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main.tex
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\documentclass[12pt, a4paper]{report}
\usepackage{Custom}
\makenomenclature
\makeindex[intoc=true,options=-s index]
\begin{document}
\hypersetup{allcolors = black}
\title{Compendium of Mathematics \& Physics}
\author{Nicolas Dewolf}
\maketitle
\dominitoc
\doparttoc
\tableofcontents
\hypersetup{allcolors = UGent}
\chapter*{Introduction}
\markboth{Introduction}
\mtcaddchapter
\section*{Goals}
This compendium originated out of the necessity for a compact summary of important theorems and formulas during physics and mathematics classes at university. When the interest in more (and more exotic) subjects grew, this collection lost its compactness and became the chaos it now is. Although there should exist some kind of overall structure, it was not always possible to keep every section self-contained or respect the order of the chapters.
It should definitely not be used as a formal introduction to any subject. It is neither a complete work nor a fact-checked one, so the usefulness and correctness is not guaranteed. However, it can be used as a look-up table for theorems and formulas, and as a guide to the literature. To this end, each chapter begins with a list of useful references. At the same time, only a small number of statements are proven in the text (or appendices). This was done to keep the text as concise as possible (a failed endeavour). However, in some cases the major ideas underlying the proofs are provided.
\section*{Structure and conventions}
Sections and statements that require more advanced concepts, in particular concepts from later chapters or (higher) category theory, will be labelled by the \textit{clubs} symbol~$\clubsuit$. Some definitions, properties or formulas are given with a proof or an extended explanation whenever I felt like it. These are always contained in a blue frame to make it clear that they are not part of the general compendium. When a section uses notions or results from a different chapter at its core, this will be recalled in a green box at the beginning of the section.
Definitions in the body of the text will be indicated by the use of \textbf{bold font}. Notions that have not been defined in this summary but that are relevant or that will be defined further on in the compendium (in which case a reference will be provided) are indicated by \textit{italic text}. Names of authors are also written in \textit{italic}.
Objects from a general category will be denoted by a lower-case letter (depending on the context, upper-case might be used for clarity), functors will be denoted by upper-case letters and the categories themselves will be denoted by symbols in \textbf{bold font}. In the later chapters on physics, specific conventions for the different types of vectors will often be adopted. Vectors in Euclidean space will be denoted by a bold font letter with an arrow above, e.g.~$\vector{a}$, whereas vectors in Minkowski space (4-vectors) and differential forms will be written without the arrow, e.g.~$\symbf{a}$. Matrices and tensors will always be represented by capital letters and, dependent on the context, a specific font will be adopted.
% \part{Set Theory, Algebra \& Category Theory}
% \insertparttoc
% \input{Math/Set/SetTheory}
% \input{Math/Algebra/Algebra}
% \input{Math/Set/CategoryTheory}
% \input{Math/Algebra/HDA}
% \input{Math/Algebra/HomologicalAlgebra}
% \part{Topology}
% \insertparttoc
% \input{Math/Topology/Topology}
% \input{Math/Topology/AlgebraicTopology}
% \input{Math/Topology/SheafTheory}
% \input{Math/Topology/Metric}
% \input{Math/Topology/AlgebraicGeometry}
% \input{Math/Set/NumberTheory}
% \part{Higher Set Theory}
% \insertparttoc
% \input{Math/Set/ToposTheory}
% \input{Math/Set/ModelCategory}
% \input{Math/Set/HigherCategoryTheory}
% \input{Math/Set/TypeTheory}
% \part{Calculus}
% \insertparttoc
% \input{Math/Calculus/Calculus}
% \input{Math/Calculus/ComplexCalculus}
% \input{Math/Calculus/MeasureTheory}
% \input{Math/Calculus/DistributionTheory}
% \input{Math/Calculus/ODE}
\input{Math/Calculus/PDE}
% \part{Linear Algebra}
% \insertparttoc
% \input{Math/LinearAlgebra/LinearAlgebra}
% \input{Math/LinearAlgebra/TensorAlgebra}
% \input{Math/Algebra/RepresentationTheory}
% \input{Math/Calculus/FunctionalAnalysis}
% \input{Math/Algebra/OperatorTheory}
% \input{Math/LinearAlgebra/CliffordAlgebra}
% \input{Math/Algebra/NCA}
% \part{(Differential) Geometry}\label{part:diffgeom}
% \insertparttoc
\input{Math/Geometry/ElementaryGeometry}
% \input{Math/Geometry/Curves}
% \input{Math/Geometry/Manifolds}
% \input{Math/Geometry/LieTheory}
% \input{Math/Geometry/FibreBundles}
% \input{Math/Geometry/VectorBundles}
% \input{Math/Geometry/PrincipalBundles}
% \input{Math/Geometry/RiemannianGeometry}
% \input{Math/Geometry/SymplecticGeometry}
% \input{Math/Geometry/ContactGeometry}
% \input{Math/Geometry/ComplexGeometry}
% \input{Math/Geometry/VariationalCalculus}
% \input{Math/Topology/K}
% \input{Math/Geometry/NCG}
% \input{Math/Geometry/SyntheticGeometry}
% \input{Math/Geometry/HDG}
% \part{Probability Theory \& Statistics}
% \insertparttoc
% \input{Math/Statistics/ProbabilityTheory}
% \input{Math/Statistics/StochasticCalculus}
% \input{Math/Statistics/InformationGeometry}
% \input{Math/Statistics/Statistics}
% \input{Math/Statistics/CausalInference}
% \input{Math/Statistics/DataAnalysis}
% \input{Math/Statistics/FuzzyLogic}
\part{Classical Physics}
\insertparttoc
% \input{Physics/ClassicalPhysics/ClassicalMechanics}
% \input{Physics/ClassicalPhysics/Constraints}
\input{Physics/ClassicalPhysics/Electromagnetism}
\input{Physics/ClassicalPhysics/Optics}
% \part{Quantum Theory}\label{part:qm}
% \insertparttoc
% \input{Physics/QuantumMechanics/QuantumMechanics}
% \input{Physics/QuantumMechanics/QuantumInformationTheory}
% \input{Physics/QuantumMechanics/Quantization}
% \input{Physics/QuantumMechanics/Perturbation}
% \input{Physics/QuantumMechanics/Scattering}
% \part{Statistical Mechanics \& Condensed Matter Physics}
% \insertparttoc
% \input{Physics/ClassicalPhysics/Thermodynamics}
% \input{Physics/ClassicalPhysics/StatisticalMechanics}
% \input{Physics/ClassicalPhysics/SolidState}
% \input{Physics/QuantumMechanics/TensorNetworks}
% \part{Relativity \& Quantum Field Theory}
% \insertparttoc
% \input{Physics/FieldTheory/SpecialRelativity}
% \input{Physics/FieldTheory/GeneralRelativity}
% \input{Physics/QuantumMechanics/Dirac}
% \input{Physics/FieldTheory/FieldTheory}
% \input{Physics/FieldTheory/GaugeTheory}
% \input{Physics/FieldTheory/QFT}
% \input{Physics/FieldTheory/Particles}
% \input{Physics/FieldTheory/AxiomaticQFT}
% \input{Physics/FieldTheory/CFT}
% \input{Physics/FieldTheory/Extension}
% \input{Physics/FieldTheory/Cosmology}
% \part{Beyond Math \& Physics}
% \insertparttoc
% \input{Math/Set/ComputerScience}
% \input{Math/Statistics/Finance}
% \input{Philosophy}
% \input{Epilogue}
% \part{Appendices}
% \insertparttoc
% \begin{appendices}
% \input{Math/Geometry/GStruct}
% \input{Notes}
% \end{appendices}
\input{Nomenclature}
\printnomenclature
\bibliographystyle{apacite}
\bibliography{Biblio}
\adjustmtc
\printindex
\printindex[author]
\end{document}