├── README.md ├── LICENSE ├── defs.tex ├── .gitignore ├── Proofs.tex ├── CollectionOfFormulas.tex └── References.bib /README.md: -------------------------------------------------------------------------------- 1 | # Formulas 2 | A collection of formulas needed for everyday calculations in geometric algebra and calculus 3 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. 2 | -------------------------------------------------------------------------------- /defs.tex: -------------------------------------------------------------------------------- 1 | %Better, emptier empty set 2 | \renewcommand{\emptyset}{\varnothing} 3 | 4 | \newcommand{\GA}[2][]{\ensuremath{\mathcal{G}_{#1}({#2})}} 5 | \newcommand{\grade}[2]{\ensuremath{\langle#2\rangle_{#1}}} 6 | \newcommand{\deriv}[2]{\ensuremath{\frac{\mathrm{d}{#1}}{\mathrm{d}{#2}}}} 7 | \newcommand{\cev}[1]{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{#1}}}}}} 8 | \newcommand{\reverse}[1]{\tilde{#1}} %The Clifford reverse 9 | \newcommand{\reals}[1]{\ensuremath{\mathbb{R}^{#1}}} 10 | \newcommand{\vol}[1]{\ensuremath{\mathrm{vol}(#1)}} 11 | \newcommand{\linner}{\mathbin{\ensuremath{\scalebox{1.4}{$\lrcorner$}}}} 12 | \newcommand{\rinner}{\mathbin{\ensuremath{\scalebox{1.4}{$\llcorner$}}}} 13 | 14 | \newcommand{\comment}[1]{} 15 | 16 | \providecommand{\abs}[1]{\lvert#1\rvert} 17 | \providecommand{\norm}[1]{\left\lVert#1\right\rVert} 18 | \providecommand{\normed}[1]{\hat{#1}} 19 | \DeclareMathOperator{\sign}{sgn} 20 | \DeclareMathOperator{\tr}{Tr} 21 | \providecommand{\signX}[1]{\ensuremath{s_{#1}}} -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | 10 | ## Intermediate documents: 11 | *.dvi 12 | *-converted-to.* 13 | # these rules might exclude image files for figures etc. 14 | # *.ps 15 | # *.eps 16 | # *.pdf 17 | 18 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 19 | *.bbl 20 | *.bcf 21 | *.blg 22 | *-blx.aux 23 | *-blx.bib 24 | *.brf 25 | *.run.xml 26 | 27 | ## Build tool auxiliary files: 28 | *.fdb_latexmk 29 | *.synctex 30 | *.synctex.gz 31 | *.synctex.gz(busy) 32 | *.pdfsync 33 | 34 | ## Auxiliary and intermediate files from other packages: 35 | 36 | # algorithms 37 | *.alg 38 | *.loa 39 | 40 | # achemso 41 | acs-*.bib 42 | 43 | # amsthm 44 | *.thm 45 | 46 | # beamer 47 | *.nav 48 | *.snm 49 | *.vrb 50 | 51 | #(e)ledmac/(e)ledpar 52 | *.end 53 | *.[1-9] 54 | *.[1-9][0-9] 55 | *.[1-9][0-9][0-9] 56 | *.[1-9]R 57 | *.[1-9][0-9]R 58 | *.[1-9][0-9][0-9]R 59 | *.eledsec[1-9] 60 | *.eledsec[1-9]R 61 | *.eledsec[1-9][0-9] 62 | *.eledsec[1-9][0-9]R 63 | *.eledsec[1-9][0-9][0-9] 64 | *.eledsec[1-9][0-9][0-9]R 65 | 66 | # glossaries 67 | *.acn 68 | *.acr 69 | *.glg 70 | *.glo 71 | *.gls 72 | 73 | # gnuplottex 74 | *-gnuplottex-* 75 | 76 | # hyperref 77 | *.brf 78 | 79 | # knitr 80 | *-concordance.tex 81 | *.tikz 82 | *-tikzDictionary 83 | 84 | # listings 85 | *.lol 86 | 87 | # makeidx 88 | *.idx 89 | *.ilg 90 | *.ind 91 | *.ist 92 | 93 | # minitoc 94 | *.maf 95 | *.mtc 96 | *.mtc0 97 | 98 | # minted 99 | _minted* 100 | *.pyg 101 | 102 | # morewrites 103 | *.mw 104 | 105 | # nomencl 106 | *.nlo 107 | 108 | # sagetex 109 | *.sagetex.sage 110 | *.sagetex.py 111 | *.sagetex.scmd 112 | 113 | # sympy 114 | *.sout 115 | *.sympy 116 | sympy-plots-for-*.tex/ 117 | 118 | # todonotes 119 | *.tdo 120 | 121 | # xindy 122 | *.xdy 123 | 124 | # WinEdt 125 | *.bak 126 | *.sav 127 | CollectionOfFormulas.pdf 128 | Proofs.pdf 129 | -------------------------------------------------------------------------------- /Proofs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[a4paper,12pt]{article} 2 | \usepackage{amsmath, amsthm} 3 | \usepackage{amsfonts} 4 | \usepackage{amssymb} 5 | \usepackage[utf8]{inputenc} 6 | \usepackage[T1]{fontenc} 7 | \usepackage{slashed} 8 | \usepackage{commath} 9 | \usepackage[style=phys, biblabel=brackets, eprint=true, sorting=none, backend=biber, maxnames=99]{biblatex} 10 | \usepackage{graphicx} 11 | \usepackage{xr} 12 | \externaldocument[form]{CollectionOfFormulas} 13 | 14 | %Bibliographies 15 | \addbibresource{References.bib} 16 | 17 | %Page Settings 18 | \setlength{\textwidth}{14.5cm} 19 | \setlength{\textheight}{22.5cm} 20 | \setlength{\topmargin}{-5mm}%{-1.5cm} 21 | %\setlength{\evensidemargin}{0.36cm} 22 | \setlength{\oddsidemargin}{5mm}%{0.36cm} 23 | 24 | \input{defs} 25 | 26 | \newcommand{\meqref}[1]{M.\eqref{form#1}} 27 | 28 | \newtheorem{thm}{Theorem} 29 | \newtheorem{lem}[thm]{Lemma} 30 | \newtheorem{cor}[thm]{Corollary} 31 | 32 | \theoremstyle{definition} 33 | \newtheorem{defn}{Definition} 34 | 35 | \newenvironment{minipeqn}[1][]{\begin{minipage}[#1]{.45\textwidth}\begin{equation}}{\nonumber\end{equation}\end{minipage}} 36 | 37 | %opening 38 | \title{Proofs for Formulas in Geometric Algebra and Geometric Calculus} 39 | 40 | \author{} 41 | 42 | \begin{document} 43 | 44 | \maketitle 45 | 46 | \section{Introduction} 47 | 48 | This file shall contain proofs and derivations for such formulas in the main collection which are not found in published literature. If you want to submit a pull request for a formula which you derived yourself, and have not published an article exposing the proof that could be cited, this is the place to submit the derivation. 49 | 50 | Formulae in this document will be referenced as usual, (xx), and formulae in the main document as M.(xx). 51 | \section{Multivector directed derivatives} 52 | 53 | Many of the following are present and proven in the literature, however, as they follow in a nice logical sequence, we present the proofs here. 54 | We wish to remind the reader that $\signX{X} = \sign{X * \reverse{X}}$, as defined in the main text. 55 | 56 | \paragraph{Proof of \meqref{eq:normsqrdif}:} write $\norm{X}^2 = \signX{X} X * \reverse{X}$, use the product rule (Proposition 8. in \cite{HitzerCalculus}), (1.20a) in \cite{CA2GC} and \meqref{eq:scalpdiff}. 57 | 58 | \paragraph{Proof of \meqref{eq:normdif}:} Write $\norm{X} = \sqrt{\norm{X}^2}$ and use the chain rule with $x \mapsto \sqrt{x}$ ($x$ a real number) and \meqref{eq:normsqrdif}. 59 | 60 | \paragraph{Proof of \meqref{eq:normeddif}:} write 61 | \begin{equation} 62 | A*\partial \normed{X} = A*\dot{\partial} \frac{\dot{X}}{\norm{X}} + A*\dot{\partial} X \frac{1}{\dot{\norm{X}}} = \frac{A}{\norm{X}} - X \frac{\signX{X} \frac{A * \reverse{X}}{\norm{X}}}{\norm{X}^2} 63 | \end{equation} 64 | and simplify, where we used the product rule and chain rule on $x \mapsto 1/x$ and $\meqref{eq:normdif}$. 65 | The second line follows immediately from the definition of the inverse \meqref{eq:inversedef} and the projection \meqref{eq:projdef}. 66 | 67 | \paragraph{Proof of \meqref{eq:normtokdif}:} use the chain rule and \meqref{eq:normdif} on $\norm{X}$ composed with $x \mapsto x^k$ for real $x$. 68 | 69 | \paragraph{Proof of \meqref{eq:evenbladedif}:} for a blade, $X_k^2 = X_k * X_k$. 70 | Apply the product rule to get $A_k*\partial X_k^2 = 2 A_k * X_k$, and the chain rule to $(X_k^2)^{n/2}$ to get the result. 71 | The second line follows from $X_k^{-1} = \frac{X_k}{X_k^2}$, when $X_k$ is invertible. 72 | 73 | \paragraph{Proof of \meqref{eq:oddbladedif}:} Write $X_k^n = X_k X_k^{n-1}$, where now $n-1$ is even. 74 | Use the product rule and the fact that $X_k^{n-1}$ commutes, and apply \meqref{eq:evenbladedif}. 75 | The second line again follows from the definition of the projection when $X_k$ is invertible. 76 | 77 | \paragraph{Proof of \meqref{eq:bladeexpdif}:} for $X_k^2 < 0$: use \meqref{eq:bladeexptrig} to write 78 | \begin{equation} 79 | A_k * \partial e^{X_k} = A_k * \partial(\cos(\norm{X_k}) + \normed{X_k} \sin(\norm{X_k})), 80 | \end{equation} 81 | apply the chain rule and \meqref{eq:normdif} to differentiate the sine and cosine terms, and \meqref{eq:normeddif} ($X_k$ is invertible since $X_k^2 < 0$) to differentiate $\normed{X_k}$ to get 82 | \begin{equation} 83 | \begin{split} 84 | A_k * \partial e^{X_k} =& -\signX{X_k}\frac{A_k*\reverse{X_k}}{\norm{X_k}}\sin(\norm{X_k}) + \signX{X_k}\frac{A_k*\reverse{X_k}}{\norm{X_k}} \normed{X_k} \cos(\norm{X_k})\\ 85 | &+ \frac{A- P_{X}(A)}{\norm{X}} \sin{\norm{X_k}}. 86 | \end{split} 87 | \end{equation} 88 | Since $\normed{X_k}^2 = -1$, we can pull out a factor of 89 | \begin{equation*} 90 | \signX{X_k}\frac{A_k*\reverse{X_k}}{\norm{X_k}}\normed{X_k} = P_{X_k}(A_k) 91 | \end{equation*} 92 | from the trigonometric terms and rewrite that as an exponential. 93 | 94 | The case $X_k^2 > 0$ goes identically, with the appropriate substitutions of hyperbolic functions for the ordinary ones. 95 | 96 | The case $X_k^2 = 0$ goes as follows: first observe that $(X_k + \epsilon A_k)^2 = \epsilon (X_k A_k + A_k X_k) + \mathcal{O}(\epsilon^2)$. 97 | Then by associativity $(X_k + \epsilon A_k)^{2n} = \mathcal{O}(\epsilon^n)$ and $(X_k + \epsilon A_k)^{2n + 1} = \mathcal{O}(\epsilon^n)$, for $n \geq 1$. 98 | Therefore we can write the power series 99 | \begin{equation*} 100 | e^{X_k + \epsilon A_k} = 1 + X_k + \epsilon A_k + \frac{1}{2} \epsilon (A_k X_k + X_k A_k) + \frac{1}{3!} \epsilon X_k A_k X_k + \mathcal{O}(\epsilon^2), 101 | \end{equation*} 102 | and the claim follows immediately from the definition $A_k * \partial e^{X_k} = \frac{\dif}{\dif \epsilon} e^{X_k + \epsilon A_k} \vert_{\epsilon = 0}$. 103 | 104 | \printbibliography[heading=bibintoc, title={References}] 105 | 106 | \end{document} 107 | -------------------------------------------------------------------------------- /CollectionOfFormulas.tex: -------------------------------------------------------------------------------- 1 | \documentclass[a4paper,12pt]{article} 2 | \usepackage{amsmath, amsthm} 3 | \usepackage{amsfonts} 4 | \usepackage{amssymb} 5 | \usepackage[utf8]{inputenc} 6 | \usepackage[T1]{fontenc} 7 | \usepackage{slashed} 8 | \usepackage{commath} 9 | \usepackage[style=phys, biblabel=brackets, eprint=true, sorting=none, backend=biber, maxnames=99]{biblatex} 10 | \usepackage{graphicx} 11 | 12 | %Bibliographies 13 | \addbibresource{References.bib} 14 | 15 | %Page Settings 16 | \setlength{\textwidth}{14.5cm} 17 | \setlength{\textheight}{22.5cm} 18 | \setlength{\topmargin}{-5mm}%{-1.5cm} 19 | %\setlength{\evensidemargin}{0.36cm} 20 | \setlength{\oddsidemargin}{5mm}%{0.36cm} 21 | 22 | \input{defs} 23 | 24 | \newenvironment{minipeqn}[1][]{\begin{minipage}[#1]{.45\textwidth}\begin{equation}}{\nonumber\end{equation}\end{minipage}} 25 | 26 | %opening 27 | \title{Formulas in Geometric Algebra and Geometric Calculus} 28 | 29 | \author{} 30 | 31 | \begin{document} 32 | 33 | \maketitle 34 | 35 | \section{Notation} 36 | 37 | Unless otherwise stated, lower case latin alphabet stand for vectors, capital latin alphabet for multivectors, and $A_k$ for a multivector with only grade $k$ components, $A_k = \grade{k}{A_k}$. 38 | 39 | We use both the left and right inner products and the "grade symmetric" inner product, which ever is more convenient: 40 | \begin{eqnarray} 41 | A_r \linner B_s &=& \grade{s-r}{A_r B_s}\\ 42 | A_r \rinner B_s &=& \grade{r-s}{A_r B_s}\\ 43 | A_r \cdot B_s &=& \grade{\abs{r-s}}{A_r B_s},\ \textrm{if}\ r, s \neq 0\\ 44 | A_r \cdot B_s &=& 0,\ \textrm{if}\ r = 0\ \textrm{or}\ s = 0\\ 45 | a \linner b &=& a \rinner b = a \cdot b 46 | \end{eqnarray} 47 | 48 | We define $\signX{X} := \sign{X * \reverse{X}}$ for multivector $X$. 49 | In a space with positive definite signature $\signX{X} = 1$ for all multivectors. 50 | 51 | Scalar product and norm: 52 | \begin{align} 53 | A * B &= \grade{}{AB}\\ 54 | \norm{A} &= \sqrt{|\reverse{A} * A|}\\ 55 | \normed{A} &= \frac{A}{\norm{A}} 56 | \end{align} 57 | 58 | Inverse, when $A_k$ a $k$ -blade and $\norm{A_k} \neq 0$, \emph{i.e.} $A_k$ is an invertible blade: 59 | \begin{equation} 60 | \label{eq:inversedef} 61 | A_k^{-1} = \frac{A_k}{A_k^2} = \frac{\signX{A_k}\reverse{A_k}}{\norm{A_k}^2}. 62 | \end{equation} 63 | 64 | Projection to subspace defined by $k$ -blade $B_k$, when $\norm{B_k} \neq 0$: 65 | \begin{eqnarray} 66 | \label{eq:projdef} 67 | P_{B_k}(A) &=& B^{-1}_{k}(B_k \rinner A) 68 | \end{eqnarray} 69 | 70 | \section{Geometric Algebra} 71 | 72 | \subsection{Basic identities} 73 | 74 | Inner, wedge and geometric products with vectors\cite{CA2GC}: 75 | %page 8, (1.27a, b), (1.28) 76 | \begin{eqnarray} 77 | a\cdot A_r &=& \frac{1}{2}(a A_r - (-1)^r A_r a) = a\linner A_r\\ 78 | a\wedge A_r &=& \frac{1}{2}(a A_r + (-1)^r A_r a)\\ 79 | a A_r &=& a \cdot A_r + a \wedge A_r 80 | \end{eqnarray} 81 | Reordering \cite{CA2GC, Chisolm:2012aa}: 82 | %GA2GC page 8, (1.23a, b), Chisolm page 83, (10-11) 83 | \begin{eqnarray} 84 | A_r\cdot B_s &=& (-1)^{r(s-1)} B_s \cdot A_r\ \textrm{for}\ r \leq s\\ 85 | A_r\linner B_s &=& (-1)^{r(s-1)} B_s \rinner A_r\\ 86 | A_r\wedge B_s &=& (-1)^{rs} B_s \wedge A_r\\ 87 | \end{eqnarray} 88 | More generally \cite{CA2GC, Chisolm:2012aa}: 89 | %Chisolm p. 83, (2), GA2GC p. 6, (1.20d) 90 | \begin{eqnarray} 91 | \grade{r+s-2j}{A_r B_s} &=& (-1)^{rs - j} \grade{r + s - 2j}{B_s A_r}\\ 92 | \grade{q}{A_r B_s C_t} &=& (-1)^{(q^2 + r^2 + s^2 + t^2 - q - r - s - t)/2} \grade{q}{C_t B_s A_r} 93 | \end{eqnarray} 94 | 95 | \subsection{Functions} 96 | 97 | The exponential: 98 | \begin{align} 99 | \label{eq:expdef} 100 | \exp(A) := e^{A} = \sum_{k = 0}^\infty \frac{1}{k!} A^k. 101 | \end{align} 102 | Via the standard proof, the exponential converges in the $\norm{\cdot}$ -norm for all $A$. 103 | 104 | For blade $A_k$, 105 | \begin{align} 106 | \label{eq:bladeexptrig} 107 | e^{A_k} = \left\{\begin{aligned} 108 | &\cos(\norm{A_k}) + \normed{A_k} \sin(\norm{A_k}), &\sign{A_k^2} < 0&\\ 109 | &\cosh(\norm{A_k}) + \normed{A_k} \sinh(\norm{A_k}), &\sign{A_k^2} > 0&\\ 110 | &1 + A_k, &A_k^2 = 0& 111 | \end{aligned} \right. 112 | \end{align} 113 | 114 | \section{Directed derivatives} 115 | 116 | The variable being differentiated is denoted $X$ when the formula is valid for multivectors, and $x$ when it applies only to vectors. Correspondingly, the direction being differentiated in is denoted $a$ and $A$ for vector and multivector directions, respectively. 117 | 118 | We assume that the direction $A$ only contains the grades in $X$. If not, the $A$ on the right hand side becomes $P(A)$, where $P$ is the projection to the grades of $X$. 119 | 120 | Note that we use the definition $A * B = \grade{}{A B}$ for the scalar product. Changing this would give slight differences in the results. 121 | 122 | Elementary derivatives \cite{HitzerCalculus}: 123 | \begin{eqnarray} 124 | %Hitzer page 6, 7 125 | \label{eq:iddiff} 126 | A * \partial X &=& A\\ 127 | \label{eq:scalpdiff} 128 | A * \partial (X * B) &=& A * B 129 | \end{eqnarray} 130 | 131 | Given multivector valued functions $f(X)$ and $F(X)$, the function $G(X) := F(f(X))$ obeys the chain rule \cite{HitzerCalculus}: 132 | \begin{equation} 133 | %Hitzer page 7: 134 | \label{eq:chainrule} 135 | A * \partial G(X) = (A * \partial f(X))*\partial_Y F(Y)\vert_{Y = f(X)}. 136 | \end{equation} 137 | 138 | The norm and the unit multivector, for $\norm{X} \neq 0$: 139 | \begin{align} 140 | %proofs document 141 | \label{eq:normsqrdif} 142 | A * \partial \norm{X}^2 &= \signX{X}2 A * \reverse{X}\\ 143 | \label{eq:normdif} 144 | A * \partial \norm{X} &= \signX{X} \frac{A * \reverse{X}}{\norm{X}}\\ 145 | \label{eq:normeddif} 146 | A * \partial \normed{X} &= \frac{A \norm{X} - \signX{X} A * \normed{X} \reverse{X}}{\norm{X}^2}\quad \textrm {for all $X$}\nonumber\\ 147 | &= \frac{A - A * X X^{-1}}{\norm{X}} = \frac{A- P_{X}(A)}{\norm{X}}\quad \textrm{when $X$ an invertible blade}\\ 148 | \label{eq:normtokdif} 149 | A * \partial \norm{X}^k &= \signX{X} k A * \normed{\reverse{X}} \norm{X}^{k-1} 150 | \end{align} 151 | 152 | Powers of a $k$-blade $X_k$ in the direction of $k$-vector $A_k$: 153 | %Should we come up with a notation for a blade, or is the distinction $X_k$ enough? 154 | \begin{align} 155 | %proofs document 156 | \label{eq:evenbladedif} 157 | \begin{split} 158 | A_k * \partial X_k^n &= n A_k *X_k X_k^{n - 2} \quad n\ \mathrm{even}\\ 159 | &= n A_k * X_k^{-1} X_k^{n},\quad n \textrm{ even and invertible} 160 | \end{split}\\ 161 | \label{eq:oddbladedif} 162 | \begin{split} 163 | A_k * \partial X_k^n &= A_k X_k^{n-1} + (n-1) A_k*X_k X_k^{n-2} 164 | \quad n \textrm{ odd}\\ 165 | &= \left(A_k + (n-1) P_{X_k}(A_k)\right) X_k^{n-1},\quad n\ \textrm{odd and invertible} 166 | \end{split} 167 | \end{align} 168 | 169 | Exponential of a blade: 170 | \begin{equation} 171 | %proofs document 172 | \label{eq:bladeexpdif} 173 | A_k *\partial e^{X_k} = \left\{\begin{aligned} 174 | &P_{X_k}(A_k) e^{X_k} + (A_k - P_{X_k}(A_k)) \frac{\sin(\norm{X_k})}{\norm{X_k}}, &\sign{X_k^2} < 0&\\ 175 | &P_{X_k}(A_k) e^{X_k} + (A_k - P_{X_k}(A_k)) \frac{\sinh(\norm{X_k})}{\norm{X_k}}, &\sign{X_k^2} > 0&\\ 176 | &A_k + \frac{1}{2}(A_k X_k + X_k A_k) + \frac{1}{6}X_k A_k X_k, &X_k^2 = 0& 177 | \end{aligned} \right. 178 | \end{equation} 179 | 180 | \section{Vector derivatives} 181 | 182 | \section{Antiderivatives} 183 | 184 | 185 | \printbibliography[heading=bibintoc, title={References}] 186 | 187 | \end{document} 188 | -------------------------------------------------------------------------------- /References.bib: -------------------------------------------------------------------------------- 1 | %% Saved with string encoding Unicode (UTF-8) 2 | @article{FundamentalTheorem, 3 | year={2011}, 4 | issn={0188-7009}, 5 | journal={Advances in Applied Clifford Algebras}, 6 | volume={21}, 7 | number={1}, 8 | doi={10.1007/s00006-010-0242-8}, 9 | title={Fundamental Theorem of Calculus}, 10 | url={http://dx.doi.org/10.1007/s00006-010-0242-8}, 11 | publisher={SP Birkhäuser Verlag Basel}, 12 | keywords={32A26; 58A05; 58A10; 58A15; Geometric algebra; geometric calculus; Green’s theorem; monogenic functions; Stokes’ theorem}, 13 | author={Sobczyk, Garret and Sánchez, OmarLeón}, 14 | pages={221-231}, 15 | language={English} 16 | } 17 | 18 | @article{UnifiedLanguage, 19 | Author = {Joan Lasenby and Anthony N. 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