Document 2 - Basics (5-6)

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

\title{Document 2 - Basics}
\author{Thomas Boufikos}
\date{September 2021}

\begin{document}

\maketitle
\section{Calculus notation} \\ \\
One example of a polynomial is: $ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$. \\
One example of exponentials is: $ g(x) = c_1e^{r_1x} + c_2e^{r_2x}$. \\
Don't forget the basic identity: \\
\[
\sin^2x + \cos^2x = 1 \implies \sin x = \pm \sqrt{1 - \cos^2x}
\] \\ \\ \\
\underline{\textbf {Limits notation}} \\
\[
\begin{align*}
    \lim _{x \to 0} \frac{x^2 + 1}{x^2 - 1} = 1 \\
     \lim _{x \to \infty} \frac{x^2 + 1}{x^2 - 1} = 1
\end{align*}
\] \\ \\
If $ \lim _{x \to a} f(x) = M $ and $ \lim _{x \to a} g(x) = M $ and we also know that: \\
$ f(x) \leq h(x) \leq g(x)$, then: \\
\[
    \lim _{x \to a} h(x) = M
\] \\ \\ \\
\underline{\textbf {Summation notation}} \\ \\
We know that this sum diverges: $ \sum _{n=1}^{\infty} \frac{1}{n} = \infty $, while: \\
\[
    \sum _{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
\] \\
converges to a real number. \\ \\
\[
    \sum _{n=0}^N \sum _{m=0}^N a_n a_m= \left( \sum _{n=0}^N a_n \right) ^2
\] \\ \\ \\
\underline{\textbf {Integrals notation}} \\ \\
\[
\int _{-\infty}^{\infty} f(x) \, dx = \lim _{x \to \infty} f(x) -  \lim _{x \to -\infty} f(x) \\
\]
If f is the derivative of F, then: $ \int_R f(x) \, dx = F(x) $ \\ \\
Volume = $ \iiint _D f(x,y,z) \, dx \, dy \, dz$. \\ \\

Now, we are gonna see the vertical bar: \\
\[
    \int _a^b f(x) \, dx = F(x) \bigg\vert _a^b
\] \\
In inline notation, we have: $ \int _a^b f(x) \, dx = F(x) \big\vert _a^b $. \\ \\ \\
If a function $ f = f(x,y) $ has these 2 variables $x$ and $y$, then we can define: \\
\begin{center}
$ \frac{\partial f}{\partial x} $ and $ \frac{\partial f}{\partial x} $
\end{center}. \\ \\ \\
\textbf\textit{{Example: }} $f(x,y) = x^2 + 2xy + e^{2y}$, then: \\
\[
\begin{align}
    \frac{\partial f}{\partial x} = 2x + 2y \\
    \frac{\partial f}{\partial y} = 2y + 2e^{2y}
\end{align}
\] \\ \\
\underline{\textbf {More derivatives}} \\ \\
If $ f(x) = x^3 $, then $ f'(x) = 3x^2, f''(x) = 6x$. \\
If f is a time variable-based function $f(t)$, then we can do this: \\
\[
\begin{align}
    f(t) = \sin t + 2t \\
    \dot{f}(t) = \cos t + 2 \\
    \ddot{f}(t) = - \sin t
\end{align}
\] \\ \\ \\
\underline{\textbf {Vectors (esvect package)}} \\ \\
\[
    \vv{r}(t) = \langle x(t), y(t), z(t) \rangle = \langle \sin t, \cos t, t \rangle
\]
\[
    \dot{\vv{r}}(t) = \langle x'(t), y'(t), z'(t) \rangle = \langle \cos t, -\sin t, 1 \rangle
\] \\ \\
Now, the last thing to do is to remember the Faraday's Law of Injuction: \\
\[
\begin{align}
    \vv{\nabla} \times \vv{E} = - \frac{\partial \vv{B}}{\partial t} \\
    \oint \vv{E} \cdot d\vv{S} = \frac{d\Phi_B}{dt}
\end{align}
\] \\ \\ \\ \\
\section{Miscellaneous notation} \\ \\
\underline{\textbf {Roots}} \\ \\
$ \sqrt{144} = 12 $, $ \sqrt{a} + \sqrt{b} > \sqrt{a+b} $ \\
$ \sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}} $ \\
$ \sqrt[n]{a} = a^{\frac{1}{n}} $, for every n. \\ \\ \\
\underline{\textbf {The quadratic formula: }}
If $ ax^2 + bx + c = 0$, then solutions:   $x = \frac{-b \pm \sqrt{\Delta}}{2a}$, where: \\
\[
    \Delta = b^2 - 4ac
\] \\ \\ \\ \\
\underline{\textbf {Set Relationships}} \\ \\
When A is subset of B, we write: $ A \subset B $ and the probabilities go on: \\
\[
    Pr(A) < Pr(B)
\] \\
But if $ A \supseteq B $, then it must be: $ Pr(A) \geq Pr(B) $. \\ \\ \\
\underline{\textbf {Probability rules}} \\ \\
\begin{enumerate}
    \item If $A$ is a set of the space $\Omega$, then we know that if an event $x \in A$
    exists in our set, its possibility follows the rule: $Pr(x) = Pr(A)$.
    \item If $\Omega$ is the space, then $Pr(\Omega) = 1$ and there is the empty set, where: $Pr(\emptyset) = 0$.
    \item If $A$ and $B$ are two events with no intersection ($A \cap B = \emptyset$), we have that $Pr(A \cup B) = Pr(A) + Pr(B)$.
\end{enumerate} \\ \\ \\ \\
\underline{\textbf {Logic Symbols}} \\ \\
\begin{itemize}
    \item If we want to say P AND Q, we write: $ P \land Q$.
    \item If we want to say P OR Q, we write: $ P \lor Q$.
    \item If we want to say  NOT Q, we write: $ \lnot Q$.
    \item If we want to say P IMPLIES Q, we write: $ P \implies Q$.
    \item If we want to say P IMPLIES Q AND Q IMPLIES P: \\
    \[
        P \iff Q
    \]
    \item If $f$ is continuous $\therefore$ \\
    $ \forall  a, b$, with $f(a) > 0 \land f(b) < 0, \exists \rho \in [a,b]$ with:
    $f(\rho) = 0.$ This is the Bolzano theorem.
\end{itemize} \\ \\ \\ \\
\emph{Multiple integrals reminder: }:
$ \int \int \cdots \int f(x,y, \ldots, z) \, dx \, dy \, \ldots \,dz = \ldots$
\\ \\ \\
\underline{\textbf {Product}} \\ \\
\[
    \log (\prod _{i=1}^n x_i) = \sum _{i=1}^n \log (x_i)
\] \\ \\ \\

Now, we gonna show the \textbf{Newton's identity} about polynomials of the form:
$(x+1)^n$. Let's go: \\
\[
  (a+b)^n = \sum _{k=0}^n \binom{n}{k} a^k b^{n-k}
\] \\ \\ \\
\underline{\textbf {Abstract Algebra}} \\ \\
$
\kappa =
\begin{pmatrix}
    1 & 2 & 3 \\
    4 & 5 & 6 \\
    7 & 8 & 9 \\
\end{pmatrix}$, \text{    }
$
\sigma =
\begin{pmatrix}
    -1 & -2 & -3 \\
    -4 & -5 & -6 \\
    -7 & -8 & -9 \\
\end{pmatrix}
$ \\ \\ \\
$
\kappa + \sigma =
\begin{pmatrix}
    0 & 0 & 0 \\
    0 & 0 & 0 \\
    0 & 0 & 0 \\
\end{pmatrix} = 0
$ \\ \\ \\
\textit{\textbf{Reminder about limits (with underset): }} \\ \\
Instead of writing this: $ \lim _{x \to \infty} f(x) = 0$, we can write the following: \\
\[
    f(x) \underset{x \to \infty}{\longrightarrow} 0
\] \\ \\ \\




\end{document}