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z>`hQT1)9^)kl;y9gQ{)XNB7A`>0~iNSuUt(80pY6t<13RRu6u3Qe_diau3`Je~5<_rCenNpgo)MMxKi5 ztkPZ_rr8)_DqKA0?X~@5ns!Tnj`aDB-d1lxC(2*xZD3<>^k06z&DsARVKG`3evH`u zzj_X1+`|`{^9zjtSH=Dvc(_e)14|!gm@H+4rr%6!Kqr)OTKbxfxv{ME^ zL3}o`LnXWGJ8Xw}-2GH8ndaLRdy_tDS70duZRY@>1B6>Gi5%xd{=B5PvubiZDGAVY zYRdUi>^qT$LnxhmAk7SJRUsDu#|WA6c?Aboe`VIjqt9n0)A*#JtNM|;3%nf+Ko2O^ zjGOD}GH4M|!l6*ZF3^jaF-z%FBVbr-0%_Z6s_63P=(~evk~(mhKuKKq>H?|=nc*3A zfcr1+3JON|R)_lMw%>ov_P?(G&;%?i@$Uft-jVn(!(Z3@H<$gF{>0x6|Gi7=uZBZ! zm9qcbv-La9?=3oiB4NJeYQHt?{BHbv2f&}kSMdMe_;39HzoY!VoBSur3F1HQE&mSi zdp-P504vOYe2hP;|JRUyH~l?9 z{?ily_{UfHBVGO-<@a#@Pn0_1e?<8sxc?pH--CufF+e~P$w5GV4 Date: Mon, 25 Feb 2019 09:38:25 +1100 Subject: [PATCH 03/16] Women of Troy background info --- english/troy.md | 44 ++++++++++++++++++++++++++++++++++++++++---- 1 file changed, 40 insertions(+), 4 deletions(-) diff --git a/english/troy.md b/english/troy.md index bc36083..d3e4eee 100644 --- a/english/troy.md +++ b/english/troy.md @@ -2,19 +2,55 @@ **Euripides (Athens, c. 450 BC)** +## Background + +- Set in 415 BC, after Trojan War +- Inspired by Homer's Iliad - mythical poem by Homer (Greek poet). Final weeks of war. +- 416 BC - Greeks invade Melos (Peloponnesian War). Melians neutral (relationship to Sparta) +- Athenians believed they were most civilised society + +### Trojan War + +- Eris, goddess of discord, gives golden apple to Hera, Athena, Aphrodite +- Zeus sends Eris to decide who has apple +- Paris/Alexander steals Helen from her husband Menelaus with help of Aphrodite +- King Menelaus gathers army +- Helen of Sparta becomes Helen of Troy +- war lasted 10 years +- Achilles & Hector killed +- gods take sides & end war +- Trojan horse - invented by Athena, executed by Odysseus & Athenians +- Trojans had banquet at end of war when Trojan horse arrived + +### Euripides + +- Athenian, 450 BC +- Golden Age - political/artistic/intellectual reneissance +- progressive representation of female characters, graphic/uncouth plays +- critical of society +- Athens attacked by Sparta >5 times + + + + ## Characters -**Poseidon** - god of the sea -**Athene** - goddess of wisdom +**Poseidon** - (careless) god of the sea, built Troy w/ Apollo +**Apollo** - god, affair w/ Cassandra in exchange for persistent foresight +**Athene** - goddess of wisdom - supported Greeks in Battle of Troy **Hecuba** - widow of Priam, King of Troy. Old & "grey-haired". +**Priam** - husband of Hecuba. 50 kids. +**Alexander** - aka Paris. Son of Priam & Hecuba. Kidnaps Helen from Sparta (most beautiful) **Cassandra** - their daughter, a prophetess. Manic. **Andromache** - their daughter-in-law, widow of Hector **Talthybius** - Greek officer - assigns women **Menelaus** - King of Sparta -**Helen** - his wife, Trojan, imprisoned +**Helen** - his wife, Trojan, imprisoned **Chorus** of Trojan women, captured **Astryanax** - small boy, Hector and Andromache's son -**Greek soldiers** +**Greek soldiers** + + - *trojan* - native of Troy - Epeios (Greek) built Trojan horse to defeat King Priam of Troy (city) -- 2.43.2 From 0bc414aebbaa28ce4e58338defea9aa3e9696386 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 25 Feb 2019 11:05:45 +1100 Subject: [PATCH 04/16] start exponential functions notes --- methods/stuff.md | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) diff --git a/methods/stuff.md b/methods/stuff.md index 37878dd..0424391 100644 --- a/methods/stuff.md +++ b/methods/stuff.md @@ -8,7 +8,7 @@ classoption: twocolumn header-includes: \pagenumbering{gobble} --- -# random methods shit +# Exponential and Index Functions ## Index laws @@ -33,4 +33,11 @@ Used for equations without common base exponent Or change base: $$\log_b c = {{\log_a c} \over {\log_a b}}$$ -If $a<1, \quad \log_{b} a < 0$ (flip inequality operator) \ No newline at end of file +If $a<1, \quad \log_{b} a < 0$ (flip inequality operator) + +## Exponential functions + +$e^x$ - natural exponential function + + +$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$ \ No newline at end of file -- 2.43.2 From dfc1aecf3c2278e043617d2b60fa767b7f454fc8 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 25 Feb 2019 22:41:54 +1100 Subject: [PATCH 05/16] complex roots --- spec/complex.md | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/spec/complex.md b/spec/complex.md index 871e1d4..d111921 100755 --- a/spec/complex.md +++ b/spec/complex.md @@ -150,3 +150,10 @@ ${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divid ## de Moivres' Theorem $(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mathbb{Z}$ + +## Roots of complex numbers + +$n$th roots of $r \operatorname{cis} \theta$ are: +$z={r^{1 \over n}} \cdot (\cos ({{\theta + 2k \pi} \over n}) + i \sin ({{\theta + 2 k \pi} \over n}))$ + +Same modulus for all solutions. 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From: Andrew Lorimer Date: Tue, 26 Feb 2019 22:03:09 +1100 Subject: [PATCH 07/16] [spec] 4f - conjugate root theorem --- spec/complex.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/spec/complex.md b/spec/complex.md index d111921..959c241 100755 --- a/spec/complex.md +++ b/spec/complex.md @@ -110,6 +110,8 @@ $z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$ ## Solving complex polynomials +Include $\pm$ for all solutions, including imaginary. + #### Dividing complex polynomials Dividing $P(z)$ by $D(z)$ gives quotient $Q(z)$ and remainder $R(z)$ such that: @@ -122,7 +124,7 @@ Let $\alpha \in \mathbb{C}$. Remainder of $P(z) \div (z - \alpha)$ is $P(\alpha) ## Conjugate root theorem -Let $P(z)$ be a polynomial with real coefficients. If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, the the conjugate $a-bi$ is also a solution. +If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, the the conjugate $a-bi$ is also a solution. ## Polar form -- 2.43.2 From 9de4a0c785d0bec4dcaebd628467d5cbc67d4c85 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Wed, 27 Feb 2019 17:23:42 +1100 Subject: [PATCH 08/16] sketching complex graphs --- spec/complex.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/spec/complex.md b/spec/complex.md index 959c241..c5ad6be 100755 --- a/spec/complex.md +++ b/spec/complex.md @@ -158,4 +158,10 @@ $(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mat $n$th roots of $r \operatorname{cis} \theta$ are: $z={r^{1 \over n}} \cdot (\cos ({{\theta + 2k \pi} \over n}) + i \sin ({{\theta + 2 k \pi} \over n}))$ -Same modulus for all solutions. Arguments are separated by ${2 \pi} \over n$ \ No newline at end of file +Same modulus for all solutions. Arguments are separated by ${2 \pi} \over n$ + +## Sketching complex graphs + +- **Straight line:** $\operatorname{Re}(z) = c$ or $\operatorname{Im}(z) = c$ (perpendicular bisector) or $\operatorname{Arg}(z) = \theta$ +- **Circle:** $|z-z_1|^2 = c^2 |z_2+2|^2$ or $|z-(a + bi)| = c$ +- **Locus:** $\operatorname{Arg}(z) \lt \theta$ -- 2.43.2 From 8cf574c0abfcc83eb077cc9a2c5684ae8cf54a63 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Fri, 1 Mar 2019 10:00:51 +1100 Subject: [PATCH 09/16] update planner, create complex reference --- planner.xlsx | Bin 13676 -> 13677 bytes spec/complex-ref.pdf | Bin 0 -> 36486 bytes spec/complex.md | 102 ++++++++++++++++++++++++------------------- 3 files changed, 57 insertions(+), 45 deletions(-) create mode 100644 spec/complex-ref.pdf diff --git a/planner.xlsx b/planner.xlsx index 62283a5499d59cdbc83735901b4d5c025228cd9e..b0c6861879baf50b22b1bbb6d6a4fab7bbd32440 100644 GIT binary patch delta 5573 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zvC(mnYR^@hn%d{%ak)FCioXVT6~TXhuxl~C(J}GSfk6P6Arz(yP*FiU*#Q0t_hg%j literal 0 HcmV?d00001 diff --git a/spec/complex.md b/spec/complex.md index c5ad6be..b70737f 100755 --- a/spec/complex.md +++ b/spec/complex.md @@ -1,81 +1,90 @@ +--- +geometry: margin=2cm + +graphics: yes +tables: yes +author: Andrew Lorimer +classoption: twocolumn +header-includes: +- \usepackage{harpoon} +- \usepackage{amsmath} +- \pagenumbering{gobble} + +--- + + # Complex & Imaginary Numbers ## Imaginary numbers -$i^2 = -1 \quad \therefore i = \sqrt {-1}$ +$$i^2 = -1 \quad \therefore i = \sqrt {-1}$$ ### Simplifying negative surds -$\sqrt{-2} = \sqrt{-1 \times 2}$ -$= \sqrt{2}i$ +\begin{equation}\begin{split}\sqrt{-2} & = \sqrt{-1 \times 2} \\ & = \sqrt{2}i\end{split}\end{equation} ## Complex numbers -$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$ +$$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$$ General form: $z=a+bi$ $\operatorname{Re}(z) = a, \quad \operatorname{Im}(z) = b$ ### Addition -If $z_1 = a+bi$ and $z_2=c+di$, then -$z_1+z_2 = (a+c)+(b+d)i$ +If $z_1 = a+bi$ and $z_2=c+di$, then + +$$z_1+z_2 = (a+c)+(b+d)i$$ ### Subtraction -If $z_1=a+bi$ and $z_2=c+di$, then $z_1−z_2=(a−c)+(b−d)i$ +If $z_1=a+bi$ and $z_2=c+di$, then + +$$z_1−z_2=(a−c)+(b−d)i$$ ### Multiplication by a real constant -If $z=a+bi$ and $k \in \mathbb{R}$, then $kz=ka+kbi$ +If $z=a+bi$ and $k \in \mathbb{R}$, then -### Powers of $i$ -$i^0=1$ -$i^1=i$ -$i^2=-1$ -$i^3=-i$ -$i^4=1$ -$\dots$ +$$kz=ka+kbi$$ -Therefore.. +### Powers of $i$ - $i^{4n} = 1$ - $i^{4n+1} = i$ - $i^{4n+2} = -1$ - $i^{4n+3} = -i$ -For $i^n$, divide $n$ by 4 and let remainder $= r$. Then $i^n = i^r$. +For $i^n$, find remainder $r$ when $n \div 4$. Then $i^n = i^r$. ### Multiplying complex expressions -If $z_1 = a+bi$ and $z_2=c+di$, then -$z_1 \times z_2 = (ac-bd)+(ad+bc)i$ +If $z_1 = a+bi$ and $z_2=c+di$, then -### Conjugates +$$z_1 \times z_2 = (ac-bd)+(ad+bc)i$$ -If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator) +### Conjugates -Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2 = |z|^2$ +If $z=a+bi$, conjugate is -- Multiplication and addition are associative +$$\overline{z} = a-bi$$ -#### Properties +##### Properties - $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$ - $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$ - $\overline{kz} = k \overline{z}, \text{ for } k \in \mathbb{R}$ -- $z \overline{z} = |z|^2$ +- $z \overline{z} = = (a+bi)(a-bi) = a^2+b^2 = |z|^2$ - $z + \overline{z} = 2 \operatorname{Re}(z)$ ### Modulus Distance from origin. -$|{z}|=\sqrt{a^2+b^2}$ -$\therefore z \overline{z} = |z|^2$ +$$|{z}|=\sqrt{a^2+b^2} \quad \therefore z \overline{z} = |z|^2$$ -#### Properties +###### Properties - $|z_1 z_2| = |z_1| |z_2|$ - $|{z_1 \over z_2}| = {|z_1| \over |z_2|}$ @@ -83,11 +92,11 @@ $\therefore z \overline{z} = |z|^2$ ### Multiplicative inverse -$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$ +\begin{equation}\begin{split}z^{-1} & = {1 \over z} \\ & = {{a-bi} \over {a^2+B^2}} \\ & = {\overline{z} \over {|z|^2}}\end{split}\end{equation} ### Dividing complex numbers -${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$ +$${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}} \quad \text{multiplicative inverse}$$ (using multiplicative inverse) @@ -97,20 +106,18 @@ ${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$ ## Argand planes - Geometric representation of $\mathbb{C}$ -- Horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$ +- horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$ - Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$ -## Solving complex quadratics - -To solve $z^2+a^2=0$ (sum of two squares): +## Solving complex polynomials -$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$ +**Include $\pm$ for all solutions, including imaginary** -*Must include $\pm$ in solutions* +## Solving complex quadratics -## Solving complex polynomials +To solve $z^2+a^2=0$ (sum of two squares): -Include $\pm$ for all solutions, including imaginary. +$$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$$ #### Dividing complex polynomials @@ -124,13 +131,13 @@ Let $\alpha \in \mathbb{C}$. Remainder of $P(z) \div (z - \alpha)$ is $P(\alpha) ## Conjugate root theorem -If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, the the conjugate $a-bi$ is also a solution. +If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, then the conjugate $\overline{z}=a-bi$ is also a solution. ## Polar form -$$\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation}$$ +\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation} -- $r=|z|$, given by Pythagoras ($r=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}$) +- $r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}$ - $\theta=\operatorname{arg}(z)$ (on CAS: `arg(a+bi)`) - **principal argument** is $\operatorname{Arg}(z) \in (-\pi, \pi]$ (note capital $\operatorname{Arg}$) @@ -151,17 +158,22 @@ ${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divid ## de Moivres' Theorem -$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mathbb{Z}$ +$$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}$$ ## Roots of complex numbers -$n$th roots of $r \operatorname{cis} \theta$ are: -$z={r^{1 \over n}} \cdot (\cos ({{\theta + 2k \pi} \over n}) + i \sin ({{\theta + 2 k \pi} \over n}))$ +$n$th roots of $z = r \operatorname{cis} \theta$ are + +$$z={r^{1 \over n}} \operatorname{cis}({{\theta + 2 k \pi} \over n})$$ Same modulus for all solutions. Arguments are separated by ${2 \pi} \over n$ +The solutions of $z^n=a \text{ where } a \in \mathbb{C}$ lie on circle + +$$x^2 + y^2 = (|a|^{1 \over n})^2$$ + ## Sketching complex graphs - **Straight line:** $\operatorname{Re}(z) = c$ or $\operatorname{Im}(z) = c$ (perpendicular bisector) or $\operatorname{Arg}(z) = \theta$ - **Circle:** $|z-z_1|^2 = c^2 |z_2+2|^2$ or $|z-(a + bi)| = c$ -- **Locus:** $\operatorname{Arg}(z) \lt \theta$ +- **Locus:** $\operatorname{Arg}(z) < \theta$ -- 2.43.2 From bd487b9356352199ce02b9c512ce5352379dff92 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 3 Mar 2019 19:16:31 +1100 Subject: [PATCH 10/16] add transformations-ref source --- methods/transformations-ref.tex | 288 ++++++++++++++++++++++++++++++++ 1 file changed, 288 insertions(+) create mode 100644 methods/transformations-ref.tex diff --git a/methods/transformations-ref.tex b/methods/transformations-ref.tex new file mode 100644 index 0000000..22532c4 --- /dev/null +++ b/methods/transformations-ref.tex @@ -0,0 +1,288 @@ +\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere +\PassOptionsToPackage{hyphens}{url} +% +\documentclass[]{article} +\usepackage{lmodern} +\usepackage{amssymb,amsmath} +\usepackage{ifxetex,ifluatex} +\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex + \usepackage[T1]{fontenc} + \usepackage[utf8]{inputenc} + \usepackage{textcomp} % provides euro and other symbols +\else % if luatex or xelatex + \usepackage{unicode-math} + \defaultfontfeatures{Scale=MatchLowercase} + \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} +\fi +% use upquote if available, for straight quotes in verbatim environments +\IfFileExists{upquote.sty}{\usepackage{upquote}}{} +\IfFileExists{microtype.sty}{% use microtype if available + \usepackage[]{microtype} + \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts +}{} +\makeatletter +\@ifundefined{KOMAClassName}{% if non-KOMA class + \IfFileExists{parskip.sty}{% + \usepackage{parskip} + }{% else + \setlength{\parindent}{0pt} + \setlength{\parskip}{6pt plus 2pt minus 1pt}} +}{% if KOMA class + \KOMAoptions{parskip=half}} +\makeatother +\usepackage{xcolor} +\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available +\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}} +\urlstyle{same} % don't use monospace font for urls +\usepackage{fullpage} +\usepackage{longtable,booktabs} +% Allow footnotes in longtable head/foot +\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}} +\makesavenoteenv{longtable} +\usepackage{graphicx,grffile} +\makeatletter +\makeatother + +% set default figure placement to htbp +\makeatletter +\def\fps@figure{htbp} +\makeatother + + +\author{Andrew Lorimer} +\date{} + +\begin{document} + +\hypertarget{transformation}{% +\section{Transformation}\label{transformation}} + +\textbf{Order of operations:} DRT - Dilations, Reflections, Translations + +\hypertarget{transforming-xn-to-ax-hnk}{% +\subsection{\texorpdfstring{Transforming \(x^n\) to +\(a(x-h)^n+K\)}{Transforming x\^{}n to a(x-h)\^{}n+K}}\label{transforming-xn-to-ax-hnk}} + +\begin{itemize} +\tightlist +\item + \(|a|\) is the dilation factor of \(|a|\) units parallel to \(y\)-axis + or from \(x\)-axis +\item + if \(a<0\), graph is reflected over \(x\)-axis +\item + \(k\) - translation of \(k\) units parallel to \(y\)-axis or from + \(x\)-axis +\item + \(h\) - translation of \(h\) units parallel to \(x\)-axis or from + \(y\)-axis +\item + for \((ax)^n\), dilation factor is \(1 \over a\) parallel to + \(x\)-axis or from \(y\)-axis +\item + when \(0 < |a| < 1\), graph becomes closer to axis +\end{itemize} + +\hypertarget{translations}{% +\subsection{Translations}\label{translations}} + +For \(y = f(x)\), these processes are equivalent: + +\begin{itemize} +\tightlist +\item + applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the + graph of \(y = f(x)\) +\item + replacing \(x\) with \(x − h\) and \(y\) with \(y − k\) to obtain + \(y − k = f (x − h)\) +\end{itemize} + +\hypertarget{dilations}{% +\subsection{Dilations}\label{dilations}} + +For the graph of \(y = f(x)\), there are two pairs of equivalent +processes: + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\item + \begin{itemize} + \tightlist + \item + Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\) + \item + Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\) + \end{itemize} +\item + \begin{itemize} + \tightlist + \item + Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\) + \item + Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\) + \end{itemize} +\end{enumerate} + +For graph of \(y={1 \over x}\), horizontal \& vertical dilations are +equivalent (symmetrical). If \(y={a \over x}\), graph is contracted +rather than dilated. + +\hypertarget{transforming-fx-to-yafnxcb}{% +\subsection{\texorpdfstring{Transforming \(f(x)\) to +\(y=Af[n(x+c)]+b\)}{Transforming f(x) to y=Af{[}n(x+c){]}+b}}\label{transforming-fx-to-yafnxcb}} + +Applies to exponential, log, trig, power, polynomial functions.\\ +Functions must be written in form \(y=Af[n(x+c)] + b\) + +\(A\) - dilation by factor \(A\) from \(x\)-axis (if \(A<0\), reflection +across \(y\)-axis)\\ +\(n\) - dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\), +reflection across \(x\)-axis)\\ +\(c\) - translation from \(y\)-axis (\(x\)-shift)\\ +\(b\) - translation from \(x\)-axis (\(y\)-shift) + +\hypertarget{power-functions}{% +\subsection{Power functions}\label{power-functions}} + +\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\) +(including \(x=0\)) + +\hypertarget{odd-and-even-functions}{% +\subsubsection{Odd and even functions}\label{odd-and-even-functions}} + +Even when \(f(x) = -f(x)\)\\ +Odd when \(-f(x) = f(-x)\) + +Function is even if it can be reflected across \(y\)-axis +\(\implies f(x)=f(-x)\)\\ +Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd + +\hypertarget{xn-where-n-in-mathbbz}{% +\subsubsection{\texorpdfstring{\(x^n\) where +\(n \in \mathbb{Z}^+\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xn-where-n-in-mathbbz}} + +\begin{longtable}[]{@{}ll@{}} +\toprule +\(n\) is even: & \(n\) is odd:\tabularnewline +\midrule +\endhead +\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/parabola.png} +& +\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cubic.png}\tabularnewline +\bottomrule +\end{longtable} + +\hypertarget{xn-where-n-in-mathbbz-}{% +\subsubsection{\texorpdfstring{\(x^n\) where +\(n \in \mathbb{Z}^-\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}-}}\label{xn-where-n-in-mathbbz-}} + +\begin{longtable}[]{@{}ll@{}} +\toprule +\(n\) is even: & \(n\) is odd:\tabularnewline +\midrule +\endhead +\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/truncus.png} +& +\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/hyperbola.png}\tabularnewline +\bottomrule +\end{longtable} + +\hypertarget{x1-over-n-where-n-in-mathbbz}{% +\subsubsection{\texorpdfstring{\(x^{1 \over n}\) where +\(n \in \mathbb{Z}^+\)}{x\^{}\{1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x1-over-n-where-n-in-mathbbz}} + +\begin{longtable}[]{@{}ll@{}} +\toprule +\(n\) is even: & \(n\) is odd:\tabularnewline +\midrule +\endhead +\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/square-root-graph.png} +& +\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cube-root-graph.png}\tabularnewline +\bottomrule +\end{longtable} + +\hypertarget{x-1-over-n-where-n-in-mathbbz}{% +\subsubsection{\texorpdfstring{\(x^{-1 \over n}\) where +\(n \in \mathbb{Z}^+\)}{x\^{}\{-1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x-1-over-n-where-n-in-mathbbz}} + +Mostly only on CAS. + +We can write +\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\ +Domain is: +\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\) + +If \(n\) is odd, it is an odd function. + +\hypertarget{xp-over-q-where-p-q-in-mathbbz}{% +\subsubsection{\texorpdfstring{\(x^{p \over q}\) where +\(p, q \in \mathbb{Z}^+\)}{x\^{}\{p \textbackslash{}over q\} where p, q \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xp-over-q-where-p-q-in-mathbbz}} + +\[x^{p \over q} = \sqrt[q]{x^p}\] + +\begin{itemize} +\tightlist +\item + if \(p > q\), the shape of \(x^p\) is dominant +\item + if \(p < q\), the shape of \(x^{1 \over q}\) is dominant +\item + points \((0, 0)\) and \((1, 1)\) will always lie on graph +\item + Domain is: + \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\) +\end{itemize} + +\hypertarget{combinations-of-functions-piecewisehybrid}{% +\subsection{Combinations of functions +(piecewise/hybrid)}\label{combinations-of-functions-piecewisehybrid}} + +\[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\] + +Open circle - point included\\ +Closed circle - point not included + +\hypertarget{sum-difference-product-of-functions}{% +\subsubsection{Sum, difference, product of +functions}\label{sum-difference-product-of-functions}} + +\begin{longtable}[]{@{}lll@{}} +\toprule +\endhead +sum & \(f+g\) & domain +\(= \text{dom}(f) \cap \text{dom}(g)\)\tabularnewline +difference & \(f-g\) or \(g-f\) & domain +\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline +product & \(f \times g\) & domain +\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline +\bottomrule +\end{longtable} + +Addition of linear piecewise graphs - add \(y\)-values at key points + +Product functions: + +\begin{itemize} +\tightlist +\item + product will equal 0 if one of the functions is equal to 0 +\item + turning point on one function does not equate to turning point on + product +\end{itemize} + +\hypertarget{matrix-transformations}{% +\subsection{Matrix transformations}\label{matrix-transformations}} + +Find new point \((x^\prime, y^\prime)\). Substitute these into original +equation to find image with original variables \((x, y)\). + +\hypertarget{composite-functions}{% +\subsection{Composite functions}\label{composite-functions}} + +\((f \circ g)(x)\) is defined iff +\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\) + +\end{document} -- 2.43.2 From 146c682596915cb38d0caa97ec80e0dd97cc7c86 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 3 Mar 2019 19:16:44 +1100 Subject: [PATCH 11/16] start electrochemistry notes --- chem/electrochemistry.md | 46 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 46 insertions(+) create mode 100644 chem/electrochemistry.md diff --git a/chem/electrochemistry.md b/chem/electrochemistry.md new file mode 100644 index 0000000..2d465e2 --- /dev/null +++ b/chem/electrochemistry.md @@ -0,0 +1,46 @@ +--- +header-includes: +- \usepackage[version=4]{mhchem} +--- + +# Electrochemistry + +## Oxidation states + +Indicates charge (ionisation) of an element + +**Oxidation** - loss of e- +**Reduction** - gain of e- + +Main group elements (i.e. group 2) - generally one oxidation state: + +| elements | valence config | oxidation state | +| ------------- | -------------- | --------------- | +| alkali metals | $s^1$ | +1 | +| alkali earths | $s^2$ | +2 | +| aluminium | $s^2 p^1$ | +2 | +| nitrogen | $s^2 p^3$ | -3 | +| oxygen | $s^2 p^4$ | -2 | +| halogens | $s^2 p^5$ | -1 | +| noble gases | $s^2,\>s^2p^5$ | n/a | + +Transition metals (d shell) may have several oxidation states. + +$$\ce{H_2}$$ + +Common oxidation numbers: + +| elements | common ox. no. in compounds | exceptions | +| ----------------- | --------------------------- | ------------------- | +| main group metals | valency | no | +| hydrogen | +1 | metal hydrides (-1) | +| oxygen | -2 | + + +### Rules for oxidation states + +- oxidation states >3 may only exist in compounds +- oxidation number of free element is 0 +- oxidation number of simple ion is the charge of the ion +- sum of oxidation numbers in polyatomic ion is the charge of the ion +- sum of oxidation numbers of a neutral compound is zero -- 2.43.2 From 7c695a8cb476f2f5ad02acfc5cba77eeb988b42c Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 3 Mar 2019 19:17:44 +1100 Subject: [PATCH 12/16] minor formatting corrections in complex notes --- spec/complex.md | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/spec/complex.md b/spec/complex.md index b70737f..ce71511 100755 --- a/spec/complex.md +++ b/spec/complex.md @@ -96,12 +96,11 @@ $$|{z}|=\sqrt{a^2+b^2} \quad \therefore z \overline{z} = |z|^2$$ ### Dividing complex numbers -$${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}} \quad \text{multiplicative inverse}$$ - -(using multiplicative inverse) +$${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}} \quad \text{(multiplicative inverse)}$$ In practice, rationalise denominator: -${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$ + +$${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$$ ## Argand planes -- 2.43.2 From 62aa2fae7b0fd1d5f01ddcd813caec3a018b38de Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 3 Mar 2019 19:20:36 +1100 Subject: [PATCH 13/16] tidy up latex sidecar files --- chem/hdpe.aux | 35 -- chem/hdpe.bbl | 42 --- chem/hdpe.blg | 48 --- chem/hdpe.dvi | Bin 4340 -> 0 bytes chem/hdpe.fdb_latexmk | 169 --------- chem/hdpe.fls | 252 -------------- chem/hdpe.log | 776 ------------------------------------------ chem/hdpe.out | 4 - chem/hdpe.synctex.gz | Bin 25163 -> 0 bytes chem/hdpe.xdv | Bin 6952 -> 0 bytes chem/latextest.aux | 1 - chem/latextest.log | 387 --------------------- chem/latextest.pdf | Bin 9858 -> 0 bytes chem/latextest.tex | 5 - 14 files changed, 1719 deletions(-) delete mode 100644 chem/hdpe.aux delete mode 100644 chem/hdpe.bbl delete mode 100644 chem/hdpe.blg delete mode 100644 chem/hdpe.dvi delete mode 100644 chem/hdpe.fdb_latexmk delete mode 100644 chem/hdpe.fls delete mode 100644 chem/hdpe.log delete mode 100644 chem/hdpe.out delete mode 100644 chem/hdpe.synctex.gz delete mode 100644 chem/hdpe.xdv delete mode 100644 chem/latextest.aux delete mode 100644 chem/latextest.log delete mode 100644 chem/latextest.pdf delete mode 100644 chem/latextest.tex diff --git a/chem/hdpe.aux b/chem/hdpe.aux deleted file mode 100644 index e381f12..0000000 --- a/chem/hdpe.aux +++ /dev/null @@ -1,35 +0,0 @@ -\relax -\citation{gmi} -\citation{fig1} -\citation{fig1} -\citation{nzmfg} -\@writefile{toc}{\contentsline {section}{\numberline {1}Invention and Manufacture}{1}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces amorphous vs crystalline polymers\cite {fig1}}}{1}} -\newlabel{fig:structure}{{1}{1}} -\@writefile{toc}{\contentsline {paragraph}{}{1}} -\@writefile{toc}{\contentsline {paragraph}{}{1}} -\@writefile{toc}{\contentsline {paragraph}{}{1}} -\@writefile{toc}{\contentsline {paragraph}{}{1}} -\@writefile{toc}{\contentsline {paragraph}{}{1}} 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index 89a7fc5..0000000 --- a/chem/hdpe.bbl +++ /dev/null @@ -1,42 +0,0 @@ -\begin{thebibliography}{1} - -\bibitem{gmi} -{Global Market Insights}. -\newblock High density polyethylene (hdpe) market size. -\newblock - \url{https://www.gminsights.com/industry-analysis/high-density-polyethylene-HDPE-market}, - 2017. -\newblock [Online; accessed 20-May-2018]. - -\bibitem{fig1} -Thorsten Graunke, Katrin Schmitt, and J~Wöllenstein. -\newblock Organic membranes for selectivity enhancement of metal oxide gas - sensors. -\newblock 2016:1--22, 01 2016. - -\bibitem{pubchem} -{National Center for Biotechnology Information}. -\newblock Pubchem compound database; cid=6325. -\newblock \url{https://pubchem.ncbi.nlm.nih.gov/compound/6325}. -\newblock [Online; accessed 20-May-2018]. - -\bibitem{nzmfg} -{New Zealand Institute of Chemistry}. -\newblock The manufacture of polyethylene. -\newblock - \url{https://web.archive.org/web/20180223141558/http://www.nzic.org.nz/ChemProcesses/polymers/10J.pdf}. -\newblock [Online; accessed 23-May-2018]. - -\bibitem{macrog} -{Polymer Science Learning Centre}. -\newblock Polyethylene. -\newblock \url{http://www.pslc.ws/macrog/pe.htm}. -\newblock [Online; accessed 23-May-2018]. - -\bibitem{wiki:xxx} -{Wikipedia contributors}. -\newblock High-density polyethylene --- {Wikipedia}{,} the free encyclopedia. -\newblock \url{https://en.wikipedia.org/wiki/High-density_polyethylene}, 2018. -\newblock [Online; accessed 20-May-2018]. - -\end{thebibliography} diff --git a/chem/hdpe.blg b/chem/hdpe.blg deleted file mode 100644 index 31fc6ed..0000000 --- a/chem/hdpe.blg +++ /dev/null @@ -1,48 +0,0 @@ -This is BibTeX, Version 0.99d (TeX Live 2017/Arch Linux) -Capacity: max_strings=100000, hash_size=100000, hash_prime=85009 -The top-level auxiliary file: hdpe.aux -The style file: plain.bst -Database file #1: hdpe.bib -Warning--empty journal in fig1 -You've used 6 entries, - 2118 wiz_defined-function locations, - 527 strings with 4932 characters, -and the 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zi+v_Q1`?rt#%G<3Ewm*L0em;9mQ)<98P8rE;%hrNYQjmIeX`wn_|nz?)WxLa8R_<#K#0-&dtQjrGy EFKgyHR{#J2 diff --git a/chem/latextest.tex b/chem/latextest.tex deleted file mode 100644 index 6939b59..0000000 --- a/chem/latextest.tex +++ /dev/null @@ -1,5 +0,0 @@ -\documentclass{article} -\usepackage{mhchem} -\begin{document} -Hello world! -\end{document} -- 2.43.2 From c78bf500ad582fe6b02a531f9df7090e86fe89ee Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 3 Mar 2019 19:21:40 +1100 Subject: [PATCH 14/16] electrochemical series and galvanic cells --- chem/electrochemistry.md | 48 ++++++++++++++++++++++++++++------------ 1 file changed, 34 insertions(+), 14 deletions(-) diff --git a/chem/electrochemistry.md b/chem/electrochemistry.md index 2d465e2..20de01e 100644 --- a/chem/electrochemistry.md +++ b/chem/electrochemistry.md @@ -1,16 +1,11 @@ ---- -header-includes: -- \usepackage[version=4]{mhchem} ---- - # Electrochemistry ## Oxidation states Indicates charge (ionisation) of an element -**Oxidation** - loss of e- -**Reduction** - gain of e- +**Oxidation** - loss of e- (at anode) +**Reduction** - gain of e- (at cathode) Main group elements (i.e. group 2) - generally one oxidation state: @@ -26,15 +21,14 @@ Main group elements (i.e. group 2) - generally one oxidation state: Transition metals (d shell) may have several oxidation states. -$$\ce{H_2}$$ - Common oxidation numbers: -| elements | common ox. no. in compounds | exceptions | -| ----------------- | --------------------------- | ------------------- | -| main group metals | valency | no | -| hydrogen | +1 | metal hydrides (-1) | -| oxygen | -2 | +| elements | common ox. state | exceptions | +| ----------------- | ---------------- | ------------------- | +| main group metals | valency | | +| hydrogen | +1 | metal hydrides (-1) | +| oxygen | -2 | ce{H2O2} (-1) | +| halogens | -1 | | ### Rules for oxidation states @@ -44,3 +38,29 @@ Common oxidation numbers: - oxidation number of simple ion is the charge of the ion - sum of oxidation numbers in polyatomic ion is the charge of the ion - sum of oxidation numbers of a neutral compound is zero + +## Electrochemical series + +- Top is most likely to be reduced +- Strongest reductants are bottom right +- Strongest oxidants are top left +- Strong oxidants have weak conjugate reductants +- $E^0$ values are measured relative to ce{H2} / ce{H^+} = 0V + +## Conjugate redox pairs + +Oxidant and conjugate *reduced form* +e.g. ce{Cu^2+} / ce{Cu}, $\quad$ ce{Zn^2+} / ce{Zn} + +Usually one member of pair is used as electrode (except for *inert electrodes*, e.g. platinum) + +## Electrochemical/galvanic cells + +1. Find two half reactions involved (between electrode and solution) +2. Higher equation will proceed left to right +3. Lower equation will proceed right to left + +emf for each cell is calculated as $E^0(\text{red}) - E^0(\text{ox})$ +Then total emf is $\sum_{i=1}^2 \Sigma E^0({i})$ + +For a spontaneous reaction to occur, species on left must be in electrical contact with species on lower right \ No newline at end of file -- 2.43.2 From d7d5be9aa432f163c9fc958491b3d57f7763571d Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 4 Mar 2019 11:20:43 +1100 Subject: [PATCH 15/16] [chem] primary cells and fuel cells --- chem/electrochemistry.md | 22 +++++++++++++++++++--- 1 file changed, 19 insertions(+), 3 deletions(-) diff --git a/chem/electrochemistry.md b/chem/electrochemistry.md index 20de01e..e439add 100644 --- a/chem/electrochemistry.md +++ b/chem/electrochemistry.md @@ -60,7 +60,23 @@ Usually one member of pair is used as electrode (except for *inert electrodes*, 2. Higher equation will proceed left to right 3. Lower equation will proceed right to left -emf for each cell is calculated as $E^0(\text{red}) - E^0(\text{ox})$ -Then total emf is $\sum_{i=1}^2 \Sigma E^0({i})$ +emf for each cell is calculated as $E^0(\text{red}) - E^0(\text{ox})$ + +For a *spontaneous* (primary/fuel cell) reaction to occur, species on left must be in electrical contact with species on lower right + +### Primary cells + +Used for low-current electronic devices. Fixed quantity of reactants. + +- **Zinc-carbon dry cell** - carbon rod cathode and zinc anode (case) in ammonium chloride/zinc chloride electrolyte +- **Alkaline cell** - steel cathode (case) and steel/brass rod anode in potassium hydroxide electrolyte +- **Silver-zinc cell** - zinc anode, graphite/silver-oxide electrolyte, potassium hydroxide electrolyte +- **Lithium cell** - magnesium oxide anode, nickel/steel cathode (case), lithium, electrolyte. Lithium is low on electrochemical series enables higher voltage + +### Fuel cells + +Used for vehicles/long-lasting applications. ce{H2} may be replenished. + +- **Alkaline fuel cell** - ce{KOH(aq)} electrolyte, ce{H2(g)} from tank, ce{O2(g)} from atmosphere, water out of hydrogen side +- **Acid fuel cell** - ce{H3PO4} electrolyte, water out of oxygen side -For a spontaneous reaction to occur, species on left must be in electrical contact with species on lower right \ No newline at end of file -- 2.43.2 From e030c9908b972635590cf883bebc23553b564387 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 4 Mar 2019 11:34:48 +1100 Subject: [PATCH 16/16] [spec] render vectors cheatsheet --- spec/vectors-ref.pdf | Bin 0 -> 60473 bytes spec/vectors.md | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) create mode 100644 spec/vectors-ref.pdf diff --git a/spec/vectors-ref.pdf b/spec/vectors-ref.pdf new file mode 100644 index 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to $b$ is equal to $\vec{a} - \vec{u} ## Vector proofs **Concurrent lines -** $\ge$ 3 lines intersect at a single point -**Collinear points -** $\ge$ 3 points lie on the same line ($\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 \lt \mu \lt 1$) +**Collinear points -** $\ge$ 3 points lie on the same line ($\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 < \mu < 1$) Useful vector properties: -- 2.43.2