\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{esint}

\title{Derevyannykh.1}
\author{Dervyannykh Mikhail Olegovich }
\date{July 2018}

\begin{document}

\maketitle{Day 1}

\section{Task 1}
\begin{equation}
\sin(n\alpha) = \sum_{k=0}^{[(n-1)/2]} (-1)^k \binom{n}{2k+1} \cos^{n-2k-1}\alpha \sin^{2k+1}\alpha.
\end{equation}

\section{Task 2}

\begin{equation}
    f(x) = \frac{\alpha_0}{2} + \sum_{k=1}^{+\infty}A_k\cos(k\frac{2\pi}{r}x+\theta_k).
\end{equation}


\section{Task 3}
\begin{equation}
    (\exists n \in \mathbb{N})(\forall\alpha \in L : |\alpha| \geqslant n)(\exists u,v,w \in V^*) : [\alpha = uvw \; \land\; |uv|\; \leqslant n \;\land\; |v| \geqslant 1\; \land\; (\forall i \in \mathbb{N}\;\cup\; \{0\}, uv^{i}w \in L)].
\end{equation}

\section{Task 4}
\begin{equation}
\iint\limits_{G}\sqrt{r^2 + r^2\Big(\frac{\partial z}{\partial p}\Big)^2 +
\Big(\frac{\partial z}{\partial \varphi}\Big)^2}
\end{equation}

\section{Task 5}
\begin{equation}
    \iiint\limits_{V}(\nabla\cdot\bold F)dV = \oiint\limits_{S}\Hat{\bold n}\times\bold F dS.
\end{equation}

\section{Task 6}

\pi = \displaystyle \sqrt{\frac{6}{\lim\limits_{x\to\infty}
\displaystyle
\prod_{\substack{k=1\\p_k\in\bold P}}^{n}
(1-\frac{1}{p^2_k})}}.


\section{Task 7}

\begin{equation}
    F(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \int\displaylimits_{-\infty}^{x}
    \displaystyle e^{\textstyle -\frac{(t-\mu)^2}{2\sigma^2}}dt.
\end{equation}

\section{Task 8}

\begin{equation}
    \Gamma(z) = \frac{1}{e^{i2\piz}-1}\int\limits_{L}t^{z-1}e^{-t}dt,\quad z\in \mathbb{C}\backslash\mathbb{Z}.
\end{equation}

\section{Task 9}
\begin{equation}
 D\Bigg[\sum_{i=1}^{n}c_i X_i\Bigg] = \sum_{i=1}^{n}c^2_i D\big[X_i\big] + 2 \quad \sum_{1\leqslant i<j \leqslant n} c_i c_j cov(X_i, X_j).
\end{equation}

\section{Task 10}
\begin{equation}
    \sqrt[n]{z} = r^{1/n}\bigg(\cos{\frac{\varphi + 2\pi k}{n}} + i\sin{\frac{\varphi + 2\pi k}{n}}\bigg),\quad k=0,\ldots,n-1.
\end{equation}

\section{Task 11}
\begin{tabular}{rcccccccccc}
& & & & & 1 \\\noalign{\smallskip\smallskip}
& & & & 1 & & 1 \\\noalign{\smallskip\smallskip}
& & & 1 & & 2 & & 1 \\\noalign{\smallskip\smallskip}
& & 1 & & 3 & & 3 & & 1\\\noalign{\smallskip\smallskip}
& 1 & & 4 & & 6 & & 4 & & 1 \\\noalign{\smallskip\smallskip}
1 & & 5 & & 10 & & 10 & & 5 && 1 \\\noalign{\smallskip\smallskip}
. & . & . & . & . & . & . & . & . & .&.& \\\noalign{\smallskip\smallskip}

\end{tabular}



\section{Task 12}
\begin{equation}
rot \, \vec{F} = \nabla \times \Bar{F} =
 \begin{vmatrix} \bar{e_x} & \bar{e_y} & \bar{e_z} \\  \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}} \\ F_x &  F_y  & F_z  \end{vmatrix}
\end{equation}

\section{Task 13}
\begin{equation}
\tau_{m\times n} = \begin{Vmatrix}
\frac{\partial f_1}{\partial x_1} \dots  \frac{\partial f_1}{\partial x_n} \\
\vdots \ddots \vdots \\
\frac{\partial f_m}{\partial x_1} \dots  \frac{\partial f_m}{\partial x_n} \\
\end{Vmatrix}
\end{equation}


\end{document}