diff --git a/labs-set-1/template.pdf b/labs-set-1/template.pdf index c04ee06..65c394c 100644 Binary files a/labs-set-1/template.pdf and b/labs-set-1/template.pdf differ diff --git a/labs-set-2/Lab 10 - 14.5 cont.pdf b/labs-set-2/Lab 10 - 14.5 cont.pdf new file mode 100644 index 0000000..0bbf14e Binary files /dev/null and b/labs-set-2/Lab 10 - 14.5 cont.pdf differ diff --git a/labs-set-2/Lab 7 - 14.1, 14.2.pdf b/labs-set-2/Lab 7 - 14.1, 14.2.pdf new file mode 100644 index 0000000..18fb0f6 Binary files /dev/null and b/labs-set-2/Lab 7 - 14.1, 14.2.pdf differ diff --git a/labs-set-2/Lab 8 - 14.3.pdf b/labs-set-2/Lab 8 - 14.3.pdf new file mode 100644 index 0000000..9d7f9b1 Binary files /dev/null and b/labs-set-2/Lab 8 - 14.3.pdf differ diff --git a/labs-set-2/Lab 9 - 14.4, 14.5.pdf b/labs-set-2/Lab 9 - 14.4, 14.5.pdf new file mode 100644 index 0000000..1bb3869 Binary files /dev/null and b/labs-set-2/Lab 9 - 14.4, 14.5.pdf differ diff --git a/labs-set-2/letterfonts.tex b/labs-set-2/letterfonts.tex new file mode 100644 index 0000000..cf91649 --- /dev/null +++ b/labs-set-2/letterfonts.tex @@ -0,0 +1,136 @@ +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} +\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}} +\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}} +\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}} +\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}} +% expected value +\newcommand{\EE}{\ensuremath{\mathbb{E}}} + +%--------------------------------------- +% BlackBoard Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} +\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} +\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} +\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} +\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} +\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} +\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} +\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} +\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} +\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} +\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} +\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} + +%--------------------------------------- +% MathCal Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}} +\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}} +\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}} +\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}} +\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}} +\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}} +\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}} +\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}} +\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}} +\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}} +\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}} +\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}} +\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}} + + + +%--------------------------------------- +% Bold Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}} +\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}} +\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}} +\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}} +\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}} +\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}} +\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}} +\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}} +\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}} +\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}} +\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}} +\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}} +\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}} +%Small Letters +\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}} +\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}} +\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}} +\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}} +\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}} +\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}} +\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}} +\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}} +\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}} +\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}} +\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}} +\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}} +\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}} + +%--------------------------------------- +% Scr Math Fonts :- +%--------------------------------------- + +\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}} +\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}} +\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}} +\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}} +\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/labs-set-2/macros.tex b/labs-set-2/macros.tex new file mode 100644 index 0000000..5ebc567 --- /dev/null +++ b/labs-set-2/macros.tex @@ -0,0 +1,33 @@ +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} + +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} + +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\der}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} +% \newcommand{\ddd}[3]{\frac{\mathrm{d}^{#3} #1}{\mathrm{d}^{#3} #2}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} +\newcommand{\id}{\text{id}} + + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/labs-set-2/preamble.tex b/labs-set-2/preamble.tex new file mode 100644 index 0000000..9f070cd --- /dev/null +++ b/labs-set-2/preamble.tex @@ -0,0 +1,754 @@ +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage{gensymb} + +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} +\usepackage{chngcntr} +\usepackage{tikz} +\usepackage{titletoc} + +% add this to the preamble (near your other \usepackage lines) +\usepackage{pgfplots} +\pgfplotsset{compat=1.18} % or compat=newest if you prefer +\usepgfplotslibrary{colormaps} % enables 'viridis' etc. + + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + + + +\usepackage{tikzsymbols} +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} +\tikzstyle{c} = [circle,fill=black,scale=0.5] +\tikzstyle{b} = [draw, thick, black, -] +\tikzset{ + vertex/.style={ + circle, + draw, + minimum size=6mm, + inner sep=0pt + } +} + +\renewcommand\qedsymbol{$\Laughey$} + +% \usepackage{xparse} +% \usepackage{pgffor} +% \usepackage{emoji} +% +% \NewDocumentCommand{\memoji}{m o}{% +% \IfNoValueTF{#2} +% {\ifmmode \text{\emoji{#1}} \else \emoji{#1}\fi} % Single emoji case +% {\foreach \index in {1,...,#2}{\ifmmode \text{\emoji{#1}} \else \emoji{#1}\fi}} % Repeated emoji case +% } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\definecolor{doc}{RGB}{0,60,110} +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylemmabg}{HTML}{FFFAF8} +\definecolor{mylemmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{1cm} +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{Theorem} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{Theorem} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + +%================================ +% Corollery +%================================ +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{Corollary} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{Corollary} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LEMMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lemma}{Lemma} +{% + enhanced, + breakable, + colback = mylemmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylemmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylemmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylemmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lemma}{lemma} +{% + enhanced, + breakable, + colback = mylemmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylemmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylemmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylemmafr}, +} +{th} + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{Exercise} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{Exercise} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{Proposition} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{Proposition} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{Claim} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{Example} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{Example} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + +%================================ +% EXERCISE BOX +%================================ + +\newcounter{questioncounter} +\counterwithin{questioncounter}{chapter} +% \counterwithin{questioncounter}{section} + +\makeatletter +\newtcbtheorem[use counter=questioncounter]{question}{Question}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{Question}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +\newtcbtheorem[number within=chapter]{wconc}{Wrong Concept}{ + breakable, + enhanced, + colback=white, + colframe=myr, + arc=0pt, + outer arc=0pt, + fonttitle=\bfseries\sffamily\large, + colbacktitle=myr, + attach boxed title to top left={}, + boxed title style={ + enhanced, + skin=enhancedfirst jigsaw, + arc=3pt, + bottom=0pt, + interior style={fill=myr} + }, + #1 +}{def} + + +%================================ +% NOTE BOX +%================================ + +\usetikzlibrary{arrows,calc,shadows.blur} +\tcbuselibrary{skins} +\newtcolorbox{note}[1][]{% + enhanced jigsaw, + colback=gray!20!white,% + colframe=gray!80!black, + size=small, + boxrule=1pt, + title=\textbf{Note:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=1.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}} +\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}} +\newcommand{\mlemma}[2]{\begin{Lemma}{#1}{}#2\end{Lemma}} +\newcommand{\mer}[2]{\begin{Exercise}{#1}{}#2\end{Exercise}} +\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}} +\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}} +\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}} +\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}} +\newcommand{\ex}[2]{\begin{Example}{#1}{}#2\end{Example}} +\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}} +\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}} +\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}} +\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}} +\newcommand{\nt}[1]{\begin{note}#1\end{note}} + +\newcommand*\circled[1]{\tikz[baseline=(char.base)]{ + \node[shape=circle,draw,inner sep=1pt] (char) {#1};}} +\newcommand\getcurrentref[1]{% + \ifnumequal{\value{#1}}{0} + {??} + {\the\value{#1}}% +} +\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}} +\newenvironment{myproof}[1][\proofname]{% + \proof[\bfseries #1: ]% +}{\endproof} + +\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}} +\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{} +\newenvironment{iclaim}[1][\claimname]{\bfseries #1\mdseries:}{} +\newcommand{\iclm}[2]{\begin{iclaim}[#1]#2\end{iclaim}} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + +% deliminators +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} + +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} +\DeclarePairedDelimiter{\round}{\lfloor}{\rceil} + +\newsavebox\diffdbox +\newcommand{\slantedromand}{{\mathpalette\makesl{d}}} +\newcommand{\makesl}[2]{% +\begingroup +\sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}% +\pdfsave +\pdfsetmatrix{1 0 0.2 1}% +\rlap{\usebox{\diffdbox}}% +\pdfrestore +\hskip\wd\diffdbox +\endgroup +} +\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{% +\slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}% +{\slantedromand\hspace{0.2ex}^{#1}}}} +\ProvideDocumentCommand\dv{o m g}{% + \ensuremath{% + \IfValueTF{#3}{% + \IfNoValueTF{#1}{% + \frac{\dd #2}{\dd #3}% + }{% + \frac{\dd^{#1} #2}{\dd #3^{#1}}% + }% + }{% + \IfNoValueTF{#1}{% + \frac{\dd}{\dd #2}% + }{% + \frac{\dd^{#1}}{\dd #2^{#1}}% + }% + }% + }% +} +\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}} +% - others +\DeclareMathOperator{\Lap}{\mathcal{L}} +\DeclareMathOperator{\Var}{Var} % varience +\DeclareMathOperator{\Cov}{Cov} % covarience +\DeclareMathOperator{\E}{E} % expected + +% Since the amsthm package isn't loaded + +% I prefer the slanted \leq +\let\oldleq\leq % save them in case they're every wanted +\let\oldgeq\geq +\renewcommand{\leq}{\leqslant} +\renewcommand{\geq}{\geqslant} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TABLE OF CONTENTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\contentsmargin{0cm} +\titlecontents{chapter}[3.7pc] +{\addvspace{30pt}% + \begin{tikzpicture}[remember picture, overlay]% + \draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);% + \pgftext[left,x=-3.7cm,y=0.2cm]{\color{white}\Large\sc\bfseries Chapter\ \thecontentslabel};% + \end{tikzpicture}\color{doc!60}\large\sc\bfseries}% +{} +{} +{\;\titlerule\;\large\sc\bfseries Page \thecontentspage + \begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/labs-set-2/template.pdf b/labs-set-2/template.pdf new file mode 100644 index 0000000..df51600 Binary files /dev/null and b/labs-set-2/template.pdf differ diff --git a/labs-set-2/template.tex b/labs-set-2/template.tex new file mode 100644 index 0000000..7705ed8 --- /dev/null +++ b/labs-set-2/template.tex @@ -0,0 +1,132 @@ +\documentclass{report} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\Huge{Calculus III}} +\author{\huge{Krishna Ayyalasomayajula}} +\date{} + +\begin{document} + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[]{}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\chapter{Lab 7} +\section{Work} + +\qs{}{ + \begin{align*} + \vec{r} \in \mathbb{R} \therefore t \ge 0 \because e^{\sqrt{t}}\notin\mathbb{R} \; t|t<0 \\ + \text{This corresponds with answer choice D} + \end{align*} +} + +\qs{}{ + \begin{align*} + \forall t>4, \quad \vec{r}\cdot\hat{j}\notin\mathbb{R} \\ + \ln(t-1) \text{ is not defined } \forall t\le1 \\ + \text{This corresponds with answer choice D} \\ + \end{align*} +} + +\qs{}{ + \begin{align*} + \vec{r}\cdot\hat{j}\in[-4,4] \quad \land \quad \vec{r}\cdot\hat{i}\in[-3,3] \\ + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}=\langle -3\sin t ,4\cos t \rangle \\ + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \right |_{t=\tfrac{\pi}{2}}= \langle -3,0 \rangle \quad \vec{r}(0)=\langle 3,0 \rangle \\ + \text{This corresponds with answer choice B} + \end{align*} +} + +\qs{}{ + \begin{align*} + \text{This can be directly evaluated to: } \\ + \langle 28,-49 \rangle \\ + \text{This corresponds with answer choice D} + \end{align*} +} + +\qs{}{ + \begin{align*} + \vec{r}\cdot\hat{j} \text{ is not defined for } t=1 \text{ but the limit as $t\to6$ can be evaluated without affecting the proceess} \\ + \vec{r}\cdot\hat{i} \text{ is not defined for } t=1 \text{ but the limit as $t\to6$ can be evaluated without affecting the proceess} \\ + \text{Using directevaluation: } \\ + \langle \frac{5}{35},-\frac{36+12-3}{5} \rangle = \langle \tfrac{1}{7},-9 \rangle \\ + \text{This corresponds with answer choice B} + \end{align*} +} + +\qs{}{ + \begin{align*} + \implies \langle 0,-6 \rangle \\ + \text{This corresponds with answer choice B} + \end{align*} +} + +\qs{}{ + \begin{align*} + e^{-\ln 6}=6^{-1}=\tfrac{1}{6} \\ + \implies \lim_{t\to\ln 6}{\langle 6e^{-t}, 3e^{-t} \rangle}=\langle 1,\tfrac{1}{2} \rangle \\ + \text{This corresponds witha nswer choice B} + \end{align*} +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle -14t,\tfrac{1}{3}t^2 \rangle \\ + \text{This corresponds with answer choice C} + \end{align*} +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle -\csc^2 t, -\cot t\csc t \rangle \\ + \text{This corresponds with answer choice A} + \end{align*} +} +\qs{}{ +\begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 8te^{t^2}, -3, 2t \rangle \\ + \text{This corresponds with answer choice B} +\end{align*} +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 18\frac{1}{6t},6t^2 \rangle \\ + \frac{\mathrm{d}^2\vec{r}}{\mathrm{d}t^2} = \langle \frac{-108}{36t^2}, 12t \rangle \langle \frac{-3}{t^2}, 12t \rangle \\ + \text{This corresponds with answer choice C} + \end{align*} +} + +\qs{}{ + + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 15t^4,-60t^4,20t^4 \rangle \\ + \| \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \| \\ + &= \sqrt{15^2\cdot t^8 +60^2\cdot t^8 +20^2\cdot t^8} \\ + &= t^4 \sqrt{15^2+60^2+20^2} = 65t^4 \\ + \hat{T}=\langle \tfrac{15}{65},\tfrac{-60}{65},\tfrac{20}{65} \rangle \\ + \text{This corresponds with answer choice B} + \end{align*} + +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 36\sin^2(2t)\cos(2t), -36\cos^2(2t)\sin(2t) \rangle \\ + \| \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \| \\ + &= \sqrt{36^2 \sin^4(2t) \cos^2(2t)+36^2\cos^4(2t)\sin^2(2t)} \\ + &=\sqrt{(36^2 \sin^2(2t) \cos^2(2t))\cdot (\sin^2(2t)+cos^2(2t))} \\ + &=36\sin(2t)\cos(2t) \\ + &=18\sin(4t) + \end{align*} +} + +\end{document} diff --git a/labs-set-2/template.toc b/labs-set-2/template.toc new file mode 100644 index 0000000..72be2c9 --- /dev/null +++ b/labs-set-2/template.toc @@ -0,0 +1,3 @@ +\contentsline {chapter}{\numberline {1}Lab 7}{2}{chapter.1}% +\contentsline {section}{\numberline {1.1}Work}{2}{section.1.1}% +\contentsfinish