeters_figures} - -% \mathImageFigure{path}{caption}{label}{width}{nbpath} -% nbpath like: ps2b:countItersAndPlot.m -\newcommand{\mathImageFigure}[5]{% -\imageFigure{#1}{\nbcite{#5}{#2}}{#3}{#4} -} - -% \mathImageTwoFigures{path1}{path2}{caption}{fig:fff}{scale=0.3}{nbpath} -\newcommand{\mathImageTwoFigures}[6]{% -\imageTwoFigures{#1}{#2}{\nbcite{#6}{#3}}{#4}{#5} -} - -% \mathImageThreeFiguresOneLine{path1}{path2}{path3}{caption}{fig:fff}{scale=0.3}{nbpath} -\newcommand{\mathImageThreeFiguresOneLine}[7]{% -\imageThreeFiguresOneLine{#1}{#2}{#3}{\nbcite{#7}{#4}}{#5}{#6} -} - -% \mathImageFourFiguresTwoLines{path1}{path2}{path3}{path4}{caption}{fig:fff}{scale=0.3}{nbpath} -\newcommand{\mathImageFourFiguresTwoLines}[8]{% -\imageFourFiguresTwoLines{#1}{#2}{#3}{#4}{\nbcite{#8}{#5}}{#6}{#7} -} - -\newcommand{\matlabFunc}[1]{% -\textbf{#1}% -} - -% {func}{path} -\newcommand{\matlabFuncPath}[2]{% -\nbcite{#2}{\textbf{#1}}% -} - -%\usepackage{macros_bm} -%\usepackage{macros_mathematica} -% -%\newcommand{\hatcalP}[0]{% -%\widehat{\calP}% -%} diff --git a/notes/.GAelectrodynamics/GAmotivation.tex b/notes/.GAelectrodynamics/GAmotivation.tex deleted file mode 100644 index eafc7c42d..000000000 --- a/notes/.GAelectrodynamics/GAmotivation.tex +++ /dev/null @@ -1,168 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -A new student of vector algebra will first learn -%first see vectors in two or three dimensions as sets of coordinates -% -%\begin{equation}\label{eqn:GAmotivation:20} -%\Ba = -%\begin{bmatrix} -%a_1 \\ -%a_2 \\ -%a_3 \\ -%\end{bmatrix}, \qquad -%\Bb = -%\begin{bmatrix} -%b_1 \\ -%b_2 \\ -%b_3 \\ -%\end{bmatrix}, -%\end{equation} -% -%or perhaps explicitly in terms of a basis \( \setlr{ \Be_1, \Be_2 \Be_3 } \) -% -%\begin{dmath}\label{eqn:GAmotivation:40} -%\begin{aligned} -%\Ba &= a_1 \Be_1 + a_2 \Be_2 + a_3 \Be_3 \\ -%\Bb &= b_1 \Be_1 + b_2 \Be_2 + b_3 \Be_3 -%\end{aligned}. -%\end{dmath} -% -%You will learn the -rules for addition and subtraction of such vectors. -%, and then how to operate on them with rotation matrices or other representations of linear transformations. -This demonstrates to the student that the vector is an algebraic object that generalize numbers, and the question of how to -multiply vectors soon follows. - -Given the toolbox of traditional vector algebra, the best answer that a new student will obtain from such a line of questioning is to learn of the dot and cross products, the multiplication like operations that we are all so familiar with - -%\begin{itemize} -%\item ``You can not multiply vectors.'', or -%\item ``Vector multiplication is not well defined.'', or -%\item ``We will get to that.'', or -%\item ``There are multiplication like operations.'' The -%\end{itemize} - -\begin{dmath}\label{eqn:GAmotivation:60} -\begin{aligned} -\Ba \cdot \Bb &= a_1 b_1 + a_2 b_2 + a_3 b_3 = \Abs{\Ba} \Abs{\Bb} \cos \theta_{ab} \\ -\Ba \cross \Bb &= -\begin{vmatrix} -\Be_1 & \Be_2 & \Be_3 \\ -a_1 & a_2 & a_3 \\ -b_1 & b_2 & b_3 \\ -\end{vmatrix} -= \ncap_{ab} \Abs{\Ba} \Abs{\Bb} \sin\theta_{ab}. -\end{aligned} -\end{dmath} - -Both of these multiplication like operations live in very different spaces, one scalar, and the other a vector that lies outside of the span of its two vector factors. Observe that the magnitudes of these two product operations are related to the product of the vectors in a Pythagorean sense - -\begin{dmath}\label{eqn:GAmotivation:180} -\lr{ \Ba \cdot \Bb }^2 + \lr{ \Ba \cross \Bb }^2 -= -\Abs{\Ba}^2 \Abs{\Bb}^2 \cos^2 \theta_{ab} -+\Abs{\Ba}^2 \Abs{\Bb}^2 \sin^2 \theta_{ab} -= -\Abs{\Ba}^2 \Abs{\Bb}^2. -\end{dmath} - -This can be seen as a hint that the dot and cross products might be components of a single vector product operation, but the precise form of that product is not obvious. - -Vector products that have the same form as the scalar magnitudes of the dot and cross products can be found in other algebraic systems. Given a complex number representation of two vectors in a 2D space - -\begin{dmath}\label{eqn:GAmotivation:200} -\begin{aligned} -z &= r e^{i \theta} \leftrightarrow (a, b) \\ -w &= \rho e^{i \alpha} \leftrightarrow (a', b'), -\end{aligned} -\end{dmath} - -the inner product of such a complex vector representation can be seen to have the same structure as the dot and cross products - -\begin{equation}\label{eqn:GAmotivation:100} -\begin{aligned} -\Real( z w^\conj ) &= r \rho \cos(\theta - \alpha) \\ -\Imag( z w^\conj ) &= r \rho \sin(\theta - \alpha). -\end{aligned} -\end{equation} - -It can be shown -(\cref{problem:introGAproblems:ComplexInnerProductVsDotAndCrossProduct}) -that this inner product has the following vector isomorphism - -\begin{dmath}\label{eqn:GAmotivation:220} -z w^\conj \leftrightarrow ( a a' + b b', a' b - a b' ). -\end{dmath} - -One component is completely symmetric, whereas the other component of this product has a component that is completely antisymmetric. -The 3D cross product also has this antisymmetry, and that antisymmetry will be seen later to be the key to the generalization of the cross product. In this particular case, one can view this antisymmetric sum \( a' b - a b' \) as one -answer of how the cross product ``generalizes'' from 3D to 2D without requiring the introduction of a normal dimension. - -The answer to questions of exactly how the vector products, in particular the cross product, should generalize to higher dimensional spaces are still outstanding. It should be expected that this cross product generalization will involve antisymmetry, just as the dot product generalization in higher dimensional spaces is completely symmetric. - -Many current students of science never see the exact structure of this generalization. Should studies happen to include -enough of the right esoteric physics and mathematics (quantum mechanics, QED, calculus on manifolds, ...) then -some answers to those questions may be found. Unfortunately, there are many such answers, and many of them each only provide -one part of the picture. - -For example, a student of non-relativistic quantum mechanics will learn of Pauli matrices, when studying spin operators. The dot and cross products will be seen to be components of a more general vector multiplication operation - -\begin{equation}\label{eqn:GAmotivation:120} -\lr{\Bsigma \cdot \Bx } -\lr{\Bsigma \cdot \By } -= -I \lr{ \Bx \cdot \By } + i \Bsigma \cdot \lr{ \Bx \cross \By }. -\end{equation} - -In quaternion algebra, a generalization of complex algebra, when a quaternion is represented as a scalar vector pair \( q = (r, \Bv) \), the quaternion product of two vectors also shows that the dot and cross products are respective components of a product of vectors - -\begin{dmath}\label{eqn:GAmotivation:240} -(0, \Bx) -(0, \By) = (-\Bx \cdot \By, \Bx \cross \By). -\end{dmath} - -A student of quantum field theory will encounter Dirac matrices, a algebraic structure that allows for the multiplication of four-vectors - -\begin{dmath}\label{eqn:GAmotivation:140} -\aslash \bslash -= -\inv{2} \symmetric{ \aslash}{ \bslash } -+ -\inv{2} \antisymmetric{ \aslash}{ \bslash } -= -a^\mu b_\mu + \inv{2} a^\mu b^\nu \antisymmetric{\gamma_\mu}{\gamma_\nu} -= -a^\mu b_\mu + \inv{2} a^\mu b^\nu \lr{ -\gamma_\mu \gamma_\nu -- -\gamma_\nu \gamma_\mu -}. -\end{dmath} - -A product of ``Dirac'' vectors has symmetric and antisymmetric components that generalize the dot and cross products. -Unfortunately, this algebra comes with still another different notation. -One interesting take away from this particular vector product is the fact that one component is a scalar, and other other -involves products of vectors, something that will require further interpretation. Since the Dirac basis typically has a matrix representation, such a product can be dismissed as just being another matrix. The products of mutually orthonormal vectors will show up again later in a context where there is no requirement to assume a matrix representation of the underlying basis. - -Another common and important context that contains generalizations of the dot and cross products is the subject of differential forms. -A student of differential forms will learn how to compute the wedge products of forms, and of duality operations, which can be used to construct generalized multiplication operations that have the structure of the 3D dot and cross products - -\begin{equation}\label{eqn:GAmotivation:160} -\begin{aligned} -df \wedge * dg &= \lr{ \sum_{i=1}^3 \PD{x_i}{f} \PD{x_i}{g} } dx_1 \wedge dx_2 \wedge dx_3 \\ -df \wedge dg &= \sum_{1 \le i < j \le 3} \lr{ -\PD{x_i}{f} \PD{x_j}{g} --\PD{x_j}{f} \PD{x_i}{g} -} -dx_i \wedge dx_j. -\end{aligned} -\end{equation} - -It is possible to express vectors as a differential form, and some advocate for this \citep{flanders1989dfa}, but this can also seem unnatural. Regardless, differential forms do highlight the existence of more general concepts of vector multiplication. %In this particular case, this generality comes with the cost of using yet another notation, one that is considerably different than the vector notation that we are comfortable with. - -It should not be surprising that all of these ideas are special cases of a more general algebraic system. - -The aim of the material to follow is to provide the instruction manual for an enhanced toolbox of vector algebra techniques that can be used to gain an integrated view of many seemingly disparate mathematical methods. These are tools that can be learned without having to first study the esoteric arts of quantum mechanics or differential forms, and have many applications once learned. These notes will focus on applications to the study of electromagnetism. - diff --git a/notes/.GAelectrodynamics/GNUmakefile b/notes/.GAelectrodynamics/GNUmakefile deleted file mode 100644 index abad861b8..000000000 --- a/notes/.GAelectrodynamics/GNUmakefile +++ /dev/null @@ -1,43 +0,0 @@ -THISDIR := GAelectrodynamics -THISBOOK := GAelectrodynamics - -include ../latex/make.bookvars - -# Override my default: -#MY_CLASSICTHESIS_FRONTBACK_FILES := $(filter-out ../classicthesis_mine/FrontBackmatter/Dedication.tex,$(MY_CLASSICTHESIS_FRONTBACK_FILES)) - -#ONCEFLAGS := -justonce - -SOURCE_DIRS += appendix -FIGURES := ../../figures/$(THISBOOK) -SOURCE_DIRS += $(FIGURES) - -PRIMARY_SOURCES := $(shell grep input chapters.tex | sed 's/%.*//;s/.*{//;s/}.*//;') -PRIMARY_SOURCES += FrontBackmatter/preface.tex - -#GENERATED_SOURCES += matlab.tex -#GENERATED_SOURCES += mathematica.tex -#GENERATED_SOURCES += julia.tex - -EPS_FILES := $(wildcard $(FIGURES)/*.eps) -PDFS_FROM_EPS := $(subst eps,pdf,$(EPS_FILES)) - -THISBOOK_DEPS += $(PDFS_FROM_EPS) -#THISBOOK_DEPS += macros_mathematica.sty - -CLEAN_TARGETS += *.sp FrontBackmatter/*.sp - -include ../latex/make.rules - -.PHONY: spellcheck -spellcheck: $(patsubst %.tex,%.sp,$(PRIMARY_SOURCES)) - -%.sp : %.tex - spellcheck $^ - touch $@ - -.PHONY: copy -copy : $(HOME)/Dropbox/$(THISDIR)/$(THISBOOK).pdf - -$(HOME)/Dropbox/$(THISDIR)/$(THISBOOK).pdf : $(THISBOOK).pdf - cp $^ $@ diff --git a/notes/.GAelectrodynamics/METADATA b/notes/.GAelectrodynamics/METADATA deleted file mode 100644 index 4967b05a7..000000000 --- a/notes/.GAelectrodynamics/METADATA +++ /dev/null @@ -1,10 +0,0 @@ -sub GAelectrodynamicsMeta -{ -my @GAelectrodynamics = -( -) ; # @GAelectrodynamics - -return @GAelectrodynamics ; -} - -1 ; diff --git a/notes/.GAelectrodynamics/R3PseudoscalarCommutation.tex b/notes/.GAelectrodynamics/R3PseudoscalarCommutation.tex deleted file mode 100644 index b2f06c3c3..000000000 --- a/notes/.GAelectrodynamics/R3PseudoscalarCommutation.tex +++ /dev/null @@ -1,49 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{\R{3} pseudoscalar commutation.}{problem:gradeselection:R3PseudoscalarCommutation}{ -Show that \( I \) given by \cref{eqn:definitions:340} -commutes with any grade \R{3} multivector. -} % problem - -\makeanswer{problem:gradeselection:R3PseudoscalarCommutation}{ - -Showing that \( I \) commutes with each of the basis vectors is sufficient - -\begin{dmath}\label{eqn:gradeselectionProblems:620} -\Be_1 I -= -\Be_1 (\Be_1 \Be_2 \Be_3) -= -\Be_1 (-\Be_2 \Be_1) \Be_3 -= --\Be_1 \Be_2 (-\Be_3 \Be_1) -= -I \Be_1 -\end{dmath} -\begin{dmath}\label{eqn:gradeselectionProblems:640} -\Be_2 I -= -\Be_2 (\Be_1 \Be_2 \Be_3) -= -\Be_2 \Be_1 (-\Be_3 \Be_2) -= --(-\Be_1 \Be_2) \Be_3 \Be_2 -= -I \Be_2. -\end{dmath} -\begin{dmath}\label{eqn:gradeselectionProblems:660} -\Be_3 I -= -\Be_3 (\Be_1 \Be_2 \Be_3) -= -(\Be_3 \Be_1 \Be_2) \Be_3 -= --(\Be_1 \Be_3) \Be_2 \Be_3 -= --\Be_1 (-\Be_2 \Be_3) \Be_3 -= -I \Be_3. \qedmarker -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/R3PseudoscalarSquare.tex b/notes/.GAelectrodynamics/R3PseudoscalarSquare.tex deleted file mode 100644 index 827c81108..000000000 --- a/notes/.GAelectrodynamics/R3PseudoscalarSquare.tex +++ /dev/null @@ -1,41 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{\R{3} pseudoscalar square}{problem:gradeselection:R3PseudoscalarSquare}{ -With the \R{3} pseudoscalar of \cref{eqn:definitions:340} show that \( I^2 = -1 \). -} % problem - -\makeanswer{problem:gradeselection:R3PseudoscalarSquare}{ - -Squaring the pseudoscalar gives - -\begin{dmath}\label{eqn:gaTutorial:160} -I^2 -= -(\Be_1 \Be_2 \Be_3) -(\Be_1 \Be_2 \Be_3) -= -\Be_1 \Be_2 (\Be_3 -\Be_1) \Be_2 \Be_3 -= --\Be_1 \Be_2 \Be_1 -\Be_3 \Be_2 \Be_3 -= --\Be_1 (\Be_2 \Be_1) -(\Be_3 \Be_2) \Be_3 -= --\Be_1 (\Be_1 \Be_2) -(\Be_2 \Be_3) \Be_3 -= -- -\Be_1^2 -\Be_2^2 -\Be_3^2 -= --1, -\end{dmath} - -as expected, showing that this quantity also has characteristics of an imaginary number. -} % answer diff --git a/notes/.GAelectrodynamics/README.md b/notes/.GAelectrodynamics/README.md deleted file mode 100644 index f859dc739..000000000 --- a/notes/.GAelectrodynamics/README.md +++ /dev/null @@ -1,2 +0,0 @@ -# GAelectrodynamics -A book on Geometric Algebra applications to electromagnetism diff --git a/notes/.GAelectrodynamics/RnDotProduct.tex b/notes/.GAelectrodynamics/RnDotProduct.tex deleted file mode 100644 index 7e7ea9f21..000000000 --- a/notes/.GAelectrodynamics/RnDotProduct.tex +++ /dev/null @@ -1,49 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{\R{n} dot product.}{problem:gradeselection:RnDotProduct}{ -Show that \ref{dfn:gradeselection:100} when applied to two vectors -is equivalent to the traditional \R{n} dot product. -} % problem - -\makeanswer{problem:gradeselection:RnDotProduct}{ -Let -\begin{dmath}\label{eqn:gradeselectionProblems:180} -\begin{aligned} -\Bx &= \sum_{i=1}^N x_i \Be_i \\ -\By &= \sum_{i=1}^N y_i \Be_i. -\end{aligned} -\end{dmath} - -The dot product of these two vectors is -\begin{dmath}\label{eqn:gradeselectionProblems:200} -\Bx \cdot \By -\equiv -\gpgradezero{ \Bx \By } -= -\gpgradezero{ -\lr{ \sum_{i=1}^N x_i \Be_i} -\lr{ \sum_{j=1}^N y_j \Be_j} -} -= -\sum_{1 \le i = j \le N} -x_i y_j -\gpgradezero{ \Be_i \Be_j } -+ -\sum_{1 \le i \ne j \le N} -x_i y_j -\gpgradezero{ \Be_i \Be_j } -\end{dmath} - -In the \( i = j \) sum, the term \( \Be_i \Be_j = \Be_i^2 = 1 \), so the scalar grade selection of that multivector product is just 1. In the \( i = j \) term, each of the \( \Be_i \Be_j \) products is a bivector, so each of those scalar grade selections is zero. - -That leaves - -\begin{dmath}\label{eqn:gradeselectionProblems:220} -\Bx \cdot \By -= -\sum_{i =1}^N x_i y_i. \qedmarker -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/TODO b/notes/.GAelectrodynamics/TODO deleted file mode 100644 index f7d916b95..000000000 --- a/notes/.GAelectrodynamics/TODO +++ /dev/null @@ -1,19 +0,0 @@ -- have boilerplate copyright. - -- dedication.tex got cleaned out on make clean. Rule that was in makefile doesn't appear to be right. Revisit. - -- strip document version info - -- exercise headers are bigger than the section headers (partially fixed) - -- have inconsistent use of qedmarker. - -- regenerate Oblique and recip. figure: - with the e_1 label over the vector at the end of it. - with the e_2 label over the vector at the end of it. - with the upper e_2 label and arrow removed. - with the e^1 lable moved to the end of the arrow (colinear with it). - with the vectors in bold. - with x -> x = (4,2) - with a point labling x more clearly. - with the text of the projections centered over the lines. diff --git a/notes/.GAelectrodynamics/WedgeRelationshipToCrossProduct.tex b/notes/.GAelectrodynamics/WedgeRelationshipToCrossProduct.tex deleted file mode 100644 index 4c558e577..000000000 --- a/notes/.GAelectrodynamics/WedgeRelationshipToCrossProduct.tex +++ /dev/null @@ -1,82 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Wedge relationship to the cross product.}{problem:gradeselection:WedgeRelationshipToCrossProduct}{ -For a pair of \R{3} vectors \( \Bx, \By \), show that the wedge and cross products are related by -\begin{dmath}\label{eqn:gradeselectionProblems:560} -\Bx \wedge \By = I (\Bx \cross \By), -\end{dmath} - -where \( I = \Be_1 \Be_2 \Be_3 \) is the \R{3} pseudoscalar. -} % problem - -\makeanswer{problem:gradeselection:WedgeRelationshipToCrossProduct}{ -Writing out \cref{eqn:gradeselectionProblems:580} explicitly gives - -\begin{dmath}\label{eqn:gradeselectionProblems:600} -\Bx \wedge \By -= -\begin{vmatrix} -x_1 & x_2 \\ -y_1 & y_2 -\end{vmatrix} -\Be_1 \Be_2 -+ -\begin{vmatrix} -x_1 & x_3 \\ -y_1 & y_3 -\end{vmatrix} -\Be_1 \Be_3 -+ -\begin{vmatrix} -x_2 & x_3 \\ -y_2 & y_3 -\end{vmatrix} -\Be_2 \Be_3 -= -\begin{vmatrix} - \Be_2 \Be_3 -& \Be_3 \Be_1 -& \Be_1 \Be_2 \\ -x_1 & x_2 & x_3 \\ -y_1 & y_2 & y_3 -\end{vmatrix} -= -\begin{vmatrix} - (\Be_1 \Be_1) \Be_2 \Be_3 -& \Be_3 \Be_1 (\Be_2 \Be_2) -& \Be_1 \Be_2 (\Be_3 \Be_3) \\ -x_1 & x_2 & x_3 \\ -y_1 & y_2 & y_3 -\end{vmatrix} -= -\Be_1 \Be_2 \Be_3 -\begin{vmatrix} -\Be_1 & \Be_2 & \Be_3 \\ -x_1 & x_2 & x_3 \\ -y_1 & y_2 & y_3 -\end{vmatrix} -= I (\Bx \cross \By). -\end{dmath} -%\begin{aligned} -%(x_1 y_2 - x_2 y_1) \Be_1 \Be_2 \\ -%&\quad+ -%(x_1 y_3 - x_3 y_1) \Be_1 \Be_3 \\ -%&\quad+ -%(x_2 y_3 - x_3 y_2) \Be_2 \Be_3 \\ -%&= -%(x_1 y_2 - x_2 y_1) \Be_1 \Be_2 (\Be_3 \Be_3) \\ -%&\quad+ -%(x_1 y_3 - x_3 y_1) \Be_1 \Be_3 (\Be_2 \Be_2) \\ -%&\quad+ -%(x_2 y_3 - x_3 y_2) \Be_2 \Be_3 (\Be_1 \Be_1) \\ -%&= -%(x_1 y_2 - x_2 y_1) I \Be_3 \\ -%&\quad+ -%(x_1 y_3 - x_3 y_1) (-I) \Be_2 \\ -%&\quad+ -%(x_2 y_3 - x_3 y_2) I \Be_1 \\ -%\end{aligned} -} % answer diff --git a/notes/.GAelectrodynamics/appendixchapters.tex b/notes/.GAelectrodynamics/appendixchapters.tex deleted file mode 100644 index 7de6f7892..000000000 --- a/notes/.GAelectrodynamics/appendixchapters.tex +++ /dev/null @@ -1,14 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%\chapter{Useful formulas and review} -% \input{usefulFormulas.tex} -%\chapter{Geometric Algebra} -% \input{gaQuickTutorial.tex} -%\chapter{Mathematica notebooks} -% \input{mathematica.tex} -%\chapter{Matlab notebooks} -% \input{matlab.tex} -%\chapter{Julia notebooks} -% \input{julia.tex} diff --git a/notes/.GAelectrodynamics/backmatter.tex b/notes/.GAelectrodynamics/backmatter.tex deleted file mode 100644 index 5a0186c48..000000000 --- a/notes/.GAelectrodynamics/backmatter.tex +++ /dev/null @@ -1,25 +0,0 @@ -% -% Copyright © 2016 Peeter Joot, Mauro Mongiardo. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -% ******************************************************************** -% Backmatter -%******************************************************* -\part{Appendices} -\appendix -\cleardoublepage -\input{appendixchapters} - -%******************************************************************** -% Other Stuff in the Back -%******************************************************* - -\part{Index} -%\part{Back matter} -%\chapter{Index} -\printindex - -\part{Bibliography} -\cleardoublepage\include{FrontBackmatter/Bibliography} -%\cleardoublepage\include{FrontBackmatter/Colophon} -%\cleardoublepage\include{FrontBackmatter/Declaration} diff --git a/notes/.GAelectrodynamics/bivectorDot.tex b/notes/.GAelectrodynamics/bivectorDot.tex deleted file mode 100644 index 57a0d8280..000000000 --- a/notes/.GAelectrodynamics/bivectorDot.tex +++ /dev/null @@ -1,99 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{Dot product expansion of two bivectors.}{problem:multiplication:bivectorDot}{ -Show that - -\boxedEquation{eqn:bivectorDot:20}{ -(\Ba \wedge \Bb) \cdot (\Bc \wedge \Bd) -= -((\Ba \wedge \Bb) \cdot (\Bc) \cdot \Bd, -} - -and hence -\boxedEquation{eqn:bivectorDot:40}{ -(\Ba \wedge \Bb) \cdot (\Bc \wedge \Bd) -= -(\Bb \cdot \Bc) (\Ba \cdot \Bd) --(\Ba \cdot \Bc)( \Bb \cdot \Bd). -} -} % problem - -\makeanswer{problem:multiplication:bivectorDot}{ - -\begin{dmath}\label{eqn:bivectorDot:60} -(\Ba \wedge \Bb) \cdot (\Bc \wedge \Bd) -= -\gpgradezero{ -(\Ba \wedge \Bb) (\Bc \wedge \Bd) -} -= -\gpgradezero{ -(\Ba \wedge \Bb) (\Bc \Bd - \cancel{\Bc \cdot \Bd}) -} -= -\gpgradezero{ -\lr{ -(\Ba \wedge \Bb) \cdot \Bc -+ \cancel{(\Ba \wedge \Bb) \wedge \Bc } -} -\Bd -} -= -\gpgradezero{ -((\Ba \wedge \Bb) \cdot \Bc ) \cdot \Bd -+ -\cancel{((\Ba \wedge \Bb) \cdot \Bc ) \wedge \Bd} -} -= -((\Ba \wedge \Bb) \cdot \Bc ) \cdot \Bd. -\end{dmath} - -Above, any product that could not possibly contribute a scalar grade has been cancelled. The remains are now straightforward to expand - -\begin{dmath}\label{eqn:bivectorDot:80} -((\Ba \wedge \Bb) \cdot \Bc ) \cdot \Bd -= -( -\Ba (\Bb \cdot \Bc) -- -\Bb (\Ba \cdot \Bc) -) -\cdot \Bd -= -(\Ba \cdot \Bd) (\Bb \cdot \Bc) -- -(\Bb \cdot \Bd) (\Ba \cdot \Bc). -\end{dmath} - -Alternatively, this result can be obtained compactly using tensort contraction techniques, first making a duality transformation and then expanding in coordinates - -\begin{dmath}\label{eqn:bivectorDot:100} -(\Ba \wedge \Bb) \cdot (\Bc \wedge \Bd) -= -\gpgradezero{ --I (\Ba \cross \Bb) (-I) (\Bc \cross \Bd) -} -= -- (\Ba \cross \Bb) \cdot (\Bc \cross \Bd) -= --(\epsilon_{ijk} \Be_i a_j b_k) \cdot (\epsilon_{r s t} \Be_r c_s d_t) -= --\epsilon_{ijk} a_j b_k \epsilon_{i s t} c_s d_t -= --\delta_{jk}^{[st]} -a_j b_k c_s d_t -= --a_s b_t c_s d_t -+ -a_t b_s c_s d_t -= --(\Ba \cdot \Bc)(\Bb \cdot \Bd) -+ -(\Ba \cdot \Bd)(\Bb \cdot \Bc). -\end{dmath} - -A student of physics might consider this a natural alternative approach. -%Some GA authors may find this alternate derivation offensive, as it contains both an expansion by coordinates and requires an alternate toolbox. -} % answer diff --git a/notes/.GAelectrodynamics/chapters.tex b/notes/.GAelectrodynamics/chapters.tex deleted file mode 100644 index 865726c36..000000000 --- a/notes/.GAelectrodynamics/chapters.tex +++ /dev/null @@ -1,119 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%---------------------------------------------------------------------------------------- -\part{Geometric Algebra} - \chapter{Basics} - \section{Did you ever ask your teacher how to multiply vectors?} - \input{GAmotivation.tex} - \subsection{Problems} - \input{ComplexInnerProductVsDotAndCrossProduct.tex} - \section{Vector multiplication} - \input{multiplication.tex} - \subsection{Problems} - \input{2dvectorsquare.tex} - \input{normalAnticommutation.tex} - \section{Definitions} - \input{definitions.tex} - \subsection{Problems} - \input{R3PseudoscalarSquare.tex} - \section{Grade selection, dot and wedge product operators} - \input{gradeselection.tex} - \subsection{Problems} - \input{RnDotProduct.tex} - \input{cyclicpermutationtwo.tex} - \input{dotprodSymmetricSum.tex} - \input{planeRotationsExponentials.tex} - \input{complexNumbers.tex} - \input{R3PseudoscalarCommutation.tex} - \input{gradeselVectorWedge.tex} - \input{WedgeRelationshipToCrossProduct.tex} - \input{vectorBivectorDot.tex} - \input{vectorTrivectorDot.tex} - \input{bivectorDot.tex} - \input{r4nonzerobivectorwedgewithself.tex} - \section{Product of two vectors} - \input{vectorproduct.tex} - \subsection{Problems} - \input{vectorproductCyclicPermutation.tex} - \input{wedgeantisym.tex} - \input{gradethreeselectionWedge.tex} - \section{Problem solutions} - \shipoutAnswer - \chapter{Geometry} - \section{Bivectors} -gabookI: 3.9 - \section{Problem solutions} - \shipoutAnswer - \section{Trivectors} - \section{Problem solutions} - \shipoutAnswer - \section{Projection and rejection} - \section{Problem solutions} - \shipoutAnswer - \section{Rotations} -gabookI: 2.5 rotations. -gabookI: 10.4.3 bivector generator of rotations. -gabookI: 29.1 - \section{Problem solutions} - \shipoutAnswer - \section{Reciprocal frames} - \input{reciprocal.tex} - \subsection{Problems} - \input{2dreciprocalMatrixCalculation.tex} - \input{2subspaceR3reciprocalExample.tex} - \section{Equivalent identities} -gabookI: 4.1+ - \section{Cramer's rule} -gabookI: 5. Generalize examples to higher dimensions. - \section{Problem solutions} - \shipoutAnswer - \chapter{Vector calculus} - \section{Curvilinear coordinates} -gabook: 31.1 - \section{Stokes theorem} - \section{Divergence theorem} - \section{Fundamental theorem of geometric calculus} - \section{Helmholtz theorem} -gabook: 45.1 - \section{Problem solutions} - \shipoutAnswer - -\part{Electromagnetism} - \chapter{Maxwell's equations} - \section{Problem solutions} - \shipoutAnswer - \chapter{Electrostatics} - \section{Problem solutions} - \shipoutAnswer - \chapter{Magnetostatics} - \section{Problem solutions} - \shipoutAnswer - \chapter{Constitutive relations} - \section{Problem solutions} - \shipoutAnswer - \chapter{Boundary value conditions} - \section{Problem solutions} - \shipoutAnswer - \chapter{Time harmonic fields} - \section{Problem solutions} - \shipoutAnswer - \chapter{Polarization} - \section{Problem solutions} - \shipoutAnswer - \chapter{Potentials} - \section{Problem solutions} - \shipoutAnswer - \chapter{Green's functions} - \section{Problem solutions} - \shipoutAnswer - \chapter{Wave equations} - \section{Problem solutions} - \shipoutAnswer - \chapter{Radiation and scattering} - \section{Problem solutions} - \shipoutAnswer -%\end{itemize} - - diff --git a/notes/.GAelectrodynamics/classicthesis-config.tex b/notes/.GAelectrodynamics/classicthesis-config.tex deleted file mode 100644 index 9991a0f92..000000000 --- a/notes/.GAelectrodynamics/classicthesis-config.tex +++ /dev/null @@ -1,364 +0,0 @@ -% **************************************************************************************************** -% classicthesis-config.tex -% formerly known as loadpackages.sty, classicthesis-ldpkg.sty, and classicthesis-preamble.sty -% Use it at the beginning of your ClassicThesis.tex, or as a LaTeX Preamble -% in your ClassicThesis.{tex,lyx} with \input{classicthesis-config} -% **************************************************************************************************** -% If you like the classicthesis, then I would appreciate a postcard. -% My address can be found in the file ClassicThesis.pdf. A collection -% of the postcards I received so far is available online at -% http://postcards.miede.de -% **************************************************************************************************** - -% **************************************************************************************************** -% 1. Configure classicthesis for your needs here, e.g., remove "drafting" below -% in order to deactivate the time-stamp on the pages -% **************************************************************************************************** -\PassOptionsToPackage{eulerchapternumbers,listings,% - pdfspacing,% - subfig,beramono,parts}{classicthesis} -% ******************************************************************** -% Available options for classicthesis.sty -% (see ClassicThesis.pdf for more information): -% drafting -% parts nochapters linedheaders -% eulerchapternumbers beramono eulermath pdfspacing minionprospacing -% tocaligned dottedtoc manychapters -% listings floatperchapter subfig -% ******************************************************************** - -% ******************************************************************** -% -% http://tex.stackexchange.com/questions/3676/too-many-math-alphabets-error -% -% adding \usepackage{txfonts} % \ointclockwise -% -% caused too many alphabets error. se workaround: - -\newcommand{\bmmax}{0} -\newcommand{\hmmax}{0} - -% ******************************************************************** -% Triggers for this config -% ******************************************************************** -\usepackage{ifthen} -\usepackage{censor} -\newboolean{enable-backrefs} % enable backrefs in the bibliography -\setboolean{enable-backrefs}{true} % true false - -\newboolean{redacted} -\usepackage{redacted} - -\newcommand{\withproblemsets}[1]{% -\ifthenelse{\boolean{redacted}}% -{% -\textcolor{Maroon}{% -PROBLEM SET RELATED MATERIAL REDACTED IN THIS DOCUMENT.% -% -PLEASE FEEL FREE TO EMAIL ME FOR THE FULL VERSION IF YOU AREN'T TAKING THE COURSE.% -}% -% -\input{censor1pg.tex}% -% -\textcolor{Maroon}{% -END-REDACTION% -}% -}% -{#1}% -} - -\newcommand{\withproblemsetsParagraph}[1]{% -\ifthenelse{\boolean{redacted}}% -{% -\textcolor{Maroon}{% -PROBLEM SET RELATED MATERIAL REDACTED IN THIS DOCUMENT.% -% -PLEASE FEEL FREE TO EMAIL ME FOR THE FULL VERSION IF YOU AREN'T TAKING THE COURSE.% -}% -% -\input{censor1p.tex}% -%\input{censor1p.tex}% -%\input{censor1p.tex}% -% -\textcolor{Maroon}{% -END-REDACTION% -}% -}% -{#1}% -} - -\newcommand{\withproblemsetsMessage}[1]{% -\ifthenelse{\boolean{redacted}}% -{% -#1% -}% -{\relax}% -} - -% **************************************************************************************************** - - -% **************************************************************************************************** - - -% **************************************************************************************************** -% 2. Personal data and user ad-hoc commands -% **************************************************************************************************** -\newcommand{\myTitle}{Geometric Algebra for Electromagnetism\xspace} -\newcommand{\mySubtitle}{Exploring GA applications to electromagnetism.\xspace} -\newcommand{\myDegree}{} -\newcommand{\myName}{Peeter Joot, Prof. Mauro Mongiardo\xspace} -\newcommand{\myProf}{} -\newcommand{\myOtherProf}{} -\newcommand{\mySupervisor}{} -\newcommand{\myFaculty}{} -\newcommand{\myDepartment}{} -\newcommand{\myUni}{} -\newcommand{\myLocation}{} -%---------------------- - \input{./.revinfo/gitCommitDateAsMyTime.tex} -%---------------------- -\newcommand{\myVersion}{version v.0\xspace} - -% ******************************************************************** -% Setup, finetuning, and useful commands -% ******************************************************************** -\newcounter{dummy} % necessary for correct hyperlinks (to index, bib, etc.) -\newlength{\abcd} % for ab..z string length calculation -\providecommand{\mLyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@} -%\newcommand{\ie}{i.\,e.} -%\newcommand{\Ie}{I.\,e.} -%\newcommand{\eg}{e.\,g.} -%\newcommand{\Eg}{E.\,g.} -% **************************************************************************************************** - - -% **************************************************************************************************** -% 3. 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Last calls before the bar closes -% **************************************************************************************************** -% ******************************************************************** -% Development Stuff -% ******************************************************************** -\listfiles -%\PassOptionsToPackage{l2tabu,orthodox,abort}{nag} -% \usepackage{nag} -%\PassOptionsToPackage{warning, all}{onlyamsmath} -% \usepackage{onlyamsmath} - -% ******************************************************************** -% Last, but not least... -% ******************************************************************** -\usepackage{classicthesis} -% **************************************************************************************************** - - -% **************************************************************************************************** -% 8. Further adjustments (experimental) -% **************************************************************************************************** -% ******************************************************************** -% Changing the text area -% ******************************************************************** -%\linespread{1.05} % a bit more for Palatino -%\areaset[current]{312pt}{761pt} % 686 (factor 2.2) + 33 head + 42 head \the\footskip -%\setlength{\marginparwidth}{7em}% -%\setlength{\marginparsep}{2em}% - -% ******************************************************************** -% Using different fonts -% ******************************************************************** -%\usepackage[oldstylenums]{kpfonts} % oldstyle notextcomp -%\usepackage[osf]{libertine} -%\usepackage{hfoldsty} % Computer Modern with osf -%\usepackage[light,condensed,math]{iwona} -%\renewcommand{\sfdefault}{iwona} -%\usepackage{lmodern} % <-- no osf support :-( -%\usepackage[urw-garamond]{mathdesign} <-- no osf support :-( -% **************************************************************************************************** - -%---------------------------------------------------------------------------------------- -% peeter's mods from from mylayout.sty -%---------------------------------------------------------------------------------------- - -% http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Layout#The_Easy_Way -% replaces manual settings (\parindent, \parskip, \topmargin, ...) -%\usepackage[margin=3.0cm]{geometry} -\usepackage[left=3cm,right=4cm]{geometry} - -\PassOptionsToPackage{answerdelayed}{exercise} -\usepackage{peeters_layout} -\usepackage{thisbook} diff --git a/notes/.GAelectrodynamics/complexNumbers.tex b/notes/.GAelectrodynamics/complexNumbers.tex deleted file mode 100644 index 8aaf9b603..000000000 --- a/notes/.GAelectrodynamics/complexNumbers.tex +++ /dev/null @@ -1,28 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{Complex numbers}{problem:gradeselection:ComplexNumbers}{ -Show that complex numbers can be represented as even grade multivectors \( z = a + \Be_1 \Be_2 b \). -} % problem - -\makeanswer{problem:gradeselection:ComplexNumbers}{ -Let \( i = \Be_1 \Be_2 \), for which we find - -\begin{dmath}\label{eqn:gradeselectionProblems:260} -i^2 -= -\lr{ \Be_1 \Be_2 } -\lr{ \Be_1 \Be_2 } -= -\Be_1 (\Be_2 \Be_1) \Be_2 -= -\Be_1 (-\Be_1 \Be_2) \Be_2 -= --(\Be_1^2) (\Be_2^2) -= --1. -\end{dmath} - -The even grade multivector \( z = a + i b \) is thus seen to have all the properties required of complex numbers. -} % answer diff --git a/notes/.GAelectrodynamics/cyclicpermutationtwo.tex b/notes/.GAelectrodynamics/cyclicpermutationtwo.tex deleted file mode 100644 index a1bf66970..000000000 --- a/notes/.GAelectrodynamics/cyclicpermutationtwo.tex +++ /dev/null @@ -1,31 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{}{problem:gradeselection:cyclicpermutationtwo}{ -Show that -\begin{dmath}\label{eqn:gradeselection:580} -\gpgradezero{ \Bx \By } -= -\gpgradezero{ \By \Bx }. -\end{dmath} -} % problem - -\makeanswer{problem:gradeselection:cyclicpermutationtwo}{ - -Expanding the vector grade zero selection in coordinates gives -\begin{dmath}\label{eqn:gradeselectionProblems:680} -\gpgradezero{ \Bx \By } -= -\sum_{ij} \gpgradezero{ x_i y_j \Be_i \Be_j } -= -\sum_{i = j} \gpgradezero{ x_i y_i \Be_i \Be_i } -= -\sum_{i = j} \gpgradezero{ (y_i \Be_i)(x_i \Be_i) } -= -\sum_{i,j} \gpgradezero{ (y_i \Be_i)(x_i \Be_i) } -= -\gpgradezero{ \By \Bx }. -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/definitions.tex b/notes/.GAelectrodynamics/definitions.tex deleted file mode 100644 index 2e34f24fc..000000000 --- a/notes/.GAelectrodynamics/definitions.tex +++ /dev/null @@ -1,192 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%{ -%%\input{../blogpost.tex} -%%\renewcommand{\basename}{multiplication} -%%%\renewcommand{\dirname}{notes/phy1520/} -%%\renewcommand{\dirname}{notes/ece1228-electromagnetic-theory/} -%%%\newcommand{\dateintitle}{} -%%%\newcommand{\keywords}{} -%% -%%\input{../peeter_prologue_print2.tex} -%% -%%\usepackage{peeters_layout_exercise} -%%\usepackage{peeters_braket} -%%\usepackage{peeters_figures} -%%\usepackage{siunitx} -%%%\usepackage{mhchem} % \ce{} -%%%\usepackage{macros_bm} % \bcM -%%%\usepackage{txfonts} % \ointclockwise -%% -%%\beginArtNoToc -%% -%%\generatetitle{Vector multiplication} -%%%\chapter{Vector multiplication} -%%%\label{chap:multiplication} -%% - -A few new GA terms have been introduced in an ad-hoc fashion as required. Here is a systematic exposition of some of the key definitions used to refer to the types of the geometric objects that will be encountered. - -\makedefinition{Scalar}{def:multiplication:scalar}{ -A real number with no implied direction. -} - -\makedefinition{Vector}{def:multiplication:vector}{ -\href{https://www.youtube.com/watch?v=bOIe0DIMbI8}{A quantity with direction and magnitude.} -} - -\makedefinition{Bivector}{def:multiplication:bivector}{ -A product of two normal vectors, or a sum thereof. -} - -The product \( \Be_1 \Be_2 \) is a bivector, as is \( \Be_2 \Be_3 + 3 \Be_4 \Be_1 \) - -\makedefinition{Trivector}{def:multiplication:trivector}{ -A product of three mutually normal vectors, or a sum thereof. -} - -The quantity \( \Be_3 \Be_1 \Be_2 \) is a trivector, as is \( \Be_1 \Be_2 \Be_3 + 3 \Be_5 \Be_4 \Be_1 \). - -\makedefinition{Blade}{def:multiplication:blade}{ -A scalar, vector, bivector, or a trivector (or higher degree analogue), that can be constructed by multiplication of a number of vectors, but not an unfactorable sum of thereof. -} - -The factorable quantity -\begin{dmath}\label{eqn:multiplication:220} -\Be_1 \Be_2 + 3 \Be_1 \Be_3 -= -\Be_1 (\Be_2 + 3 \Be_3) -\end{dmath} - -is a blade, whereas - -\begin{dmath}\label{eqn:multiplication:240} -\Be_1 \Be_2 + 3 \Be_3 \Be_4, -\end{dmath} - -and -\begin{dmath}\label{eqn:multiplication:260} -\Be_1 \Be_2 + \Be_2 \Be_3 + \Be_3 \Be_1 -\end{dmath} - -are not. - -Scalar, vector, bivector, and trivectors are also referred to as sums of 0-blades, 1-blades, 2-blades, and 3-blades respectively. - -\makedefinition{Grade.}{def:multiplication:grade}{ -The minimum number of vector products required to form a given blade. -} - -The grade of a scalar, vector, bivector, and trivector are 0, 1, 2, and 3 respectively. - -The quantities -\begin{dmath}\label{eqn:multiplication:300} -\begin{aligned} -\Be_2 + \Be_1 \Be_2 \Be_2 &= \Be_1 + \Be_2 \\ -\Be_1 \Be_2 \Be_2 \Be_2 \Be_3 &= \Be_1 \Be_2 \Be_3 \\ -\end{aligned} -\end{dmath} - -have grades 1 and 3 respectively. - -Quantities with higher grades than 3 are not generally given explicit names, but can be referred to having grade-k. When an object of grade-k is -also a blade, it can be referred to as a k-blade. - -In a three dimensional space the highest grade possible is 3. Blades can have grades higher than 3 in higher dimensional vector spaces. - -\makedefinition{Pseudoscalar.}{def:multiplication:pseudoscalar}{ -A blade with grade that matches the dimension of the space. -} - -In a two dimensional space \( \Be_2 \Be_1 \) is a pseudoscalar. In a three dimensional space -\( \Be_3 \Be_1 \Be_2 \) is a pseudoscalar, as is \( \Be_3 \Be_1 (\Be_2 + \Be_3 ) \). A pseudoscalar has an implied orientation, which can be -associated with the handedness of the underlying basis. It is conventional to refer to - -\begin{dmath}\label{eqn:definitions:320} -i = \Be_1 \Be_2, -\end{dmath} - -as ``the pseudoscalar'' for a two dimensional space, and to - -\begin{dmath}\label{eqn:definitions:340} -I = \Be_1 \Be_2 \Be_3, -\end{dmath} - -as ``the pseudoscalar'' for a three dimensional space. - -\makedefinition{Multivector.}{def:multiplication:multivector}{ -A sum of zero or more blades. -} - -Examples include -\begin{dmath}\label{eqn:multiplication:280} -\begin{aligned} -&3 \\ -& 1 + \Be_1 \Be_2 \\ -& 2 - \Be_1 \Be_2 \Be_3 \\ -& \Be_1 + 2 \Be_1 \Be_2 + \Be_2 \Be_3 - 3 \Be_3 \Be_1 + \Be_1 \Be_2 \Be_4 -\end{aligned} -\end{dmath} - -\makedefinition{Dual}{dfn:definitions:dual}{ -The dual of a multivector is the product of that multivector with a pseudoscalar for a subspace that contains the multivector. Such multiplication is referred to as a duality transformation. -} % definition - -For example, given \( M = 1 + \Be_1 \Be_2 \), multiplication by \( i = \Be_1 \Be_2 \) is a duality transformation with respect to the x-y plane, and multiplication by \( I = \Be_1 \Be_2 \Be_3 \) is a duality transformation with \R{3}. - -With the following shorthand notation is convienent for sucessive products of orthonormal basis vectors - -\begin{dmath}\label{eqn:definitions:420} -\Be_{ij\cdots k} = \Be_i \Be_j \cdots \Be_k, -\end{dmath} - -consider some concrete examples of duality transformations of blades. -The dual vectors to the basis vectors of a 2D space are those same vectors rotated by \( \pi/2 \) - -\begin{dmath}\label{eqn:definitions:360} -\begin{aligned} -\Be_1 \Be_{12} &= \Be_2 \\ -\Be_2 \Be_{12} &= -\Be_1, -\end{aligned} -\end{dmath} - -with an inverse duality transformation given by the multiplication with \( \Be_{12}^{-1} = \Be_{21} \) - -\begin{dmath}\label{eqn:definitions:440} -\begin{aligned} -\Be_2 \Be_{21} &= \Be_1 \\ --\Be_1 \Be_{21} &= \Be_2. -\end{aligned} -\end{dmath} - -The \R{3} duals to the basis vectors are bivectors - -\begin{dmath}\label{eqn:definitions:380} -\begin{aligned} -\Be_1 \Be_{123} &= \Be_{23} \\ -\Be_2 \Be_{123} &= \Be_{31} \\ -\Be_3 \Be_{123} &= \Be_{12}, -\end{aligned} -\end{dmath} - -whereas the duals to those bivectors with respect to the pseudoscalar \( I^{-1} = \Be_{321} \) are the original basis vectors - -\begin{dmath}\label{eqn:definitions:400} -\begin{aligned} -\Be_{23} \Be_{321} &= \Be_1 \\ -\Be_{31} \Be_{321} &= \Be_2 \\ -\Be_{12} \Be_{321} &= \Be_3. -\end{aligned} -\end{dmath} - -In a sense that can be defined more precisely once the general dot product operator is defined, the dual to a given blade represents an object that is normal to the original blade. - -The dual of any scalar is a pseudoscalar, whereas the dual of a pseudoscalar is a scalar. - -%When working with multivector integrals it will be useful to consider the differential volume element a volume weighted pseudoscalar. - -%%%} -%%%\EndArticle -%%\EndNoBibArticle diff --git a/notes/.GAelectrodynamics/dotprodSymmetricSum.tex b/notes/.GAelectrodynamics/dotprodSymmetricSum.tex deleted file mode 100644 index c3e389b88..000000000 --- a/notes/.GAelectrodynamics/dotprodSymmetricSum.tex +++ /dev/null @@ -1,52 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Dot product of vectors as symmetric sum}{problem:gradeselection:dotprod}{ -Show that the dot product of two vectors can be written as a symmetric sum - -\begin{dmath}\label{eqn:gradeselection:600} -\Bx \cdot \By = \inv{2} \lr{ \Bx \By + \By \Bx }. -\end{dmath} -} % problem - -\makeanswer{problem:gradeselection:dotprod}{ -There are a few ways that this can be demonstrated. The first relies on the classical definition of the dot product. Expanding the square of a vector sum gives - -\begin{dmath}\label{eqn:gradeselectionProblems:700} -(\Bx + \By)^2 = \Bx^2 + \By^2 + \Bx \By + \By \Bx. -\end{dmath} - -By comparison this must also be equal to - -\begin{dmath}\label{eqn:gradeselectionProblems:720} -\Abs{\Bx + \By}^2 = \Bx^2 + \By^2 + 2 \Bx \cdot \By, -\end{dmath} - -so -\begin{dmath}\label{eqn:gradeselectionProblems:740} -\Bx \By + \By \Bx = 2 \Bx \cdot \By. -\end{dmath} - -This might be viewed as a cheat, since it is not using the dot product as defined by grade zero selection according to \cref{dfn:gradeselection:100}. Using that definition will produce the same result - -\begin{dmath}\label{eqn:gradeselectionProblems:760} -\gpgradezero{ (\Bx + \By)^2 } -= -\gpgradezero{ \Bx^2 + \By^2 + \Bx \By + \By \Bx } -= -\Bx^2 + \By^2 -+ -\gpgradezero{ -\Bx \By } -+ \gpgradezero{ \By \Bx }. -\end{dmath} - -It was shown in \cref{problem:gradeselection:cyclicpermutationtwo} that \( \gpgradezero{ \Bx \By } = \gpgradezero{ \By \Bx } \) so -\begin{dmath}\label{eqn:gradeselectionProblems:780} -2 \gpgradezero{ \Bx \By } = \Bx \By + \By \Bx. -\end{dmath} - -Using \cref{dfn:gradeselection:100}, this completes the problem. -} % answer diff --git a/notes/.GAelectrodynamics/gradeselVectorWedge.tex b/notes/.GAelectrodynamics/gradeselVectorWedge.tex deleted file mode 100644 index 553558184..000000000 --- a/notes/.GAelectrodynamics/gradeselVectorWedge.tex +++ /dev/null @@ -1,50 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{Vector wedge coordinate expansion and antisymmetry}{problem:gradeselection:vectorwedge}{ -Show that -\begin{dmath}\label{eqn:gradeselection:620} -\Bx \wedge \By -= -%\sum_{i < j} (x_i y_j - x_j y_i) \Be_i \Be_j. -\sum_{i < j} -\begin{vmatrix} -x_i & x_j \\ -y_i & y_j -\end{vmatrix} -\Be_i \Be_j. -\end{dmath} - -Observe from this coordinate expansion that the wedge product of two vectors is antisymmetric -\boxedEquation{eqn:gradeselection:640}{ -\Bx \wedge \By = -\By \wedge \Bx. -} -} % problem - -\makeanswer{problem:gradeselection:vectorwedge}{ -Given \( \Bx = \sum_i x_i \Be_i \), and \( \By = \sum_i y_i \Be_i \), the wedge of these two vectors is a grade two selection that picks out only products that differ in index - -\begin{dmath}\label{eqn:gradeselectionProblems:580} -\Bx \wedge \By -= -\gpgradetwo{ \Bx \By } -= -\sum_{i,j} \gpgradetwo{ x_i \Be_i y_j \Be_j } -= -\sum_{i \ne j} x_i y_j \gpgradetwo{ \Be_i \Be_j } -= -\sum_{i \ne j} x_i y_j \Be_i \Be_j -= -\sum_{i < j} (x_i y_j - x_j y_i) \Be_i \Be_j -= -\sum_{i < j} -\begin{vmatrix} -x_i & x_j \\ -y_i & y_j -\end{vmatrix} -\Be_i \Be_j. -\end{dmath} - -When \( \Bx, \By \), the rows in each of the above determinants will swap, negating the sign of each. This implies \( \By \wedge \Bx = -\Bx \wedge \By \). -} % answer diff --git a/notes/.GAelectrodynamics/gradeselection.tex b/notes/.GAelectrodynamics/gradeselection.tex deleted file mode 100644 index d512d9638..000000000 --- a/notes/.GAelectrodynamics/gradeselection.tex +++ /dev/null @@ -1,219 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%{ -%\input{../blogpost.tex} -%\renewcommand{\basename}{gradeselection} -%%\renewcommand{\dirname}{notes/phy1520/} -%\renewcommand{\dirname}{notes/ece1228-electromagnetic-theory/} -%%\newcommand{\dateintitle}{} -%%\newcommand{\keywords}{} -% -%\input{../peeter_prologue_print2.tex} -% -%\usepackage{peeters_layout_exercise} -%\usepackage{peeters_braket} -%\usepackage{peeters_figures} -%\usepackage{siunitx} -%%\usepackage{mhchem} % \ce{} -%%\usepackage{macros_bm} % \bcM -%\usepackage{macros_qed} % \qedmarker -%%\usepackage{txfonts} % \ointclockwise -% -%\beginArtNoToc -% -%\generatetitle{XXX} -%%\chapter{XXX} -%%\label{chap:gradeselection} -% -Having defined the axioms and definitions of Geometric Algebra, it desirable to define the grade selection operator, the dot product operator and the wedge product operator, and consider some simple examples of each. - -\makedefinition{Grade selection operator}{dfn:gradeselection:gradeselection}{ -Given a multivector \( M \) containing k-grade components \( M_k \) - -\begin{equation*} -M = \sum_{i = 0}^N M_i, -\end{equation*} - -the grade selection operator is defined as - -\begin{equation*}\label{eqn:gradeselection:40} -\gpgrade{M}{k} \equiv M_k. -\end{equation*} - -Selection of the (scalar) zero grade is often written as -\begin{equation*} -\gpgradezero{M} \equiv \gpgrade{M}{0} = M_0. -\end{equation*} -} - -For example, if \( M = 3 - \Be_3 + 2 \Be_1 \Be_2 \), then -\begin{equation}\label{eqn:gradeselection:80} -\begin{aligned} -\gpgradezero{M} &= 3 \\ -\gpgrade{M}{1} &= - \Be_3 \\ -\gpgrade{M}{2} &= 2 \Be_1 \Be_2 \\ -\gpgrade{M}{3} &= 0. -\end{aligned} -\end{equation} - -\makedefinition{Dot product}{dfn:gradeselection:100}{ -The dot (or inner) product of two multivectors - -\begin{equation*} -\begin{aligned} -A &= \sum_{i = 0}^N A_i, \\ -B &= \sum_{i = 0}^N B_i, -\end{aligned} -\end{equation*} - -is defined as -\begin{equation*} -A \cdot B \equiv -\sum_{i,j = 0}^N \gpgrade{ A_i B_j }{\Abs{i - j}} -\end{equation*} -} % definition - -As an example, consider two vectors in a 2D space - -\begin{dmath}\label{eqn:gradeselection:140} -\begin{aligned} -\Ba &= \lr{ x \Be_1 + y \Be_2 } \\ -\Ba' &= \lr{ x' \Be_1 + y' \Be_2 }, -\end{aligned} -\end{dmath} - -for which this definition of the dot product gives - -\begin{dmath}\label{eqn:gradeselection:160} -\Ba \cdot \Ba' -= -\gpgrade{ \Ba \Ba' }{\Abs{1 - 1}} -= -\gpgradezero{ \Ba \Ba' } -= -\gpgradezero{ \lr{ x \Be_1 + y \Be_2 } \lr{ x' \Be_1 + y' \Be_2 } } -= -\gpgradezero{ x x' \Be_1^2 + y y' \Be_2^2 + (x y' - y x') \Be_1 \Be_2 } -= -x x' + y y'. -\end{dmath} - -It is left to the reader (\cref{problem:gradeselection:RnDotProduct}) to show that this definition also reduces to the traditional \R{n} dot product. - -As a second example, consider the dot product of a vector with a bivector. With \( \Ba \) as defined in \cref{eqn:gradeselection:140} and \( i = \Be_1 \Be_2 \) - -\begin{dmath}\label{eqn:gradeselection:240} -\Ba \cdot i -= -\gpgrade{ \Ba i }{1} -= -\gpgrade{ \lr{ x \Be_1 + y \Be_2 } \Be_1 \Be_2 }{1} -= -\gpgrade{ x \Be_1^2 \Be_2 + y \Be_2 (-\Be_2 \Be_1) }{1} -= -\gpgrade{ x \Be_2 - y \Be_1 }{1} -= -x \Be_2 - y \Be_1. -\end{dmath} - -This particular dot product is trivial, since the product \( \Ba i \) has only a vector component. -In this example \( i \) is the pseudoscalar for the two dimensional space, and it can be observed that multiplication of a vector from the right serves to rotate the vector by 90 degrees. It is not a coincidence that this is strikingly similar to the action of the imaginary from complex algebra. It can be shown (\cref{problem:gradeselection:PlaneRotations}) -that \( e^{i\theta} \) acts as a rotation operator as it does in complex algebra, and that a GA representation of complex numbers is possible (\cref{problem:gradeselection:ComplexNumbers}). - -For a non-trivial vector-bivector dot product, consider - -\begin{dmath}\label{eqn:gradeselection:560} -\lr{ \Be_1 + \Be_2 } \cdot \lr{ \Be_1 \Be_2 + 3 \Be_2 \Be_3 } -= -\gpgradeone{ -\lr{ \Be_1 + \Be_2 } \lr{ \Be_1 \Be_2 + 3 \Be_2 \Be_3 } -} -= -\gpgradeone{ -\Be_1^2 \Be_2 + 3 \Be_1 \Be_2 \Be_3 -+ -\Be_2 \Be_1 \Be_2 + 3 \Be_2^2 \Be_3 -} -= -\gpgradeone{ -\Be_2 + 3 \cancel{\Be_1 \Be_2 \Be_3} -- -\Be_1 + 3 \Be_3 -} -= -\Be_2 - \Be_1 + 3 \Be_3. -\end{dmath} - -The vector-bivector dot product filters out products that no common factors, since such products result in trivector components. - -\makedefinition{Wedge product.}{dfn:gradeselection:480}{ -For the multivectors \( A, B \) defined in \cref{dfn:gradeselection:100}, the wedge (or outer) product is defined as - -\begin{equation*} -A \wedge B -\equiv -\sum_{i,j = 0}^N \gpgrade{ A_i B_j }{i + j}. -\end{equation*} -} % definition - -For example, the wedge product of the 2D vectors of \cref{eqn:gradeselection:140} is - -\begin{dmath}\label{eqn:gradeselection:500} -\Ba \wedge \Bb -= -\gpgradetwo{ -\lr{ x \Be_1 + y \Be_2 } -\lr{ x' \Be_1 + y' \Be_2 } -} -= -\gpgradetwo{ -(x x' + y y') + (x y' - x' y) \Be_1 \Be_2 -} -= -(x y' - x' y) \Be_1 \Be_2. -\end{dmath} - -The wedge product of two vectors in a plane contains an antisymmetrized sum of the vector coefficients, but is weighted by a ``unit'' bivector, the pseudoscalar for the plane. - -As another example consider - -\begin{dmath}\label{eqn:gradeselection:520} -\Be_1 \wedge \lr{ 2\Be_1 + 3 \Be_2 } -= -\gpgradetwo{ -\Be_1 \lr{ 2\Be_1 + 3 \Be_2 } -} -= -\gpgradetwo{ -2 \Be_1^2 + 3 \Be_1 \Be_2 -} -= -3 \Be_1 \Be_2. -\end{dmath} - -Components of the vectors are that colinear are filtered out. In this case that is the \( \Be_1 \) component of the second vector \( \Be_1 + 3 \Be_2 \). It is not coincidence that this is also a property of the cross product. That relationship will be explored in (\cref{problem:gradeselection:WedgeRelationshipToCrossProduct}). - -As a final example, consider the wedge product of a vector with a bivector - -\begin{dmath}\label{eqn:gradeselection:540} -\Be_1 \wedge \lr{ \Be_1 \Be_2 - 7 \Be_2 \Be_3 } -= -\gpgradethree{ -\Be_1 \lr{ \Be_1 \Be_2 - 7 \Be_2 \Be_3 } -} -= -\gpgradethree{ -\Be_1^2 \Be_2 - 7 \Be_1 \Be_2 \Be_3 -} -= -- 7 \Be_1 \Be_2 \Be_3. -\end{dmath} - -Because \( \Be_1 \Be_2 \) has a common factor with \( \Be_1 \) it is filtered out of the resulting wedge product. The end result, in this case, is a 3D pseudoscalar. - -The wedge product of two bivectors in \R{3}, by this definition, is always zero, since there can be no grade 4 term in such a product. It is also the case that the components of any \R{3} bivectors wedged together will also have a common factor, which nessesarily kills the wedge product of any two \R{3} bivectors. This is not the case for arbitrary \R{N} bivectors, an example of which is \( \Be_1 \Be_2 + \Be_3 \Be_4 \). There is no common factor in this bivector, so it can be wedged with itself and still produce a non-zero result (i.e. this bivector is not a blade). - -%} -%\EndNoBibArticle diff --git a/notes/.GAelectrodynamics/gradethreeselectionWedge.tex b/notes/.GAelectrodynamics/gradethreeselectionWedge.tex deleted file mode 100644 index 2e513206b..000000000 --- a/notes/.GAelectrodynamics/gradethreeselectionWedge.tex +++ /dev/null @@ -1,83 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{Wedge of three vectors}{problem:gradethreeselectionWedge:wedgeThree}{ -Show that -\begin{dmath}\label{eqn:gradethreeselectionWedge:700} -\gpgradethree{ \Ba \Bb \Bc } -= -\Ba \wedge ( \Bb \wedge \Bc ) -= -(\Ba \wedge \Bb) \wedge \Bc -= -- -\Bb \wedge (\Ba \wedge \Bc). -\end{dmath} - -Observe that is antisymmetric in any two vectors, and thus completely antisymmetric (i.e. associative). This allows the grade three selection of any three vectors to be written more simply as - -\boxedEquation{eqn:gradethreeselectionWedge:720}{ -\gpgradethree{ \Ba \Bb \Bc } -= -\Ba \wedge \Bb \wedge \Bc. -} - -This can be considered the definition of \( \Ba \wedge \Bb \wedge \Bc \). - -Some authors will define the wedge product of \( m \) vectors as - -\begin{equation}\label{eqn:gradethreeselectionWedge:820} -\Bx_1 \wedge \Bx_2 \wedge \cdots \wedge \Bx_m -= \inv{m!} \sum \Bx_{i_1} \Bx_{i_2} \cdots \Bx_{i_m} \sgn(\pi(i_1 i_2 \cdots i_m)), -\end{equation} - -where \(\sgn(\pi(\cdots))\) is the sign of the permutation of the indices. With focus on \R{3}, such a definition is not required. A reader interested in the \R{N} case should demonstrate from the axioms and definitions that -\( \gpgrade{ \Bx_1 \Bx_2 \cdots \Bx_m}{m} \) expands as specified in \cref{eqn:gradethreeselectionWedge:820}. -} % problem - -\makeanswer{problem:gradethreeselectionWedge:wedgeThree}{ -Consider an expansion first in products of \( \Ba, \Bb \) - -\begin{dmath}\label{eqn:gradethreeselectionWedge:740} -\gpgradethree{ \Ba \Bb \Bc } -= -\gpgradethree{ (\cancel{\Ba \cdot \Bb} + \Ba \wedge \Bb) \Bc } -= -\gpgradethree{ (\Ba \wedge \Bb) \Bc }. -\end{dmath} - -The dot product was killed since it leaves only a vector product within the grade selection operator. Since a vector bivector product can have only grade 1 and grade three terms (example: \( \Be_1 (\Be_1 \wedge \Be_2) = \Be_2, \Be_1 (\Be_2 \wedge \Be_3) = \Be_1 \Be_2 \Be_3 \), this leaves just - -\begin{dmath}\label{eqn:gradethreeselectionWedge:760} -\gpgradethree{ \Ba \Bb \Bc } -= -(\Ba \wedge \Bb) \wedge \Bc. -\end{dmath} - -Similarly, expanding the \( \Bb \Bc \) product gives -\begin{dmath}\label{eqn:gradethreeselectionWedge:780} -\gpgradethree{ \Ba \Bb \Bc } -= -\gpgradethree{ \Ba (\cancel{\Bb \cdot \Bc} + \Ba \wedge \Bc) } -= -\gpgradethree{ \Ba (\Bb \wedge \Bc) } -= -\Ba \wedge (\Bb \wedge \Bc), -\end{dmath} - -and finally, expanding products of \( \Ba \Bc \) after commutation - -\begin{dmath}\label{eqn:gradethreeselectionWedge:800} -\gpgradethree{ \Ba \Bb \Bc } -= -\gpgradethree{ (\cancel{2 \Bb \cdot \Ba} - \Bb \Ba) -\Bc} -= --\gpgradethree{ \Bb \Ba \Bc } -= --\gpgradethree{ \Bb (\cancel{\Ba \cdot \Bc} + \Ba \wedge \Bc) } -= -- \Bb \wedge (\Ba \wedge \Bc). -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/multiplication.tex b/notes/.GAelectrodynamics/multiplication.tex deleted file mode 100644 index e21500db5..000000000 --- a/notes/.GAelectrodynamics/multiplication.tex +++ /dev/null @@ -1,138 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%{ -%%\input{../blogpost.tex} -%%\renewcommand{\basename}{multiplication} -%%%\renewcommand{\dirname}{notes/phy1520/} -%%\renewcommand{\dirname}{notes/ece1228-electromagnetic-theory/} -%%%\newcommand{\dateintitle}{} -%%%\newcommand{\keywords}{} -%% -%%\input{../peeter_prologue_print2.tex} -%% -%%\usepackage{peeters_layout_exercise} -%%\usepackage{peeters_braket} -%%\usepackage{peeters_figures} -%%\usepackage{siunitx} -%%%\usepackage{mhchem} % \ce{} -%%%\usepackage{macros_bm} % \bcM -%%%\usepackage{txfonts} % \ointclockwise -%% -%%\beginArtNoToc -%% -%%\generatetitle{Vector multiplication} -%%%\chapter{Vector multiplication} -%%%\label{chap:multiplication} -%% -Geometric Algebra defines a multiplication operation for vectors, forming a vector space spanned by all the possible vector products. This algebra is described by the following small set of axioms - -\makeaxiom{Associative multiplication.}{axiom:multiplication:associative}{ - -The product of any three vectors \(\Ba,\Bb,\Bc\) is associative. - -\begin{equation*}\label{eqn:multiplication:160} -\Ba (\Bb \Bc) -= (\Ba \Bb) \Bc -= \Ba \Bb \Bc. -\end{equation*} -} - -\makeaxiom{Linearity.}{axiom:multiplication:linear}{ -Vector products are linear with respect to addition and subtraction. - -\begin{dmath*}\label{eqn:multiplication:180} -\begin{aligned} -(\Ba + 3 \Bb \Bd) \Bc &= \Ba \Bb + 3 \Bb \Bd \Bc \\ -\Ba (\Bb \Bd - 2 \Bc) &= \Ba \Bb \Bd - 2 \Ba \Bc. -\end{aligned} -\end{dmath*} -} - -\makeaxiom{Contraction.}{axiom:multiplication:contraction}{ - -The square of a vector is the squared length of the vector. - -\begin{dmath*}\label{eqn:multiplication:200} -\Ba^2 = \Abs{\Ba}^2. -\end{dmath*} - -The notion of length here is metric dependent. For the problems considered in these notes -it can be assumed that there is an orthonormal Euclidean basis, where the vector length is always positive. -For special relativistic calculations, also of interest in electrodynamics, but not the focus of these notes, the length of a (four-)vector may generally be negative or positive. -} - -These axioms are simple enough, but have a rich set of consequences\footnote{Similar to Feynman on gravitation \citep{feynman1963flp} ``... have shall said everything required, for a sufficiently talented mathematician could then deduce all the consequences of these principles. However, since you are not assumed to be sufficiently talented yet, we shall discuss the consequences in more detail''.}. - -The linearity and associativity axioms need little comment, but the contraction property might be surprising. For one justification of this rule, consider a one dimensional vector space spanned by a single unit vector \( \setlr{ \Be } \). That span, for real \( x \) is all the values - -\begin{dmath}\label{eqn:multiplication:20} -\Bx = x \Be. -\end{dmath} - -FIXME: picture to demonstrate the number line isomorphism. - -This vector space is isomorphic with a number line, all the possible real values \( x \). -Given a positive number \( x \), the multiplication rules for real numbers require that \( (\pm x)^2 = x^2 \). -The square of a number provides the (squared) length of the number, its distance from the origin. -The same rule can be imposed for one dimensional vectors, -a requirement that the (squared) distance from the origin equals the square of the vector itself. Such a rule is consistent with the rules of scalar multiplication, and for the -one dimensional vectors of \cref{eqn:multiplication:20} can be stated as - -\begin{equation}\label{eqn:multiplication:40} -\Bx^2 = x^2. -\end{equation} - -This contraction axiom, justified or not, has additional implications - -\begin{dmath}\label{eqn:multiplication:80} -x^2 -= \Bx^2 -= (x \Be)(x \Be) -= x^2 \Be^2. -\end{dmath} - -This rule requires the square of a unit (Euclidean) vector to be unity - -%\begin{equation}\label{eqn:multiplication:60} -\boxedEquation{eqn:multiplication:60}{ -\Be^2 = 1. -} -%\end{equation} - -With this implication noted, now consider the square of a simple two dimensional vector - -\begin{dmath}\label{eqn:gaTutorial:80} -2 -= -(\Be_1 + \Be_2)^2 -= (\Be_1 + \Be_2)(\Be_1 + \Be_2) -= \Be_1^2 + \Be_2 \Be_1 + \Be_1 \Be_2 + \Be_2^2 -= 2 + \Be_2 \Be_1 + \Be_1 \Be_2. -\end{dmath} - -The sum above with both scalar terms and terms that are composed of products of vectors is called a multivector. -A product of two perpendicular vectors (or a sum of such products) is called a bivector, and can be used to represent an oriented plane. -Geometric Algebra allows for sums of scalars, vectors, bivectors, and higher degree products. - -Observe that for this identity to hold, the bivector terms must sum to zero. That is - -%\begin{dmath}\label{eqn:multiplication:140} -\boxedEquation{eqn:multiplication:140}{ -\Be_1 \Be_2 = -\Be_1 \Be_2. -} -%\end{dmath} - -This implies that the product of two orthonormal vectors anticommutes. In general it is also true that - -\maketheorem{Normal anticommutation}{thm:multiplication:anticommutationNormal}{ -The product of any two normal vectors \(\Bu\), and \(\Bv\) anticommute. -\begin{equation*} -\Bu \Bv = -\Bv \Bu. -\end{equation*} -} % theorem - -%%%} -%%%\EndArticle -%%\EndNoBibArticle diff --git a/notes/.GAelectrodynamics/normalAnticommutation.tex b/notes/.GAelectrodynamics/normalAnticommutation.tex deleted file mode 100644 index 9af43410c..000000000 --- a/notes/.GAelectrodynamics/normalAnticommutation.tex +++ /dev/null @@ -1,38 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Normal anticommutation}{problem:multiplication:unitsquare}{ -Prove \cref{thm:multiplication:anticommutationNormal}. -} % problem - -\makeanswer{problem:multiplication:unitsquare}{ -Consider the square of a vector \( \Bx = u \Bu + v \Bv \) with respect to a basis of unit vectors \( \setlr{ \Bu, \Bv }\). That is - -\begin{dmath}\label{eqn:introGAproblems:160} -\Bx^2 -= -\lr{ u \Bu + v \Bv } -\lr{ u \Bu + v \Bv } -= -u^2 \Bu^2 -+ v^2 \Bv^2 -+ u v \lr{ \Bu \Bv + \Bv \Bu } -= -u^2 -+ v^2 -+ u v \lr{ \Bu \Bv + \Bv \Bu }. -\end{dmath} - -If these vectors are normal \( \Bx^2 = u^2 + v^2 \), which means -\begin{dmath}\label{eqn:introGAproblems:180} -\Bu \Bv = -\Bv \Bu. -\end{dmath} - -Observe that a side effect of this computation shows that the traditional vector dot product of two unit vectors can also be written as a symmetric bivector sum - -\begin{dmath}\label{eqn:introGAproblems:200} -\Bu \cdot \Bv = \inv{2} \lr{ \Bu \Bv + \Bv \Bu }. -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/planeRotationsExponentials.tex b/notes/.GAelectrodynamics/planeRotationsExponentials.tex deleted file mode 100644 index d57dee9f8..000000000 --- a/notes/.GAelectrodynamics/planeRotationsExponentials.tex +++ /dev/null @@ -1,158 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{Plane rotations.}{problem:gradeselection:PlaneRotations}{ - -With \( i = \Be_1 \Be_2 \) for the pseudoscalar of the \( x,y \) plane, - -\makesubproblem{}{problem:gradeselection:3:b} -justify the assertion that \( e^{i \theta} = \cos\theta + i \sin\theta \), where \( theta \) is a scalar angle. - -\makesubproblem{}{problem:gradeselection:3:c} -Show that right multiplication of a 2D vector by \( e^{i\theta} \) rotates that vector by \( \theta \) radians. - -\makesubproblem{}{problem:gradeselection:3:d} -Does the rotation multivector \( e^{i\theta} \) commute with the 2D basis vectors? - -\makesubproblem{}{problem:gradeselection:3:e} -What is the action of multiplication of a vector by \( e^{i\theta} \) from the left? -} % problem - -\makeanswer{problem:gradeselection:PlaneRotations}{ -\makeSubAnswer{}{problem:gradeselection:3:b} - -Assume that the exponential of a multivector argument is represented by a Taylor series - -\begin{dmath}\label{eqn:gradeselectionProblems:280} -e^X = \sum_{k = 0}^\infty \frac{X^k}{k!}, -\end{dmath} - -and note that the pseudoscalar commutes with scalar rotation angles \( \theta \), so -\begin{dmath}\label{eqn:gradeselectionProblems:300} -e^{i\theta} -= \sum_{k = 0}^\infty \frac{(i\theta)^k}{k!} -= \sum_{k = 0}^\infty \frac{i^k\theta^k}{k!} -= -\sum_{k = 0}^\infty \frac{i^{2k}\theta^{2k}}{(2k)!} -+ -\sum_{k = 0}^\infty \frac{i^{2k + 1}\theta^{2k +1}}{(2k + 1)!} -= -\sum_{k = 0}^\infty \frac{(-1)^{k}\theta^{2k}}{(2k)!} -+ -i \sum_{k = 0}^\infty \frac{(-1)^{k}\theta^{2k +1}}{(2k + 1)!} -= \cos \theta + i \sin\theta. -\end{dmath} -\makeSubAnswer{}{problem:gradeselection:3:c} - -Consider the action of the exponential on each of the unit vectors. For \( \Be_1 \) that is - -\begin{dmath}\label{eqn:gradeselectionProblems:320} -\Be_1 e^{i \theta} -= -\Be_1 \lr{ \cos\theta + i \sin\theta } -= -\Be_1 \cos\theta + \Be_1 (\Be_1 \Be_2 )\sin\theta -= -\Be_1 \cos\theta + \Be_2 \sin\theta. -\end{dmath} - -This shows that the vector \( \Be_1 \) is rotated counterclockwise by \( \theta \) radians. Similarly for \( \Be_2 \) - -\begin{dmath}\label{eqn:gradeselectionProblems:340} -\Be_2 e^{i \theta} -= -\Be_2 \lr{ \cos\theta + i \sin\theta } -= -\Be_2 \cos\theta + \Be_1 (\Be_1 \Be_2 )\sin\theta -= -\Be_2 \cos\theta + \Be_1 (-\Be_2 \Be_1) \sin\theta. -= -\Be_2 \cos\theta - \Be_1 \sin\theta. -\end{dmath} - -This is also a rotation by \( \theta \) radians. Given a vector \( \Bx = x \Be_1 + y \Be_2 \), this gives - -\begin{dmath}\label{eqn:gradeselectionProblems:360} -\Bx' -= \Bx e^{i\theta} -= -x \lr{ \Be_1 \cos\theta + \Be_2 \sin\theta } + y \lr{ \Be_2 \cos\theta - \Be_1 \sin\theta }. -\end{dmath} - -In particular - -\begin{dmath}\label{eqn:gradeselectionProblems:380} -\begin{bmatrix} -\Bx' \cdot \Be_1 \\ -\Bx' \cdot \Be_2 \\ -\end{bmatrix} -= -\begin{bmatrix} -x \cos\theta - y \sin\theta \\ -x \sin\theta + y \cos\theta -\end{bmatrix} -= -\begin{bmatrix} -\cos\theta &- \sin\theta \\ -\sin\theta &+ \cos\theta -\end{bmatrix} -\begin{bmatrix} -x \\ -y -\end{bmatrix}. -\end{dmath} - -Observe that this is the rotation matrix that takes the points \((x, y)\) to their position \((x', y')\) rotated by \( \theta \) radians. -\makeSubAnswer{}{problem:gradeselection:3:d} - -The action from the left on \( \Be_1 \) is - -\begin{dmath}\label{eqn:gradeselectionProblems:400} -e^{i\theta} \Be_1 -= -\lr{ \cos\theta + \Be_1 \Be_2 \sin\theta} \Be_1 -= -\Be_1 \cos\theta + \Be_1 \Be_2 \Be_1 \sin\theta -= -\Be_1 \cos\theta + \Be_1 (-\Be_1 \Be_2) \sin\theta -= -\Be_1 \lr{ \cos\theta - i \sin\theta } -= -\Be_1 e^{-i\theta}, -\end{dmath} - -and the action from the left on \( \Be_2 \) is - -\begin{dmath}\label{eqn:gradeselectionProblems:420} -e^{i\theta} \Be_2 -= -\lr{ \cos\theta + \Be_1 \Be_2 \sin\theta} \Be_2 -= -\Be_2 \cos\theta + \Be_1 \sin\theta -= -\Be_2 \cos\theta + (\Be_2 \Be_2) \Be_1 \sin\theta -= -\Be_2 \lr{ \cos\theta - i \sin\theta } -= -\Be_2 e^{-i\theta}. -\end{dmath} - -This change of sign is due to the fact that the pseudoscalar anticommutes with each of the basis vectors in the plane. - -\makeSubAnswer{}{problem:gradeselection:3:e} - -Since the exponential toggles sign on commutation with both of the vectors of the plane, the rotation operation can be applied from either left or right, with sufficient care to get the direction right - -\begin{equation}\label{eqn:gradeselectionProblems:440} -\Bx e^{i\theta} = e^{-i\theta} \Bx. -\end{equation} - -It is also possible to split the rotation operation into half angle rotation operators that act from both the left and right - -\begin{dmath}\label{eqn:gradeselectionProblems:460} -\Bx' = e^{-i\theta/2} \Bx e^{i\theta/2}. -\end{dmath} - -A student who has studied computer graphics rotation theory may have seen quaternion rotation operators with this form, and a student of quantum mechanics will have seen Pauli matrix rotation operations of this form. This is, in fact, the form that is generally desirable for 3D or higher order rotations, since it rotates the portions of a vector that lie in the rotation plane, leaving the normal components untouched. -} % answer diff --git a/notes/.GAelectrodynamics/r4nonzerobivectorwedgewithself.tex b/notes/.GAelectrodynamics/r4nonzerobivectorwedgewithself.tex deleted file mode 100644 index 51952194b..000000000 --- a/notes/.GAelectrodynamics/r4nonzerobivectorwedgewithself.tex +++ /dev/null @@ -1,28 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{\R{4} wedge of a non-blade with itself.}{problem:gradeselection:r4nonzerobivectorwedgewithself}{ -With \( B = \Be_1 \Be_2 + \Be_3 \Be_4 \), show that \( B \wedge B \ne 0 \). -} % problem - -\makeanswer{problem:gradeselection:r4nonzerobivectorwedgewithself}{ -\begin{dmath}\label{eqn:r4nonzerobivectorwedgewithself:n} -B \wedge B -= -\gpgrade{ B B }{4} -= -\gpgrade{ -\lr{ \Be_1 \Be_2 + \Be_3 \Be_4 } -\lr{ \Be_1 \Be_2 + \Be_3 \Be_4 } -}{4} -= -\Be_3 \Be_4 \Be_1 \Be_2 -+ -\Be_1 \Be_2 \Be_3 \Be_4 -= -2 \Be_1 \Be_2 \Be_3 \Be_4 -\ne 0. -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/reciprocal.tex b/notes/.GAelectrodynamics/reciprocal.tex deleted file mode 100644 index 4579b2486..000000000 --- a/notes/.GAelectrodynamics/reciprocal.tex +++ /dev/null @@ -1,204 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%{ -%\input{../blogpost.tex} -%\renewcommand{\basename}{reciprocal} -%%\renewcommand{\dirname}{notes/phy1520/} -%\renewcommand{\dirname}{notes/ece1228-electromagnetic-theory/} -%%\newcommand{\dateintitle}{} -%%\newcommand{\keywords}{} -% -%\input{../peeter_prologue_print2.tex} -% -%\usepackage{peeters_layout_exercise} -%\usepackage{peeters_braket} -%\usepackage{peeters_figures} -%\usepackage{siunitx} -%%\usepackage{mhchem} % \ce{} -%%\usepackage{macros_bm} % \bcM -%%\usepackage{macros_qed} % \qedmarker -%%\usepackage{txfonts} % \ointclockwise -% -%\beginArtNoToc -% -%\generatetitle{Reciprocal frame vectors} -%%\chapter{reciprocal frame vectors} -%%\label{chap:reciprocal} -% -\makedefinition{Reciprocal frame}{dfn:reciprocal:frame}{ -Given a basis for a subspace \( \setlr{ \Bx_1, \Bx_2, \cdots \Bx_n } \), the reciprocal frame is defined as the set of vectors \( \setlr{ \Bx^1, \Bx^2, \cdots \Bx^n } \) satisfying - -\begin{dmath*} -\Bx_i \cdot \Bx^j = {\delta_i}^j. -\end{dmath*} - -The vector \( \Bx^j \) is not the jth power of \( \Bx \), but is a superscript index, the conventional way of denoting a reciprocal frame vector. -} % definition - -The concept of a reciprocal frame generalizes the notion of normal to non-orthonormal bases. - -\paragraph{Motivation:} Reciprocal frames are required to express the GA form of Stokes theorem and for more general GA integration theory, including integration with respect to both curvilinear coordinates and higher dimensions. While the focus in these notes are two and three dimensional problems, often in Cartesian coordinates, it is desirable to formulate integration theory in a way that is compatible with the four dimensional non-Euclidean vector space that describes the intrinsic relativistic nature of electrodynamics. - -One of the utilities of a reciprocal frame is that it allows for the computation of the coordinates of a vector with respect to a non-orthonormal frame. Let a vector be given in terms of coordinates \( a^i \), where \( i \) is an index not a power - -\begin{dmath}\label{eqn:reciprocal:20} -\Ba = \sum_i a^i \Bx_i. -\end{dmath} - -Dotting with the reciprocal frame vectors then trivally extracts these coordinates - -\begin{dmath}\label{eqn:reciprocal:40} -\Ba \cdot \Bx^i -= \lr{\sum_j a^j \Bx_j} \cdot \Bx^i -= \sum_j a^j {\delta_j}^i -= a^i. -\end{dmath} - -Similarly, dotting with the frame basis vectors provides the coordinates with respect to the reciprocal frame. Let those coordinates be \( a_i \), so that - -\begin{dmath}\label{eqn:reciprocal:60} -\Ba = \sum_i a_i \Bx^i. -\end{dmath} - -Dotting with the basis vectors gives - -\begin{dmath}\label{eqn:reciprocal:80} -\Ba \cdot \Bx^i -= \lr{\sum_j a_j \Bx^j} \cdot \Bx_i -= \sum_j a_j {\delta^j}_i -= a_i. -\end{dmath} - -An example of a 2D oblique Euclidean basis and a corresponding reciprocal basis is plotted in \cref{fig:obliqueReciprocal:obliqueReciprocalFig2}. Also plotted are the superposition of the projections required to arrive at a given point \( (4,2) \)) along the \( \Be_1, \Be_2 \) directions or the \( \Be^1, \Be^2 \) directions. -In this plot, neither of the reciprocal frame vectors \( \Be^i \) are normal to the corresponding basis vectors \( \Be_i \). When one of \( \Be_i \) is increased(decreased) in magnitude, there will be a corresponding decrease(increase) in the magnitude of \( \Be^i \), but if the orientation is remained fixed, the corresponding direction of the reciprocal frame vector stays the same. - -\imageFigure{../../figures/GAelectrodynamics/obliqueReciprocalFig2}{Oblique and reciprocal bases.}{fig:obliqueReciprocal:obliqueReciprocalFig2}{0.45} - -How are the reciprocal frame vectors computed? While these vectors have a natural GA representation, this is not intrinsically a GA problem, and can be solved with standand linear algebra, using a matrix inversion. For example, given a 2D basis \( \setlr{ \Bx_1, \Bx_2 } \), the reciprocal basis can be assumed to have a coordinate representation in the original basis - -\begin{dmath}\label{eqn:reciprocal:100} -\begin{aligned} -\Bx^1 &= a \Bx_1 + b \Bx_2 \\ -\Bx^2 &= c \Bx_1 + d \Bx_2. -\end{aligned} -\end{dmath} - -Imposing the constraints of \cref{dfn:reciprocal:frame} leads to a pair of 2x2 linear systems that are easily solved to find -\begin{dmath}\label{eqn:reciprocal:120} -\begin{aligned} -\Bx^1 &= \inv{ \Bx_1^2 \Bx_2^2 - \lr{ \Bx_1 \cdot \Bx_2}^2 } \lr{ \Bx_2^2 \Bx_1 - \lr{ \Bx_1 \cdot \Bx_2 } \Bx_2 } \\ -\Bx^2 &= \inv{ \Bx_1^2 \Bx_2^2 - \lr{ \Bx_1 \cdot \Bx_2}^2 } \lr{ \Bx_1^2 \Bx_2 - \lr{ \Bx_1 \cdot \Bx_2 } \Bx_1 } \\ -\end{aligned} -\end{dmath} - -The reader may notice that for \R{3} the denominator is related to the norm of the cross product \( \Bx_1 \cross \Bx_2 \). More generally, this can be expressed as the square of the bivector \( \Bx_1 \wedge \Bx_2 \) - -\begin{dmath}\label{eqn:reciprocal:140} --\lr{\Bx_1 \wedge \Bx_2 }^2 -= --\lr{\Bx_1 \wedge \Bx_2 } \cdot \lr{\Bx_1 \wedge \Bx_2 } -= --\lr{ \lr{\Bx_1 \wedge \Bx_2 } \cdot \Bx_1 } \cdot \Bx_2 -= -\Bx_1^2 \Bx_2^2 - \lr{\Bx_1 \cdot \Bx_2}^2. -\end{dmath} - -Additionally, the numerators are each dot products of \( \Bx_1, \Bx_2 \) with that same bivector - -\begin{dmath}\label{eqn:reciprocal:160} -\begin{aligned} -\Bx^1 &= \frac{\Bx_2 \cdot \lr{ \Bx_1 \wedge \Bx_2 } }{ \lr{ \Bx_1 \wedge \Bx_2}^2 } \\ -\Bx^2 &= \frac{\Bx_1 \cdot \lr{ \Bx_2 \wedge \Bx_1 } }{ \lr{ \Bx_1 \wedge \Bx_2}^2 }, -\end{aligned} -\end{dmath} - -or - -%\begin{dmath}\label{eqn:reciprocal:180} -\boxedEquation{eqn:reciprocal:180}{ -\begin{aligned} -\Bx^1 &= \Bx_2 \cdot \inv{ \Bx_1 \wedge \Bx_2 } \\ -\Bx^2 &= \Bx_1 \cdot \inv{ \Bx_2 \wedge \Bx_1 }. -\end{aligned} -} -%\end{dmath} - -Geometrically, dotting with the bivector of the plane is a hybrid rotation and scaling operation. For example, for \R{2} with \( \Bx_1 = a_1 \Be_1 + a_2 \Be_2, \Bx_2 = b_1 \Be_1 + b_2 \Be_2 \), that pseudoscalar for this basis is - -\begin{dmath}\label{eqn:reciprocal:260} -\Bx_1 \wedge \Bx_2 -= -\lr{ a_1 \Be_1 + a_2 \Be_2 } \wedge \lr{ b_1 \Be_1 + b_2 \Be_2 } -= -\lr{ a_1 b_2 - a_2 b_1 } \Be_{12}. -\end{dmath} - -This has inverse -\begin{dmath}\label{eqn:reciprocal:280} -\inv{\Bx_1 \wedge \Bx_2 } -= -\inv{ a_1 b_2 - a_2 b_1 } \Be_{21}. -\end{dmath} - -So for the \R{2} the reciprocal frame is just - -\begin{dmath}\label{eqn:reciprocal:300} -\begin{aligned} -\Bx^1 &= \Bx_2 \frac{\Be_{21}}{ a_1 b_2 - a_2 b_1 } \\ -\Bx^2 &= \Bx_1 \frac{\Be_{12}}{ a_1 b_2 - a_2 b_1 } -\end{aligned} -\end{dmath} - -The vector \( \Bx^1 \) is obtained by rotating \( \Bx_2 \) by \( -\pi/2 \), and rescaling it. -The vector \( \Bx^2 \) is similarly obtained by a scaling and a rotation of \( \Bx_1 \) by \( \pi/2 \). - -Generalizing \cref{eqn:reciprocal:180} is almost possible by inspection. Given -a subspace spanned by a three vector basis \( \setlr{ \Bx_1, \Bx_2, \Bx_3 } \) the reciprocal frame vectors can be written as dot products - -\begin{dmath}\label{eqn:reciprocal:320} -\begin{aligned} -\Bx^1 &= \lr{ \Bx_2 \wedge \Bx_3 } \cdot \lr{ \Bx^3 \wedge \Bx^2 \wedge \Bx^1 } \\ -\Bx^2 &= \lr{ \Bx_3 \wedge \Bx_1 } \cdot \lr{ \Bx^1 \wedge \Bx^3 \wedge \Bx^2 } \\ -\Bx^3 &= \lr{ \Bx_1 \wedge \Bx_2 } \cdot \lr{ \Bx^2 \wedge \Bx^1 \wedge \Bx^3 } \\ -\end{aligned} -\end{dmath} - -Each of those trivector terms equals \( - \Bx^1 \wedge \Bx^2 \wedge \Bx^3 \) and can be related to the (known) pseudoscalar \( \Bx_1 \wedge \Bx_2 \wedge \Bx_3 \) by observing that - -\begin{dmath}\label{eqn:reciprocal:340} -\lr{ \Bx^1 \wedge \Bx^2 \wedge \Bx^3 } \cdot \lr{ \Bx_3 \wedge \Bx_2 \wedge \Bx_1 } -= -\Bx^1 \cdot \lr{ \Bx^2 \cdot \lr{ \Bx^3 \cdot \lr{ \Bx_3 \wedge \Bx_2 \wedge \Bx_1 } }} -= -\Bx^1 \cdot \lr{ \Bx^2 \cdot \lr{ \Bx_2 \wedge \Bx_1 } } -= -\Bx^1 \cdot \Bx_1 -= -1, -\end{dmath} - -which means that - -\begin{dmath}\label{eqn:reciprocal:360} --\Bx^1 \wedge \Bx^2 \wedge \Bx^3 -= -\inv{ \Bx_3 \wedge \Bx_2 \wedge \Bx_1 } -= \inv{ \Bx_1 \wedge \Bx_2 \wedge \Bx_3 }, -\end{dmath} - -and - -\boxedEquation{eqn:reciprocal:380}{ -\begin{aligned} -\Bx^1 &= \lr{ \Bx_2 \wedge \Bx_3 } \cdot \inv{ \Bx_1 \wedge \Bx_2 \wedge \Bx_3 } \\ -\Bx^2 &= \lr{ \Bx_3 \wedge \Bx_1 } \cdot \inv{ \Bx_1 \wedge \Bx_2 \wedge \Bx_3 } \\ -\Bx^3 &= \lr{ \Bx_1 \wedge \Bx_2 } \cdot \inv{ \Bx_1 \wedge \Bx_2 \wedge \Bx_3 } \\ -\end{aligned} -} - -It should be clear how to generalize this to higher dimensions if desired. - - -%} -%\EndNoBibArticle diff --git a/notes/.GAelectrodynamics/redacted.sty b/notes/.GAelectrodynamics/redacted.sty deleted file mode 100644 index 9f455de10..000000000 --- a/notes/.GAelectrodynamics/redacted.sty +++ /dev/null @@ -1 +0,0 @@ -\setboolean{redacted}{false} % uncensored version. diff --git a/notes/.GAelectrodynamics/renumber b/notes/.GAelectrodynamics/renumber deleted file mode 100644 index a743062b5..000000000 --- a/notes/.GAelectrodynamics/renumber +++ /dev/null @@ -1,57 +0,0 @@ -perl -p -i ../latex/bin/latexRegex.pl 2dvectorsquare.tex -perl -p -i ../latex/bin/latexRegex.pl ComplexInnerProductVsDotAndCrossProduct.tex -perl -p -i ../latex/bin/latexRegex.pl GAmotivation.tex -perl -p -i ../latex/bin/latexRegex.pl R3PseudoscalarCommutation.tex -perl -p -i ../latex/bin/latexRegex.pl R3PseudoscalarSquare.tex -perl -p -i ../latex/bin/latexRegex.pl RnDotProduct.tex -perl -p -i ../latex/bin/latexRegex.pl WedgeRelationshipToCrossProduct.tex -perl -p -i ../latex/bin/latexRegex.pl complexNumbers.tex -perl -p -i ../latex/bin/latexRegex.pl cyclicpermutationtwo.tex -perl -p -i ../latex/bin/latexRegex.pl definitions.tex -perl -p -i ../latex/bin/latexRegex.pl dotprodSymmetricSum.tex -perl -p -i ../latex/bin/latexRegex.pl gradeselVectorWedge.tex -perl -p -i ../latex/bin/latexRegex.pl gradeselection.tex -perl -p -i ../latex/bin/latexRegex.pl gradethreeselectionWedge.tex -perl -p -i ../latex/bin/latexRegex.pl multiplication.tex -perl -p -i ../latex/bin/latexRegex.pl normalAnticommutation.tex -perl -p -i ../latex/bin/latexRegex.pl planeRotationsExponentials.tex -perl -p -i ../latex/bin/latexRegex.pl vectorBivectorDot.tex -perl -p -i ../latex/bin/latexRegex.pl vectorproduct.tex -perl -p -i ../latex/bin/latexRegex.pl vectorproductCyclicPermutation.tex -perl -p -i ../latex/bin/latexRegex.pl wedgeantisym.tex -perl -p -i ~/physicsplay/bin/latexRegex.pl GAmotivation.tex -perl -p -i ~/physicsplay/bin/latexRegex.pl definitions.tex -perl -p -i ~/physicsplay/bin/latexRegex.pl gradeselection.tex -perl -p -i ~/physicsplay/bin/latexRegex.pl multiplication.tex -perl -p -i ~/physicsplay/bin/latexRegex.pl vectorproduct.tex -../latex/bin/lgrep 2dvectorsquare.tex | tee o ; . ./o -../latex/bin/lgrep ComplexInnerProductVsDotAndCrossProduct.tex | tee o ; . ./o -../latex/bin/lgrep GAmotivation.tex | tee o ; . ./o -../latex/bin/lgrep R3PseudoscalarCommutation.tex | tee o ; . ./o -../latex/bin/lgrep R3PseudoscalarSquare.tex | tee o ; . ./o -../latex/bin/lgrep RnDotProduct.tex | tee o ; . ./o -../latex/bin/lgrep WedgeRelationshipToCrossProduct.tex | tee o ; . ./o -../latex/bin/lgrep complexNumbers.tex | tee o ; . ./o -../latex/bin/lgrep cyclicpermutationtwo.tex | tee o ; . ./o -../latex/bin/lgrep definitions.tex | tee o ; . ./o -../latex/bin/lgrep dotprodSymmetricSum.tex | tee o ; . ./o -../latex/bin/lgrep gradeselVectorWedge.tex | tee o ; . ./o -../latex/bin/lgrep gradeselection.tex | tee o ; . ./o -../latex/bin/lgrep gradethreeselectionWedge.tex | tee o ; . ./o -../latex/bin/lgrep multiplication.tex | tee o ; . ./o -../latex/bin/lgrep normalAnticommutation.tex | tee o ; . ./o -../latex/bin/lgrep planeRotationsExponentials.tex | tee o ; . ./o -../latex/bin/lgrep vectorBivectorDot.tex | tee o ; . ./o -../latex/bin/lgrep vectorproduct.tex | tee o ; . ./o -../latex/bin/lgrep vectorproductCyclicPermutation.tex | tee o ; . ./o -../latex/bin/lgrep wedgeantisym.tex | tee o ; . ./o -../latex/bin/lgrep vectorTrivectorDot.tex | tee o ; . ./o -perl -p -i ../latex/bin/latexRegex.pl vectorTrivectorDot.tex -../latex/bin/lgrep bivectorDot.tex | tee o ; . ./o -perl -p -i ../latex/bin/latexRegex.pl bivectorDot.tex -../latex/bin/lgrep reciprocal.tex | tee o ; . ./o -perl -p -i ~/physicsplay/bin/latexRegex.pl reciprocal.tex -../latex/bin/lgrep 2subspaceR3reciprocalExample.tex | tee o ; . ./o -perl -p -i ../latex/bin/latexRegex.pl 2subspaceR3reciprocalExample.tex -../latex/bin/lgrep r4nonzerobivectorwedgewithself.tex | tee o ; . ./o -perl -p -i ../latex/bin/latexRegex.pl r4nonzerobivectorwedgewithself.tex diff --git a/notes/.GAelectrodynamics/thisbook.sty b/notes/.GAelectrodynamics/thisbook.sty deleted file mode 100644 index 1ebf9bb58..000000000 --- a/notes/.GAelectrodynamics/thisbook.sty +++ /dev/null @@ -1,123 +0,0 @@ -%----------------------------------------- -%\usepackage{censor} - -%----------------------------------------- -\usepackage{macros_lcal} -\usepackage{macros_cal} - -%-------------------------------------------------------------------- -% L7: -\usepackage{mhchem} - -%-------------------------------------------------------------------- -% For ps1 problem 2: - -\usepackage{enumerate} -\usepackage{mathtools} - -%-------------------------------------------------------------------- - -\usepackage{txfonts} % \ointclockwise - -%-------------------------------------------------------------------- - -\usepackage{peeters_braket} - -%-------------------------------------------------------------------- -\usepackage{siunitx} % \ang{} - -\usepackage{matlab} -\usepackage{steinmetz} % \phase{} -%\usepackage{latexsym} % \mho - -%-------------------------------------------------------------------- - -\usepackage{GAelectrodynamics} - -%-------------------------------------------------------------------- - -% \xrightarrow - -%\usepackage{mathtools} - -% \FORALL, \IF, ... - -%\usepackage{algorithmic} - -%\usepackage{kbordermatrix} -%\usepackage{easybmat} - -%-------------------------------------------------------------------- -\newcommand{\reading}[0]{\paragraph{Reading:}} - -%-------------------------------------------------------------------- -%\newcommand{\nought}[0]{\circ} - -% \ce{...} chemical formala formatting. -%\usepackage[version=3]{mhchem} -%\usepackage{mhchem} - -%\usepackage{units} - -% \grtsim (\ge \approx hybrid), and probably other stuff: -%\usepackage[]{amssymb} - -% with abbrev, can't tell if this is example or exercise. Abbrev would be nice for -% consistecy with fig. and eq. references. -%\crefname{Exercise}{ex.}{ex.} -%\Crefname{Exercise}{Ex.}{Ex.} - -\usepackage{peeters_layout_exercise} - -% use (a), (b), (c), numbering for sub-questions -\renewcommand{\QuestionNB}{\alph{Question}.\ } -\renewcommand{\theQuestion}{\alph{Question}} - -%-------------------------------------------------------------------- -% this fixme package + classicthesis + makeidx + hyperref -% interact oddly. -% If I use it, then my index entries don't have hyperlinks anymore? - -%\usepackage[draft,english]{fixme} -%\fxusetheme{color} - -% based on: -% \newcommand*\listfigurename{List of Figures} -% "koma-script/scrreprt.cls" - -%\newcommand*\listfixmename{List of Fixmes} - -%\newcommand{\fxwarning}[2]{} - -%-------------------------------------------------------------------- -% -% index usage: -%\index{gnat!size of} -\RequirePackage{makeidx} - \makeindex -% \newcommand{\index}[1]{} - -\newcommand{\underlineAndIndex}[1]{\textunderline{#1}\index{#1}} -\newcommand{\paragraphAndIndex}[1]{\paragraph{#1}\index{#1}} - -\usepackage{book_layout} -\usepackage{macros_bm} -%-------------------------------------------------------------------- -% -% example: -%\emtheorychapcite{nonIntegralBinomialSeries} -%\newcommand{\emtheorychapcite}[1]{\citep{GAelectrodynamics:#1}} -\newcommand{\emtheorychapcite}[1]{\cref{chap:#1}} - -%-------------------------------------------------------------------- -\usepackage{macros_qed} -%-------------------------------------------------------------------- - -% GAelectrodynamics.sty: -%\newcommand{\nbref}[1]{ -%\itemRef{GAelectrodynamics-emt}{#1} -%} - -%-------------------------------------------------------------------- -%\newcommand{\mo}{Prof.\ M. Mojahedi} -%-------------------------------------------------------------------- diff --git a/notes/.GAelectrodynamics/vectorBivectorDot.tex b/notes/.GAelectrodynamics/vectorBivectorDot.tex deleted file mode 100644 index b31434b10..000000000 --- a/notes/.GAelectrodynamics/vectorBivectorDot.tex +++ /dev/null @@ -1,138 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Vector bivector dot product}{problem:gradeselection:vectorBivectorDot}{ -The dot product of a vector and bivector in \R{N} (or in fact any metric) expands as - -\boxedEquation{eqn:gradeselection:660}{ -\Ba \cdot \lr{ \Bb \wedge \Bc } -= --\lr{ \Bb \wedge \Bc } \cdot \Ba -= -( \Ba \cdot \Bb ) \Bc --( \Ba \cdot \Bc ) \Bb. -} - -Demonstrate this by coordinate expansion using an orthonormal basis for \R{N}. - -The right hand side may look familiar. Demonstrate, for \R{3} without expansion in coordinates, that - -\boxedEquation{eqn:gradeselection:680}{ -\Ba \cdot \lr{ \Bb \wedge \Bc } -= --\Ba \cross \lr{ \Bb \cross \Bc }. -} -} % problem - -\makeanswer{problem:gradeselection:vectorBivectorDot}{ -Expansion in coordinates is frowned upon in a number of GA references, but can be a quick way to the results of interest. Consider such an expansion for a \R{N} vector space - -\begin{dmath}\label{eqn:gradeselectionProblems:681} -\begin{aligned} -\Ba \cdot \lr{ \Bb \wedge \Bc } &= \sum_{i, j, k} a_i b_j c_k \Be_i \cdot (\Be_j \wedge \Be_k) \\ -\lr{ \Bb \wedge \Bc } \cdot \Ba &= \sum_{i, j, k} a_i b_j c_k (\Be_j \wedge \Be_k) \cdot \Be_i -\end{aligned} -\end{dmath} - -Observe that these sums can be restricted to indexes \( i \ne j \), since \( \Bx \wedge \Bx = 0 \) for any \(\Bx\). The dot products are - -\begin{dmath}\label{eqn:gradeselectionProblems:820} -\Be_i \cdot (\Be_j \wedge \Be_k) -= -\gpgradeone{ \Be_i (\Be_j \wedge \Be_k) } -= -\gpgradeone{ \Be_i \Be_j \Be_k }, -\end{dmath} - -and -\begin{dmath}\label{eqn:gradeselectionProblems:840} -(\Be_j \wedge \Be_k) \cdot \Be_i -= -\gpgradeone{ (\Be_j \wedge \Be_k) \Be_i } -= -\gpgradeone{ \Be_j \Be_k \Be_i }. -\end{dmath} - -In each expansion, there are three cases, one where \( i,j,k\) are all unique. In this case, the vector product is a trivector, so the grade one selection is zero. That leaves only \( i = j \ne k \), and \( i = k \ne j \). - -Consider the \( i = j \) case in the first dot product expansion - -\begin{dmath}\label{eqn:gradeselectionProblems:860} -\gpgradeone{ \Be_i \Be_j \Be_k } -= -\gpgradeone{ \Be_i \Be_i \Be_k } -= -\gpgradeone{ \Be_k } -= -\Be_k. -\end{dmath} - -For the \( i = k \) case, this is - -\begin{dmath}\label{eqn:gradeselectionProblems:880} -\gpgradeone{ \Be_i \Be_j \Be_k } -= -\gpgradeone{ \Be_i (-\Be_k \Be_j) } -= --\gpgradeone{ \Be_i \Be_i \Be_j } -= --\gpgradeone{ \Be_j } -= --\Be_j. -\end{dmath} - -Inspection shows that the general pattern is -\begin{dmath}\label{eqn:gradeselectionProblems:900} -\Be_i \cdot (\Be_j \wedge \Be_k) = -(\Be_i \cdot \Be_j) \Be_k --(\Be_i \cdot \Be_k) \Be_j, -\end{dmath} - -and -\begin{dmath}\label{eqn:gradeselectionProblems:920} -(\Be_j \wedge \Be_k) \cdot \Be_i = -(\Be_i \cdot \Be_k) \Be_j --(\Be_i \cdot \Be_j) \Be_k. -\end{dmath} - -Substitution back into \cref{eqn:gradeselectionProblems:681} proves the first result for Euclidean spaces. For the relation to the cross product in the \R{3} case - -\begin{dmath}\label{eqn:gradeselectionProblems:940} -\Ba \cdot \lr{ \Bb \wedge \Bc } -= -\gpgradeone{ -\Ba \lr{ \Bb \wedge \Bc } -} -= -\gpgradeone{ -\Ba I \lr{ \Bb \cross \Bc } -} -= -\gpgradeone{ -I \Ba \lr{ \Bb \cross \Bc } -} -= -\gpgradeone{ -I \lr{ -\Ba \wedge \lr{ \Bb \cross \Bc } -+ -\Ba \cdot \lr{ \Bb \cross \Bc } -} -}. -\end{dmath} - -The dot product leaves the vector selection of a trivector, which is zero. Expanding the wedge product as a cross product once again gives -\begin{dmath}\label{eqn:gradeselectionProblems:960} -\Ba \cdot \lr{ \Bb \wedge \Bc } -= -\gpgradeone{ -I^2 -\Ba \cross \lr{ \Bb \cross \Bc } -} -= --\Ba \cross \lr{ \Bb \cross \Bc }. -\end{dmath} - -} % answer diff --git a/notes/.GAelectrodynamics/vectorTrivectorDot.tex b/notes/.GAelectrodynamics/vectorTrivectorDot.tex deleted file mode 100644 index bd2db76e5..000000000 --- a/notes/.GAelectrodynamics/vectorTrivectorDot.tex +++ /dev/null @@ -1,73 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Vector trivector dot product}{problem:gradeselection:vectorTrivectorDot}{ -Show that - -\begin{dmath}\label{eqn:vectorTrivectorDot:20} -\Ba \cdot \lr{ \Bb \wedge \Bc \wedge \Bd} -= -\lr{ \Bb \wedge \Bc \wedge \Bd} \cdot \Ba -= -( \Ba \cdot \Bb ) (\Bc \wedge \Bd) --( \Ba \cdot \Bc ) (\Bb \wedge \Bd) -+( \Ba \cdot \Bd ) (\Bb \wedge \Bc). -\end{dmath} - -Note that this is another specific case of the more general identity - -%\begin{dmath}\label{eqn:vectorTrivectorDot:100} -\boxedEquation{eqn:vectorTrivectorDot:100}{ -\Bx \cdot \lr{ \By_1 \wedge \By_2 \wedge \cdots \wedge \By_n } -= -\sum_{i = 1}^n (-1)^i (\Bx \cdot \By_i) \lr{ \By_1 \wedge \cdots \wedge \By_{i-1} \wedge \By_{i+1} \wedge \cdots \wedge \By_n }. -} -%\end{dmath} - -This dot product is symmetric(antisymmetric) when the grade of the blade the vector is dotted with is odd(even). - -See \citep{doran2003gap} for a demonstration that this holds for any metric. -} % problem - -\makeanswer{problem:gradeselection:vectorTrivectorDot}{ -Expanding in coordinates gives - -\begin{dmath}\label{eqn:vectorTrivectorDot:40} -\Ba \cdot \lr{ \Bb \wedge \Bc \wedge \Bd} -= \sum_{j \ne k \ne l} a_i b_j c_k d_l -\gpgradetwo{ \Be_i \Be_j \Be_k \Be_l }. -\end{dmath} - -The products within the grade two selection operator can be of either grade two or grade four, so only the terms where one of -\( i = j \), \( i = k \), or \( i = l \) contributes. Repeated anticommutation of the perperdicular unit vectors can put each such pair adjacent, where they square to unity. Those are respectively - -\begin{dmath}\label{eqn:vectorTrivectorDot:60} -\begin{aligned} -\gpgradetwo{ \Be_i \Be_i \Be_k \Be_l } &= \Be_k \Be_l \\ -\gpgradetwo{ \Be_i \Be_j \Be_i \Be_l } &= -\gpgradetwo{ \Be_j \Be_i \Be_i \Be_l } = - \Be_j \Be_l \\ -\gpgradetwo{ \Be_i \Be_j \Be_k \Be_i } &= -\gpgradetwo{ \Be_j \Be_i \Be_k \Be_i } = +\gpgradetwo{ \Be_j \Be_k \Be_i \Be_i } = \Be_j \Be_k -\end{aligned} -\end{dmath} - -Substitution back into \cref{eqn:gradeselectionProblems:681} gives - -\begin{dmath}\label{eqn:vectorTrivectorDot:80} -\Ba \cdot \lr{ \Bb \wedge \Bc \wedge \Bd} -= \sum_{j \ne k \ne l} a_i b_j c_k d_l -\lr{ -\Be_i \cdot \Be_j (\Be_k \Be_l) -- -\Be_i \cdot \Be_k (\Be_j \Be_l) -+ -\Be_i \cdot \Be_l (\Be_j \Be_k) -} -= -( \Ba \cdot \Bb ) (\Bc \wedge \Bd) --( \Ba \cdot \Bc ) (\Bb \wedge \Bd) -+( \Ba \cdot \Bd ) (\Bb \wedge \Bc). -\end{dmath} - -Repeating this from the other direction gives the same result. -} % answer diff --git a/notes/.GAelectrodynamics/vectorproduct.tex b/notes/.GAelectrodynamics/vectorproduct.tex deleted file mode 100644 index e1e47def1..000000000 --- a/notes/.GAelectrodynamics/vectorproduct.tex +++ /dev/null @@ -1,176 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -%{ -%\input{../blogpost.tex} -%\renewcommand{\basename}{vectorproduct} -%%\renewcommand{\dirname}{notes/phy1520/} -%\renewcommand{\dirname}{notes/ece1228-electromagnetic-theory/} -%%\newcommand{\dateintitle}{} -%%\newcommand{\keywords}{} -% -%\input{../peeter_prologue_print2.tex} -% -%\usepackage{peeters_layout_exercise} -%\usepackage{peeters_braket} -%\usepackage{peeters_figures} -%\usepackage{siunitx} -%%\usepackage{mhchem} % \ce{} -%%\usepackage{macros_bm} % \bcM -%%\usepackage{macros_qed} % \qedmarker -%%\usepackage{txfonts} % \ointclockwise -% -%\beginArtNoToc -% -%\generatetitle{XXX} -%%\chapter{XXX} -%%\label{chap:vectorproduct} -% -Given two vectors \( \Bx, \By \) the scalar grade of the vector product \( \Bx \By \) was shown (\cref{problem:gradeselection:RnDotProduct}) to be -\begin{equation}\label{eqn:vectorproduct:20} -\gpgradezero{ \Bx \By } -= -\sum_{i = 1}^N x_i y_i -= -\Bx \cdot \By. -\end{equation} - -The grade two selection of this product was found (\cref{problem:gradeselection:vectorwedge}) to be - -\begin{equation}\label{eqn:vectorproduct:40} -\gpgradetwo{ \Bx \By } -= -\sum_{i < j} -%(x_i y_j - x_j y_i) -\begin{vmatrix} -x_i & x_j \\ -y_i & y_j -\end{vmatrix} -\Be_i \Be_j -= -\Bx \wedge \By -= --\By \wedge \Bx. -\end{equation} - -The reader should convince themself that the vector product \( \Bx \By \) has only even grades (0,2), and can therefore be expanded as - -\begin{dmath}\label{eqn:vectorproduct:60} -\Bx \By -= -\gpgradezero{ \Bx \By } -+ -\gpgradetwo{ \Bx \By }, -\end{dmath} - -or -\boxedEquation{eqn:vectorproduct:80}{ -\Bx \By -= -\Bx \cdot \By -+ -\Bx \wedge \By. -} - -This is a fundamental and very useful relationship. In these notes this is a consequence of the axioms and the generalized definitions of the dot and wedge products. Some authors will use this to define the geometric product of two vectors. - -When considering the Euclidean space \R{3}, an additional relationship follows (\cref{problem:gradeselection:WedgeRelationshipToCrossProduct}), which is also incredibly useful - -\boxedEquation{eqn:vectorproduct:100}{ -\Bx \By -= -\Bx \cdot \By -+ -I -(\Bx \cross \By). -} - -Note that this is the GA equivalent of the Pauli relationship \cref{eqn:GAmotivation:120} that will be familiar to a student of quantum spin states. -The ability to combine dot and cross product relationships into a single multivector equation is not just a theoretical nicety. This happens to also be the reason that GA is so applicable to the study of electromagnetism. To illustrate this, and provide a hint of things to come, consider the GA formulation of the electrostatic and magnetostatic Maxwell equations. - -\makeexample{Electrostatic and magnetostatics.}{example:vectorproduct:electrostatics}{ - -With no magnetic current, no magnetic sources, and no time derivatives, Maxwell's equations in simple media take the form - -\begin{dmath}\label{eqn:vectorproduct:120} -\begin{aligned} -\spacegrad \cdot \BB &= 0 \\ -\spacegrad \cross \BB &= \mu \BJ \\ -\spacegrad \cross \BE &= 0 \\ -\spacegrad \cdot \BE &= \frac{\rho}{\epsilon}. -\end{aligned} -\end{dmath} - -For electrostatic conditions \( \BJ = 0 \), so using \cref{eqn:vectorproduct:100} the first and last equations can be combined into a single first order homogeneous multivector gradient equation - -\begin{equation}\label{eqn:vectorproduct:140} -\spacegrad \BB -= -\spacegrad \cdot \BB +I (\spacegrad \cross \BB ) -= -0. -\end{equation} - -The electric gradient equation is - -\begin{equation}\label{eqn:vectorproduct:160} -\spacegrad \BE -= -\spacegrad \cdot \BE +I (\spacegrad \cross \BE ) -= -\frac{\rho}{\epsilon}. -\end{equation} - -Maxwell's equations are reduced to two multivector equations with this transformation -\begin{dmath}\label{eqn:vectorproduct:180} -\begin{aligned} -\spacegrad \BE &= \frac{\rho}{\epsilon} \\ -\spacegrad \BB &= 0. -\end{aligned} -\end{dmath} - -For magnetostatics \( \rho = 0 \), and the same assembly of Maxwell's equations gives - -\begin{dmath}\label{eqn:vectorproduct:220} -\begin{aligned} -\spacegrad \BB &= I \mu \BJ \\ -\spacegrad \BE &= 0. -\end{aligned} -\end{dmath} - -It will be seen later that it is actually more natural to express magnetic fields as a bivector \( I \BB \). Using \( I^2 = -1 \) (\cref{problem:gradeselection:R3PseudoscalarSquare}) the magnetostatic equation takes the form - -\begin{dmath}\label{eqn:vectorproduct:240} -\spacegrad (I \BB) = - \mu \BJ. -\end{dmath} - -Both the electrostatic and magnetostatic equations can be solved directly using the Green's function for the gradient, producing the Coulomb integral for the electric field and Biot-Savart's law for the magnetic field. -Before demonstrating this, the concepts required to attack multivector integrals must be formulated. -} % example - -Using \cref{problem:gradeselection:dotprod} and \cref{eqn:vectorproduct:80} it can be shown that the wedge product is an explicit antisymmetrized sum of vector products, just as the dot product is the symmetrized vector product sum - -\boxedEquation{eqn:vectorproduct:300}{ -\begin{aligned} -\Bx \cdot \By &= \inv{2} \lr{ \Bx \By + \By \Bx } \\ -\Bx \wedge \By &= \inv{2} \lr{ \Bx \By - \By \Bx } -\end{aligned} -} - -Some authors will use these as the respective definitions of the dot and wedge products. - -The non-commutative nature of the vector product was one of the first observed consequences of the axioms. The vector product is also not generally anticommutative, as was the case for normal vectors. Rearranging \cref{eqn:vectorproduct:300} provides the general commutation identity for two vectors - -%\begin{dmath}\label{eqn:vectorproduct:320} -\boxedEquation{eqn:vectorproduct:320}{ -\By \Bx = 2 \Bx \cdot \By - \Bx \By. -} -%\end{dmath} - -Observe that when the vectors are perpendicular, the strict anticommutation result follows. -This can be a handy tool for abstract multivector expression manipulation. - - -%} -%\EndNoBibArticle diff --git a/notes/.GAelectrodynamics/vectorproductCyclicPermutation.tex b/notes/.GAelectrodynamics/vectorproductCyclicPermutation.tex deleted file mode 100644 index 78849139b..000000000 --- a/notes/.GAelectrodynamics/vectorproductCyclicPermutation.tex +++ /dev/null @@ -1,28 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% -\makeproblem{Commutation within grade zero selection}{problem:vectorproduct:cyclicpermutation}{ - -It was previously shown using coordinates that - -\begin{dmath}\label{eqn:vectorproduct:260} -\gpgradezero{ \Bx \By } = \gpgradezero{ \By \Bx }. -\end{dmath} - -Repeat this proof using \cref{eqn:vectorproduct:80}. -} % problem - -\makeanswer{problem:vectorproduct:cyclicpermutation}{ -\begin{dmath}\label{eqn:vectorproduct:301} -\gpgradezero{ \By \Bx } -= -\gpgradezero{ \By \cdot \Bx + \By \wedge \Bx } -= -\gpgradezero{ \By \cdot \Bx } -= -\gpgradezero{ \Bx \cdot \By } -= -\gpgradezero{ \Bx \By } -\end{dmath} -} % answer diff --git a/notes/.GAelectrodynamics/wedgeantisym.tex b/notes/.GAelectrodynamics/wedgeantisym.tex deleted file mode 100644 index 7289971b5..000000000 --- a/notes/.GAelectrodynamics/wedgeantisym.tex +++ /dev/null @@ -1,21 +0,0 @@ -% -% Copyright © 2016 Peeter Joot. All Rights Reserved. -% Licenced as described in the file LICENSE under the root directory of this GIT repository. -% - -\makeproblem{Vector wedge antisymmetric structure}{problem:vectorproduct:wedgeantisym}{ -Prove the wedge product relationship of \cref{eqn:vectorproduct:300}. -} % problem - -\makeanswer{problem:vectorproduct:wedgeantisym}{ - -Rearranging \cref{eqn:vectorproduct:80} for the wedge product and substitution of the dot product symmetric sum from \cref{problem:gradeselection:dotprod} gives - -\begin{dmath}\label{eqn:gradeselectionProblems:800} -\Bx \wedge \By -= \Bx \By - \Bx \cdot \By -= \Bx \By - \inv{2} \lr{ \Bx \By + \By \Bx } -= \inv{2} \lr{ 2 \Bx \By - \Bx \By - \By \Bx } -= \inv{2} \lr{ \Bx \By - \By \Bx }. -\end{dmath} -} % answer