\input{head}
\input{customstuff}


\begin{document}


\title{The Geometry of Physics --\\Problem Solutions}
\author{David Luposchainsky, Ernest Yeung}
\date{\number\day.\,\number\month.\,\number\year}
\maketitle

\abstract{These are my solutions to problems given in Theodore Frankel's book ``The Geometry of Physics'' (second edition). As I could not find any other sources, I do not know whether they are correct or not, so read with care (especially the index battles). If you have a solution that is not in here already, a better way of showing something, or just some useful comment, I'd like to hear about it\footnote{e-mail: stupid underscore name at gmx dot net}.} \\

20110714 - further solutions begun by Ernest Yeung.


\section*{Conventions}
If not mentioned differently, use the following conventions:
\begin{itemize}
	\item Use Einstein summation. Sometimes I'll typeset a \(\sum\) for clarification though.
	\item The ``\(\rightharpoondown\)'' used in the book will be used implicitly, i.e. multiindices are always assumed to be in ascending order.
	\item Abbreviations concerning the metric tensor: \\ \ieq{g:=|\det{\{g_{ij}\}}|}, \ieq{s:=\sign{\det{\{g_{ij}\}}}}
\end{itemize}

\newpage

\setcounter{tocdepth}{4}
\tableofcontents

\newpage

\section*{I Manifolds, Tensors and Exterior Forms}
                   
\section{Manifolds and Vector Fields}
 \input{01ManifoldsVector}

                                                                  
\section{Tensors and Exterior Forms}
		\input{02tensors}

\section{Integration of Differential Forms}
		\input{03integration}

\section{The Lie derivative}
		\input{04liederivative}

\section{The Poincare Lemma and Potentials}
		\input{05potentials}

\section{Holonomic and Nonholonomic Constraints}

\section*{II Geometry and Topology}

\section{$\mathbb R^3$ and Minkowski Space}
		\input{07r3minkowski}

\section{The Geometry of Surfaces in $\mathbb{R}^3$}

\section{9 Covariant Differentiation and Curvature}
		\input{09covariantdifferentiation}

\section{10 Geodesics}

\section{11 Relativity, Tensors, and Curvature}

\section{12 Curvature and Topology: Synge's Theorem}

\section{13 Betti Numbers and De Rham's Theorem}

\section{14 Harmonic Forms}

\section*{III Lie Groups, Bundles, and Chern Forms}


	\section{15. Lie groups}
		\input{15liegroups}
	\section{16. Vector Bundles in Geometry and Physics}
		\input{16vectorbundles}


\section{17. Fiber Bundles, Gauss-Bonnet, and Topological Quantization  }
\input{17fiberbundles}



\section{18. Connections and Associated Bundles}

\input{18ConnectionsandAssociatedBundles}



\section{19. The Dirac Equation}

\input{19TheDiracEquation}

\section{20. Yang-Mills Fields }

             \input{20YangMillsFields}

\section{21.}









\begin{appendix}

\section{A. Elasticity}


\subsection{A.a. The Classical Cauchy Stress Tensor and Equations of Motion}

$B(t)$ compact body, might be portion of larger body in motion

mass 3 form 
\[
m^3 \equiv \rho \text{ vol }
\]

mass conservation
\[
\frac{d}{dt} \int_{B(t)} m^3 = \int_{B(t)} \mathcal{L}_{\textbf{v} + \frac{ \partial }{ \partial t}} m^3 = 0 
\]
$\mathbf{b}$ external force density (per unit mass)
\[
\frac{d}{dt} \int_{B(t)} v^i m^3 = \int_{B(t)} b^i m^3 + \int_{ \partial B(t)} t^{ij}n_j da
\]

\subsection{A.b. Stresses in Terms of Exterior Forms}

$t$ pseudo $(n-1)$ form on $M^n$ with values in tangent bundle $TM$ (vector bundle language)

\begin{equation}
        \mathbf{t} = \mathbf{e}_r \otimes \mathfrak{t}^r \equiv \mathbf{e}_r \otimes \mathfrak{t}^r_{ \, \, \underline{J}} \sigma^J 
\end{equation}

\[
\int_{ \partial B} \mathbf{t} = \int_{\partial B} \mathbf{e}_r \otimes \mathfrak{t}^r = \mathbf{e}_r \int_{\partial B} \mathfrak{t}^r = \mathbf{e}_r \int_{ \partial B} \mathfrak{t}^r_{ \underline{J}} dx^J
\]
as total traction that part of body outside $\partial B$ exerts on $B$

(2.73) $\forall \, $  Riemannian $M$, write $(n-1)$ form $\mathfrak{t}^r$ in terms of vector $t^{(r)}$

\[
\mathfrak{t}^r = i(\mathbf{t}^r) \text{vol} = i (\mathbf{t}^{(r)} ) \sqrt{g} \epsilon_{\underline{I}} dx^I = \sqrt{g} \mathbf{t}^{(r)i} \epsilon_{i \underline{J}} dx^J
\]

\begin{equation}
t^r_{ \, \underline{J}}  = \sqrt{g} t^{(r)i} \epsilon_{i \underline{J}} \quad \quad \, (A.6)
\end{equation}
\[
t^{ri} \equiv t^{(r)i} = \frac{1}{\sqrt{g}} t^r_{ \, \, \underline{J}} \epsilon^{i J }
\]
relation between stress form $t^r$ and Cauchy's stress tensor $t^{ri}$

assuming $\mathbf{t} = \mathbf{e}_r \otimes \mathfrak{t}^r$ is $(n-1)$ form section of the tangent bundle, thus from (9.31), we have

20141102 EY recall 
\[
\nabla \alpha = e \otimes (d\alpha + \omega \wedge \alpha) \quad \quad \quad \, (9.31)
\]
and 
\[
\begin{gathered}
        \nabla \alpha = \nabla( e_i \otimes \alpha^i) = (\nabla e_i) \otimes_{\wedge} \alpha^i + e_i \otimes d\alpha^i \\ 
 \text{ where }  \\
(\nabla e_i) \otimes_{\wedge} \alpha^i = (e_k \otimes \omega^k_{\, \, i}\otimes_{\wedge} \alpha^i \equiv e_k \otimes ( \omega^k_{\, \, i} \wedge \alpha^i)
\end{gathered}
\]

\begin{equation}
        \nabla t = \nabla (\mathbf{e}_r \otimes \mathfrak{t}^r) = \nabla \mathbf{e}_r \otimes_{\wedge} \mathfrak{t}^r + \mathbf{e}_r \otimes d\mathfrak{t}^r = \mathbf{e}_r \otimes (d\mathfrak{t}^r + \omega^r_{ \, \, s} \wedge \mathfrak{t}^s) = \mathbf{e}_r \otimes \nabla \mathfrak{t}^r \quad \quad \quad \, (A.7)
\end{equation}


\subsection{A.f. Hamilton's Principle in Elasticity}


\begin{equation}
        \delta U = \int_B \delta E_{RC} dX^R \wedge S^C \quad \quad \quad \, (A.24) 
\end{equation}

\begin{equation}
        \delta U = \int_B S^{CR} \delta E_{RC} \text{VOL}^n \quad \quad \quad \, (A.25)
\end{equation}

Hamilton's principle
\begin{equation}
\delta \int T dt - \int \delta U dt + \int \delta W dt = 0 \quad \quad \quad \, (A.26)
\end{equation}

We don't write $\delta \int U dt $ because we don't assume \\

We don't assume that $\exists \, $ stored energy function $U$ so we don't write $\delta \int U dt$, $\exists \, $ only differential $\delta U$

\end{appendix}
\end{document}