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Derivation-on-Manifold-CheatSheet.tex
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\documentclass[a4paper,12pt]{scrartcl}
\usepackage{pdflscape}
\usepackage{tabularx}
\usepackage{array}
\setlength{\extrarowheight}{6mm}
\usepackage{geometry}
\geometry{
a4paper,
total={185mm,277mm},
left=5mm,
right=5mm,
top=5mm,
bottom=5mm,
}
\usepackage[subpreambles=true]{standalone}
\usepackage{amssymb}
\usepackage[leqno]{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage[symbol]{footmisc}
\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
\providecommand{\ExtD}{\textrm{d}}
\providecommand{\Lie}{\mathcal{L}}
\usepackage{arydshln}
%https://tex.stackexchange.com/questions/48980/whole-page-table-with-tabularx
\begin{document}
\begin{landscape}
\thispagestyle{empty}
\noindent
\paragraph{CARTAN CALCULUS}
\mbox{}\\
$\quad$Suppose $M$ is a smooth manifold, $x^\mu$ a coordinate chart. Denote by $\Omega(M)$ the algebra of differential forms on $M$ and by $\mathfrak{X}(M)$ the $C^\infty(M)$-module of vector fields. \\
\vspace{5mm}
\begin{tabularx}{\linewidth}{|c|X|X|c|}
\hline
& $f \in C^\infty(M)$ & $\ExtD x^\mu \in \Omega^1(M)$ & $\omega^{(k)} \wedge \beta$ \\
\hline
$\ExtD$ & $\ExtD f = \left(\dfrac{\partial f }{\partial x^\nu} \right) \: \ExtD x^\nu$ & 0 & $\left( \ExtD \omega \right) \wedge \beta + (-)^k \omega \wedge \left( \ExtD\beta \right) $ \\
$\Lie_X$ & $\Lie_X f = X(f) = \left(\dfrac{\partial f }{\partial x^\nu} \right) \: X^\nu$ & $\Lie_X \ExtD x^\mu = \ExtD \left(X^a \partial_a x^\mu\right) = \ExtD \left(X^a \delta_a^\mu \right) = \ExtD\left(X^\mu\right) = \partial_\nu X^\mu \ExtD x^\nu$ & $\left( \Lie_X \omega \right) \wedge \beta + \omega \wedge \left(\Lie_X\beta \right)$ \\
$\iota_X$ & $0$ & $\iota_X \ExtD x^\mu = \ExtD x^\mu (X) = X^\mu$ & $\left( \iota_X \omega \right) \wedge \beta + (-)^k \omega \wedge \left( \iota_X\beta \right) $ \\
$g^\ast$ \footnotemark[4] & $g^\ast \left(f\right) = f \circ g $ & $ g^\ast \left(\ExtD x^\mu \right) = \ExtD\left(x^\mu \circ g \right)$ & $g^\ast\left(\omega\right) \wedge g^\ast \left( \beta \right)$ \\ %& & & & & \\
\hline
\end{tabularx}
\begin{minipage}[c][.585\textheight]{0.46 \linewidth}
\fbox{
\parbox{\textwidth}{
The \emph{Cartan calculus} involves the following three \emph{graded derivations} on $\Omega(M)$
\begin{itemize}\itemsep0em
\item the \emph{exterior derivative} $d$;
\item the \emph{Lie derivative operators} $\Lie_X$, where $X \in \mathfrak{X}(M)$;
\item the \emph{contraction operators} $\iota_X$, where $X \in \mathfrak{X}(M)$.
\end{itemize}
\begin{center}
\includestandalone[scale=0.80]{cartancalculusdiagram}
\end{center}
Satisfying the following identities:
\begin{align}
\ExtD^2 &= 0 \label{cartfirst}\\
\ExtD \Lie_X - \Lie_X d &= 0 \\
\ExtD \iota_X + \iota_X d &= \Lie_X \qquad \textrm{\small\emph{(Cartan's formula)}}\label{magic}\\
\Lie_X \Lie_Y - \Lie_Y \Lie_X &= \Lie_{[X,Y]} \\
\Lie_X \iota_Y - \iota_Y \Lie_X &= \iota_{[X,Y]}\\
\iota_X \iota_Y + \iota_Y \iota_X &= 0 \label{cartlast}
\end{align}
}
}
\vfill
\end{minipage}
\hspace{1cm}
\begin{minipage}[c][.55\textheight]{0.46 \linewidth}
\fbox{
\parbox{\textwidth}{
\begin{displaymath}
\Omega(M) = \left(\bigoplus_{k=0}^m \Omega^k(M), \wedge \right) \qquad \textrm{\emph{Grassmann Algebra} on M}
\end{displaymath}
is a (graded-)commutative graded algebra over the ring $C^\infty(M)$.
}
}
\vfill
Recall:
\vfill
\fbox{
\parbox{\textwidth}{
A \emph{(graded-)commutatative ($I$-)graded algebra} $(V, \wedge)$ is an algebra over ring $R$ such that :
\begin{displaymath}
V = \bigoplus_{i \in I} V_i
\end{displaymath}
and
\begin{displaymath}
v^{(i)} \wedge w = (-)^i w \wedge v^{(i=} \qquad \forall v^{(i)} \in V_{i}, \; w \in V
\end{displaymath}
}
}
\vfill
\fbox{
\parbox{\textwidth}{
A \emph{graded derivation} of $\Omega(M)$ is a degree $k$ linear operator $A$ on $\Omega(M)$ :
\begin{displaymath}
A:\Omega^p(M) \rightarrow \Omega^{p+k}(M)
\end{displaymath}
such that:
\begin{displaymath}
A (\omega \wedge \eta) = A(\omega) \wedge \eta + (-1)^{kp} \omega \wedge A(\eta) \qquad \forall \omega \in \Omega^k(M) , \; \eta \in \Omega^\cdot(M)
\end{displaymath}
}
}
\end{minipage}
\footnote[4]{$g:N \sim (y^A) \rightarrow M \sim (x^\mu) $}
\newpage
\paragraph{ABSOLUTE DIFFERENTIAL CALCULUS}
\mbox{}\\
$\quad$Suppose $M$ is a smooth manifold, $x^\mu$ a coordinate chart. Denote by $\Omega(M)$ the algebra of differential forms on $M$ and by $\mathfrak{X}(M)$ the $C^\infty(M)$-module of vector fields. \\
\vspace{5mm}
\begin{tabularx}{\linewidth}{|c|X|c|c|c|c|}
\hline
& $f \in C^\infty(M)$ & $\ExtD x^\mu \in \Omega^1(M)$ & $\partial_\mu \in \mathfrak{X}(M)$ & $T_1 \otimes T_2$ & $\omega^{(k)} \wedge \beta$ \\
\hline
$\ExtD$ & $\left(\dfrac{\partial f }{\partial x^\nu} \right) \: \ExtD x^\nu$ & 0 & - & - & $\left( \ExtD \omega \right) \wedge \beta + (-)^k \omega \wedge \left( \ExtD\beta \right) $\\
%
$\Lie_X$ & $X(f) = X^\nu \left(\dfrac{\partial f }{\partial x^\nu} \right)$ & $\Lie_X \ExtD x^\mu = \ExtD\left(X^\mu\right) =\left(\dfrac{\partial X^\mu }{\partial x^\nu} \right) \ExtD x^\nu$ & $\Lie_X \partial_\mu = [X, \partial_\mu]$ & $\left(\Lie_X T_1\right) \otimes T_2 + T_1 \otimes \left(\Lie_X T_2 \right)$ & $\left( \Lie_X \omega \right) \wedge \beta + \omega \wedge \left(\Lie_X\beta \right)$ \\
%
$\iota_X$ & $0$ & $\iota_X \ExtD x^\mu = \ExtD x^\mu (X) = X^\mu$ & $0$ & $\left(\iota_X T_1\right) \otimes T_2 + T_1 \otimes \left(\iota_X T_2 \right)$ & $\left( \iota_X \omega \right) \wedge \beta + (-)^k \omega \wedge \left( \iota_X\beta \right) $ \\
%
\cdashline{4-5}
$g^\ast$ \footnotemark[4] & $g^\ast \left(f\right) = f \circ g $ & $ g^\ast \left(\ExtD x^\mu \right) = \ExtD\left(x^\mu \circ g \right)$ &$(g^{-1})_\ast \partial_\mu = \dfrac{\partial[g^{-1}]^A}{\partial x^\mu}\partial_A$ \quad \footnotemark[3] & $g^\ast \left( T_1\right) \otimes g^\ast \left( T_2\right)$ \quad \footnotemark[3] & $g^\ast\left(\omega\right) \wedge g^\ast \left( \beta \right)$\\
\hdashline
%$\nabla_X$ & $\nabla_X f = X(f) $ & & & & \\
$\nabla_X$ & $\nabla_X f = X(f) = X^\nu \left(\dfrac{\partial f }{\partial x^\nu} \right)$ & $\nabla_X \ExtD x^\mu = X^\nu \left( - \Gamma^\mu_{\, \nu \, \alpha} \right) \ExtD x^\alpha$ & $\nabla_X \partial_\mu =X^\nu \Gamma^\alpha_{\, \nu \, \mu} \partial_\alpha$ & $\left(\nabla_X T_1\right) \otimes T_2 + T_1 \otimes \left(\nabla_X T_2 \right)$ & $\left( \nabla_X \omega \right) \wedge \beta + \omega \wedge \left(\nabla_X\beta \right)$\\%& & & & & \\
\hdashline
\end{tabularx}
\begin{minipage}[c][.45\textheight]{0.46 \linewidth}
The first four operation are naturally defined on every smooth manifold.
\fbox{
\parbox{\textwidth}{
\begin{displaymath}
\mathcal{T}(M) = \left(\bigoplus_{l,k=0}^\infty
T^k_l(M),
\otimes \right)
\qquad \textrm{\emph{Tensor Algebra} on M}
\end{displaymath}
is a bi-graded algebra over the ring $C^\infty(M)$.
}
}
\vfill
\fbox{
\parbox{\textwidth}{
$(\mathfrak{X}(M) , [-,-])$ form a Lie algebra over $\mathbb{R}$:
\begin{align}
[X,Y] &= -[Y,X] \\
[aX + bY, Z] &= a[X,Z] + b[Y,Z] \\
[[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] &= 0
\end{align}
}
}
\vfill
\fbox{
\parbox{\textwidth}{
Lie bracket is defined by the following (redundant) equations:
\begin{align*}
\left[\partial_i, \partial_j\right] &= 0 \\
\left[f \partial_i , g \partial_j\right] &= f \cdot (\partial_i g)\cdot\partial_j - g\cdot(\partial_j f )\cdot\partial_i \\
\left[f X , g Y \right] &= f\cdot g\cdot[X,Y] + f\cdot X(g)\cdot Y - g\cdot Y(f)\cdot X \\
\left[X ,Y\right] &= \left( X^i \partial_i Y^j - Y^i\partial_i X^j \right) \partial_j
\end{align*}
}
}
\end{minipage}
\hspace{1cm}
\setcounter{equation}{0}
\begin{minipage}[c][.485\textheight]{0.46 \linewidth}
The last one is not natural, it is an additional structure given to $M$:\\
\fbox{
\parbox{\textwidth}{
\emph{Affine connection}
\begin{align*}
\nabla: \mathfrak{X}(M) \times \mathfrak{X}(M) &\longrightarrow \mathfrak{X}(M) \\
(X,Y) &\longmapsto \nabla_X Y
\end{align*}
such that: % $\forall X,Y,Z \in \mathfrak{X}(M)$, $\forall a \in \Real$, $\forall f \in C^\infty(M)$:
\begin{align}
\nabla_{(X+ a Z)} Y &= \nabla_X Y + a \nabla_Z Y \\
\nabla_{X} \left( Y+ a Z \right) &= \nabla_X Y + a \nabla_Y Z \\
\nabla_{f\, X} Y &= f \, \nabla_X Y \\
\nabla_{X}\left( f \, Y\right) &= \left(\nabla_X f \right)\, Y + f \, \left(\nabla_X Y \right)
\end{align}
}
}
\vfill
\fbox{
\parbox{\textwidth}{
\begin{displaymath}
\Gamma^\alpha_{\, \mu \, \nu} = \ExtD x^\alpha \left( \nabla_{\mu} \partial_\nu \right)
\qquad \textrm{\emph{Christoffel symbols} of $\nabla$}
\end{displaymath}
}
}
\vfill
\fbox{
\parbox{\textwidth}{
Extension of operator $\nabla_X$ from $\mathfrak{X}(M)$ to $\Omega^1(M)$ is implemented by:
\begin{displaymath}
\left\langle \nabla_X \omega \right\vert \left. Y \right\rangle \coloneqq
\nabla_X \left\langle \omega \right\vert \left. Y \right\rangle -
\left\langle \omega \right\vert \left. \nabla_X Y \right\rangle
\end{displaymath}
}
}
\end{minipage}
\footnote[4]{$g:N \sim (y^A) \rightarrow M \sim (x^\mu) $}
\footnote[3]{Caveat: pull-back is well defined only on covariant tensors. Otherwise $g$ has to be a diffeomorphism.}
\end{landscape}
\end{document}