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\title{Equations in Physics \\ \small{A brief introduction}}
\author{Your name here \thanks{Institute of Mathematics, University of Wrocław and Wikipedia}}

\begin{document}
\maketitle

\begin{abstract}
In physics, there are equations in every field that relate physical quantities to each other and enable calculations. While entire handbooks of equations can summarize many aspects of the subject, they cannot cover everything. Some equations are highly specialized and pertain only to specific areas of study.
\end{abstract}

\section{Classical Mechanics}

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. The concepts it covers, such as \emph{mass, acceleration, and force}, are commonly used and known. This article lists equations from \textbf{Newtonian Mechanics.}

\subsubsection{Kinematic quantities}

\begin{itemize}
\item Velocity $\displaystyle v = \frac{dx}{dt}$,
\item Acceleration $\displaystyle a = \frac{dv}{dt}$.
\end{itemize}

\subsubsection{Dynamic quantities}

\begin{itemize}
\item Momentum $\displaystyle p = mv$,
\item Force $\displaystyle F = \frac{dp}{dt}$.
\end{itemize}

\subsubsection{Euler's equation for rigid body dynamics}

$$
I \cdot \alpha + \omega \times (I \cdot \omega)
 = \tau
 $$
 
\section{Thermodynamics}

This is a \emph{small} summary of common equations and quantities in thermodynamics.

\subsubsection{Thermal transfer}

\begin{itemize}
\item Thermal conduction rate $\displaystyle P = \frac{dQ}{dt}$,
\item thermal intensity $\displaystyle I = \frac{dP}{dA}$.
\end{itemize}

\subsubsection{Kinetic theory}

\begin{itemize}
\item Ideal gas law $\displaystyle pV = nRT$,
\item the pressure of an ideal gas $\displaystyle p = \frac{1}{3} nm\langle v^2 \rangle$.
\end{itemize}

\subsubsection{Entropy}

If $K_B$ is the \emph{Boltzmann constant} and $\Omega$ denotes the volume of macrostate in the phase space then

$$
S = K_B \ln \Omega.
$$

\section{Maxwell's equations}

\emph{Maxwell's equations} are a set of coupled partial differential equations that form the foundation of classical electromagnetism, classical optics and electric circuits.

\subsection{Differential equations}
\begin{gather*}
\nabla \cdot E = \frac{\rho}{\varepsilon_0} \\
\nabla \cdot B = 0 \\
\nabla \times E = -\frac{\partial B}{\partial t} \\
\nabla \times B = \mu_0 \left( J + \varepsilon \frac{\partial E}{\partial t} \right) 
\end{gather*}

\subsection{Integral equations}
\begin{gather*}
\oiint_{\partial \Omega} E \cdot dS = \frac{1}{\varepsilon_0} \iiint_\Omega \rho dV \\
\oiint_{\partial \Omega} B \cdot dS = 0\\
\oint_{\partial \Sigma} E \cdot d\ell = - \frac{d}{dt} \iint_\Sigma B \cdot dS \\
\oint_{\partial \Sigma} B \cdot d\ell  = \mu_0 \left( \iint_{\Sigma} J \cdot dS + \varepsilon \frac{d}{dt} \iint_\Sigma E \cdot dS \right) 
\end{gather*}

\end{document}