e084adbe3b00bfbf96abe5203c14a0d184bcbb29
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  72\pagestyle{fancy}
  73\fancyhead[LO,LE]{Year 12 Specialist}
  74\fancyhead[CO,CE]{Andrew Lorimer}
  75
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 101
 102\begin{document}
 103
 104\begin{multicols}{2}
 105
 106  \section{Complex numbers}
 107
 108  \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
 109
 110  \begin{align*}
 111    \text{Cartesian form: } & a+bi\\
 112    \text{Polar form: } & r\operatorname{cis}\theta
 113  \end{align*}
 114
 115  \subsection*{Operations}
 116
 117  \definecolor{shade1}{HTML}{ffffff}
 118  \definecolor{shade2}{HTML}{e6f2ff}
 119  \definecolor{shade3}{HTML}{cce2ff}
 120  \begin{tabularx}{\columnwidth}{r|X|X}
 121    & \textbf{Cartesian} & \textbf{Polar} \\
 122    \hline
 123    \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
 124    \hline
 125    \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
 126    \cline{1-1}\cline{3-3}
 127    \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
 128    \hline
 129    \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
 130    \hline
 131    \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
 132  \end{tabularx}
 133
 134  \subsubsection*{Scalar multiplication in polar form}
 135
 136  For \(k \in \mathbb{R}^+\):
 137  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
 138
 139  \noindent For \(k \in \mathbb{R}^-\):
 140  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
 141
 142    \subsection*{Conjugate}
 143
 144    \begin{align*}
 145      \overline{z} &= a \mp bi\\
 146      &= r \operatorname{cis}(-\theta)
 147    \end{align*}
 148
 149    \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
 150
 151    \subsubsection*{Properties}
 152
 153    \begin{align*}
 154      \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
 155      \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
 156      \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
 157      z\overline{z} &= (a+bi)(a-bi)\\
 158      &= a^2 + b^2\\
 159      &= |z|^2
 160    \end{align*}
 161
 162    \subsection*{Modulus}
 163
 164    \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
 165
 166    \subsubsection*{Properties}
 167
 168    \begin{align*}
 169      |z_1z_2|&=|z_1||z_2|\\
 170      \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
 171      |z_1+z_2|&\le|z_1|+|z_2|
 172    \end{align*}
 173
 174    \subsection*{Multiplicative inverse}
 175
 176    \begin{align*}
 177      z^{-1}&=\frac{a-bi}{a^2+b^2}\\
 178      &=\frac{\overline{z}}{|z|^2}a\\
 179      &=r \operatorname{cis}(-\theta)
 180    \end{align*}
 181
 182    \subsection*{Dividing over \(\mathbb{C}\)}
 183
 184    \begin{align*}
 185      \frac{z_1}{z_2}&=z_1z_2^{-1}\\
 186      &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
 187      &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
 188      & \qquad \text{(rationalise denominator)}
 189    \end{align*}
 190
 191    \subsection*{Polar form}
 192
 193    \begin{align*}
 194      z&=r\operatorname{cis}\theta\\
 195      &=r(\cos \theta + i \sin \theta)
 196    \end{align*}
 197
 198    \begin{itemize}
 199      \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
 200      \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
 201      \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
 202      \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
 203      \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
 204      \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
 205    \end{itemize}
 206
 207    \subsection*{de Moivres' theorem}
 208
 209    \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
 210
 211    \subsection*{Complex polynomials}
 212
 213    Include \(\pm\) for all solutions, incl. imaginary
 214
 215    \begin{tabularx}{\columnwidth}{ R{0.55} X  }
 216      \hline
 217      Sum of squares & \(\begin{aligned} 
 218        z^2 + a^2 &= z^2-(ai)^2\\
 219      &= (z+ai)(z-ai) \end{aligned}\) \\
 220      \hline
 221      Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
 222      \hline
 223      Division & \(P(z)=D(z)Q(z)+R(z)\) \\
 224      \hline
 225      Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
 226      \hline
 227      Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
 228      \hline
 229      Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
 230      \hline
 231    \end{tabularx}
 232
 233    \subsection*{\(n\)th roots}
 234
 235    \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
 236
 237    \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
 238
 239    \begin{itemize}
 240
 241      \item{Same modulus for all solutions}
 242      \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
 243      \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
 244      \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
 245      \item{Solutions of \(z^n=a\) where \(a \in \mathbb{C}\) lie on the circle \(x^2+y^2=\left(|a|^{\frac{1}{n}}\right)^2\) \quad (intervals of \(\frac{2\pi}{n}\))}
 246    \end{itemize}
 247
 248    \noindent For \(0=az^2+bz+c\), use quadratic formula:
 249
 250    \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
 251
 252    \subsection*{Fundamental theorem of algebra}
 253
 254    A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
 255
 256    \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
 257    \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
 258
 259    \subsection*{Argand planes}
 260
 261    \begin{center}\begin{tikzpicture}[scale=2]
 262      \draw [->] (-0.2,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 263      \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
 264      \coordinate (P) at (1,1);
 265      \coordinate (a) at (1,0);
 266      \coordinate (b) at (0,1);
 267      \coordinate (O) at (0,0);
 268      \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
 269        \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
 270        \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
 271        \begin{scope}
 272          \path[clip] (O) -- (P) -- (a);
 273          \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
 274          \node at ($(O)+(20:3mm)$) {$\theta$};
 275        \end{scope}
 276        \filldraw (P) circle (0.5pt);
 277    \end{tikzpicture}\end{center}
 278
 279    \begin{itemize}
 280      \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
 281      \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
 282    \end{itemize}
 283
 284    \subsection*{Sketching complex graphs}
 285
 286    \subsubsection*{Linear}
 287
 288    \begin{itemize}
 289      \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
 290      \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
 291      \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
 292    \end{itemize}
 293
 294    \subsubsection*{Circles}
 295
 296    \begin{itemize}
 297      \item \(|z-z_1|^2=c^2|z_2+2|^2\)
 298      \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
 299    \end{itemize}
 300
 301    \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
 302
 303    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 304      \draw [->] (0,0) -- (1,0) node [right]  {$\operatorname{Re}(z)$};
 305      \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 306      \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
 307      \draw [->, thick, blue] (0,0) -- (1,1);
 308      \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
 309      \begin{scope}
 310        \path[clip] (0,0) -- (1,1) -- (1,0);
 311        \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
 312        \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
 313      \end{scope}
 314      \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
 315      \node [blue, mydot] {};
 316    \end{tikzpicture}\end{center}
 317
 318    \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
 319
 320    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 321      \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 322      \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 323      \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
 324      \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
 325      \begin{scope}
 326        \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
 327        \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
 328      \end{scope}
 329      \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
 330      \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
 331      \node [brown, mydot] at (-0.25,0) {};
 332      \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
 333      \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
 334      \node [below, font=\footnotesize] at (1,0) {\(1\)};
 335    \end{tikzpicture}\end{center}
 336
 337    \section{Vectors}
 338    \begin{center}\begin{tikzpicture}
 339      \draw [->] (-0.5,0) -- (3,0) node [right]  {\(x\)};
 340      \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
 341      \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
 342      \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 343        \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
 344        \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
 345      \end{scope}
 346      \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
 347    \end{tikzpicture}\end{center}
 348    \subsection*{Column notation}
 349
 350    \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
 351      \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
 352
 353        \subsection*{Scalar multiplication}
 354
 355        \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
 356
 357        \noindent For \(k \in \mathbb{R}^-\), direction is reversed
 358
 359        \subsection*{Vector addition}
 360        \begin{center}\begin{tikzpicture}[scale=1]
 361          \coordinate (A) at (0,0);
 362          \coordinate (B) at (2,2);
 363          \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
 364          \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
 365          \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
 366        \end{tikzpicture}\end{center}
 367
 368        \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
 369
 370        \begin{itemize}
 371          \item Draw each vector head to tail then join lines
 372          \item Addition is commutative (parallelogram)
 373          \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
 374        \end{itemize}
 375
 376        \subsection*{Magnitude}
 377
 378        \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
 379
 380        \subsection*{Parallel vectors}
 381
 382        \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
 383
 384        For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
 385        \[\boldsymbol{a \cdot b}=\begin{cases}
 386          |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 387          -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 388        \end{cases}\]
 389        %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
 390        %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
 391
 392        \subsection*{Perpendicular vectors}
 393
 394        \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
 395
 396        \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
 397        \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
 398
 399          \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
 400
 401
 402          \begin{center}\begin{tikzpicture}[scale=2]
 403            \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
 404            \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
 405            \begin{scope}
 406              \path[clip] (1,0.5) -- (1,0) -- (0,0);
 407              \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 408              \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 409            \end{scope}
 410          \end{tikzpicture}\end{center}
 411          \begin{align*}\boldsymbol{a} \cdot \boldsymbol{b} &= a_1 b_1 + a_2 b_2 \\  &= |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta \\ &\quad (\> 0 \le \theta \le \pi) \text{ - from cosine rule}\end{align*}
 412            \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
 413
 414            \subsubsection*{Properties}
 415
 416            \begin{enumerate}
 417              \item
 418                \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
 419              \item
 420                \(\boldsymbol{a \cdot 0}=0\)
 421              \item
 422                \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
 423              \item
 424                \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
 425              \item
 426                \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
 427              \item
 428                \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
 429            \end{enumerate}
 430
 431            \subsection*{Angle between vectors}
 432
 433            \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
 434
 435            \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
 436
 437            (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
 438
 439            \subsection*{Angle between vector and axis}
 440
 441            \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
 442            which makes angles \(\alpha, \beta, \gamma\) with positive side of
 443            \(x, y, z\) axes:
 444            \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
 445
 446            \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
 447            between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
 448            \(x\)-axis
 449
 450            \subsection*{Projections \& resolutes}
 451
 452            \begin{tikzpicture}[scale=3]
 453              \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
 454              \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
 455              \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
 456              \begin{scope}
 457                \path[clip] (1,0.5) -- (1,0) -- (0,0);
 458                \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 459                \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 460              \end{scope}
 461              \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 462                \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
 463              \end{scope}
 464              \draw (0,0) coordinate (O)
 465              (1,0) coordinate (A)
 466              (1,0.5) coordinate (B)
 467              pic [draw,red,angle radius=2mm] {right angle = O--A--B};
 468            \end{tikzpicture}
 469
 470            \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
 471
 472            \begin{align*}
 473              \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
 474              & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
 475              & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
 476            \end{align*}
 477
 478            \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
 479            \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
 480
 481            \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
 482            \begin{align*}
 483              s &= |\boldsymbol{u}|\\
 484              &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
 485              &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
 486              &= |\boldsymbol{a}| \cos \theta
 487            \end{align*}
 488
 489            \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
 490
 491            \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
 492
 493
 494            \subsection*{Vector proofs}
 495
 496            \textbf{Concurrent:} intersection of \(\ge\) 3 lines
 497
 498            \begin{tikzpicture}
 499              \draw [blue] (0,0) -- (1,1);
 500              \draw [red] (1,0) -- (0,1);
 501              \draw [brown] (0.4,0) -- (0.6,1);
 502              \filldraw (0.5,0.5) circle (2pt);
 503            \end{tikzpicture}
 504
 505            \subsubsection*{Collinear points}
 506
 507            \(\ge\) 3 points lie on the same line
 508
 509            \begin{tikzpicture}
 510              \draw [purple] (0,0) -- (4,1);
 511              \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
 512              \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
 513              \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
 514              \coordinate (O) at (2.8,-0.2);
 515              \node at (O) [below] {\(O\)}; 
 516              \begin{scope}[->, orange, thick] 
 517                \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
 518                \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
 519                \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
 520              \end{scope}
 521            \end{tikzpicture}
 522
 523            \begin{align*}
 524              \text{e.g. Prove that}\\
 525              \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
 526              \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
 527              &= \overrightharp{OA} + m\overrightharp{AB}\\
 528              &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
 529              &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
 530              &=(1-m)\boldsymbol{a}+m{b}
 531            \end{align*}
 532            \begin{align*}
 533              \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
 534              \text{where } \lambda + \mu &= 1\\
 535              \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
 536            \end{align*}
 537
 538
 539            \subsubsection*{Parallelograms}
 540
 541            \begin{center}\begin{tikzpicture}
 542              \coordinate (O) at (0,0) node [below left] {\(O\)};
 543              \coordinate (A) at (4,0);
 544              \coordinate (B) at (6,2);
 545              \coordinate (C) at (2,2);
 546              \coordinate (D) at (6,0);
 547
 548              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
 549              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
 550              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
 551              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
 552
 553              \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
 554              \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
 555              \begin{scope}
 556                \path[clip] (C) -- (A) -- (O);
 557                \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
 558                \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
 559              \end{scope}
 560              \draw [gray, thick, dotted] (B) -- (D) node [pos=0.5, right, black, font=\footnotesize] {\(|\boldsymbol{c}|\sin\theta\)} (A) -- (D) node [pos=0.5, below, black, font=\footnotesize] {\(|\boldsymbol{c}|\cos\theta\)};
 561              \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
 562            \end{tikzpicture}\end{center}
 563
 564            \begin{itemize}
 565              \item
 566                Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
 567              \item
 568                If diagonals are equal length, it is a rectangle
 569              \item
 570                \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
 571              \item
 572                Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
 573            \end{itemize}
 574
 575            \subsubsection*{Useful vector properties}
 576
 577            \begin{itemize}
 578              \item
 579                \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
 580                \(k \in \mathbb{R} \setminus \{0\}\)
 581              \item
 582                If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
 583                least one point in common, then they lie on the same straight line
 584              \item
 585                \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
 586              \item
 587                \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
 588            \end{itemize}
 589
 590            \subsection*{Linear dependence}
 591
 592            \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
 593            \begin{align*}
 594              0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
 595              \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
 596            \end{align*}
 597
 598            \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
 599            independent if no vector in the set is expressible as a linear
 600            combination of other vectors in set, or if they are parallel.
 601
 602            \subsection*{Three-dimensional vectors}
 603
 604            Right-hand rule for axes: \(z\) is up or out of page.
 605
 606            \tdplotsetmaincoords{60}{120} 
 607            \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick}, 
 608              vector/.style={-stealth,red,very thick}, 
 609              vector guide/.style={dashed,gray,thick}]
 610
 611              %standard tikz coordinate definition using x, y, z coords
 612              \coordinate (O) at (0,0,0);
 613
 614              %tikz-3dplot coordinate definition using x, y, z coords
 615
 616              \pgfmathsetmacro{\ax}{1}
 617              \pgfmathsetmacro{\ay}{1}
 618              \pgfmathsetmacro{\az}{1}
 619
 620              \coordinate (P) at (\ax,\ay,\az);
 621
 622              %draw axes
 623              \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
 624              \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
 625              \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
 626
 627              %draw a vector from O to P
 628              \draw[vector] (O) -- (P);
 629
 630              %draw guide lines to components
 631              \draw[vector guide]         (O) -- (\ax,\ay,0);
 632              \draw[vector guide] (\ax,\ay,0) -- (P);
 633              \draw[vector guide]         (P) -- (0,0,\az);
 634              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 635              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 636              \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
 637              \node[tdplot_main_coords,above right]
 638              at (\ax,\ay,\az){(\ax, \ay, \az)};
 639            \end{tikzpicture}\end{center}
 640
 641            \subsection*{Parametric vectors}
 642
 643            Parametric equation of line through point \((x_0, y_0, z_0)\) and
 644            parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
 645
 646            \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
 647
 648              \section{Circular functions}
 649
 650              \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
 651
 652              \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
 653              \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
 654
 655              \subsection*{Reciprocal functions}
 656
 657              \subsubsection*{Cosecant}
 658
 659              \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
 660
 661              \begin{itemize}
 662                \item
 663                  \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
 664                \item
 665                  \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 666                \item
 667                  \textbf{Turning points} at
 668                  \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
 669                \item
 670                  \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 671              \end{itemize}
 672
 673              \subsubsection*{Secant}
 674
 675\begin{tikzpicture}
 676  \begin{axis}[ytick={-1,1}, yticklabels={\(-1\), \(1\)}, xmin=-7,xmax=7,ymin=-3,ymax=3,enlargelimits=true, xtick={-6.2830, -3.1415, 3.1415, 6.2830},xticklabels={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
 677%    \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
 678    \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
 679    \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
 680    \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 681    \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
 682    \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
 683    \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 684\end{axis}
 685    \node [black] at (7,3.5) {\(\cos x\)};
 686\end{tikzpicture}
 687
 688                \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 689
 690                \begin{itemize}
 691
 692                  \item
 693                    \textbf{Domain}
 694                    \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
 695                  \item
 696                    \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 697                  \item
 698                    \textbf{Turning points} at
 699                    \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 700                  \item
 701                    \textbf{Asymptotes} at
 702                    \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
 703                \end{itemize}
 704
 705                \subsubsection*{Cotangent}
 706
 707\begin{tikzpicture}
 708  \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
 709    \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
 710\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
 711\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
 712\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
 713\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
 714    \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 715    \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
 716    \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
 717    \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
 718    \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 719\end{axis}
 720\end{tikzpicture}
 721
 722                  \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 723
 724                  \begin{itemize}
 725
 726                    \item
 727                      \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
 728                    \item
 729                      \textbf{Range} \(= \mathbb{R}\)
 730                    \item
 731                      \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 732                  \end{itemize}
 733
 734                  \subsubsection*{Symmetry properties}
 735
 736                  \[\begin{split}
 737                    \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
 738                    \operatorname{sec} (-x) & = \operatorname{sec} x \\
 739                    \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
 740                    \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
 741                    \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
 742                    \operatorname{cot} (-x) & = - \operatorname{cot} x
 743                  \end{split}\]
 744
 745                  \subsubsection*{Complementary properties}
 746
 747                  \[\begin{split}
 748                    \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
 749                    \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
 750                    \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
 751                    \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
 752                  \end{split}\]
 753
 754                  \subsubsection*{Pythagorean identities}
 755
 756                  \[\begin{split}
 757                    1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
 758                    1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
 759                  \end{split}\]
 760
 761                  \subsection*{Compound angle formulas}
 762
 763                  \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
 764                  \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
 765                  \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
 766
 767                  \subsection*{Double angle formulas}
 768
 769                  \[\begin{split}
 770                    \cos 2x &= \cos^2 x - \sin^2 x \\
 771                    & = 1 - 2\sin^2 x \\
 772                    & = 2 \cos^2 x -1
 773                  \end{split}\]
 774
 775                  \[\sin 2x = 2 \sin x \cos x\]
 776
 777                  \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
 778
 779                  \subsection*{Inverse circular functions}
 780
 781                  \begin{tikzpicture}
 782                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
 783                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
 784                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
 785                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
 786                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
 787                      \addplot[mark=*, blue] coordinates {(1,0)};
 788                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
 789                    \end{axis}
 790                  \end{tikzpicture}\\
 791
 792                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 793
 794                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
 795                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
 796
 797                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
 798                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
 799
 800                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
 801                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 802
 803                  \begin{tikzpicture}
 804                    \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
 805                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
 806                      \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
 807                      \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
 808                    \end{axis}
 809                  \end{tikzpicture}
 810\columnbreak
 811                  \section{Differential calculus}
 812
 813                  \subsection*{Limits}
 814
 815                  \[\lim_{x \rightarrow a}f(x)\]
 816                  \(L^-,\quad L^+\) \qquad limit from below/above\\
 817                  \(\lim_{x \to a} f(x)\) \quad limit of a point\\
 818
 819                  \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
 820                  e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
 821
 822                  \subsubsection*{Limit theorems}
 823
 824                  \begin{enumerate}
 825                    \item
 826                      For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
 827                    \item
 828                      \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
 829                    \item
 830                      \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
 831                    \item
 832                      \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
 833                    \item
 834                      \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 835                    \item
 836                      \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
 837                  \end{enumerate}
 838
 839                  \subsection*{Gradients of secants and tangents}
 840
 841                  \textbf{Secant (chord)} - line joining two points on curve\\
 842                  \textbf{Tangent} - line that intersects curve at one point
 843
 844                  \subsection*{First principles derivative}
 845
 846                  \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
 847
 848                  \subsubsection*{Logarithmic identities}
 849
 850                  \(\log_b (xy)=\log_b x + \log_b y\)\\
 851                  \(\log_b x^n = n \log_b x\)\\
 852                  \(\log_b y^{x^n} = x^n \log_b y\)
 853
 854                  \subsubsection*{Index identities}
 855
 856                  \(b^{m+n}=b^m \cdot b^n\)\\
 857                  \((b^m)^n=b^{m \cdot n}\)\\
 858                  \((b \cdot c)^n = b^n \cdot c^n\)\\
 859                  \({a^m \div a^n} = {a^{m-n}}\)
 860
 861                  \subsection*{Reciprocal derivatives}
 862
 863                  \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
 864
 865                  \subsection*{Differentiating \(x=f(y)\)}
 866                  Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
 867
 868                  \subsection*{Second derivative}
 869                  \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
 870                  \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
 871
 872                  \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
 873
 874                  \subsubsection*{Points of Inflection}
 875
 876                  \emph{Stationary point} - i.e.
 877                  \(f^\prime(x)=0\)\\
 878                  \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
 879                  \(f^{\prime\prime} = 0\))
 880
 881                  \subsubsection*{Strictly increasing/decreasing}
 882
 883                  For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
 884
 885                  \textbf{Strictly increasing}\\
 886                  \hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
 887                  \textbf{strictly decreasing}\\
 888                  \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
 889                  \begin{warning}
 890                    Endpoints are included, even where gradient \(=0\)
 891                  \end{warning}
 892
 893
 894                  \begin{table*}[ht]
 895                    \centering
 896                    \begin{tabularx}{\textwidth}{rYYY}
 897                      \hline
 898                      \rowcolor{shade2}
 899                      & \adjustbox{margin=0 1ex, valign=m}{\centering\(\dfrac{d^2 y}{dx^2} > 0\)}  & \adjustbox{margin=0 1ex, valign=m}{\centering \(\dfrac{d^2y}{dx^2}<0\)} & \adjustbox{margin=0 1ex, valign=m}{\(\dfrac{d^2y}{dx^2}=0\) (inflection)} \\
 900                      \hline
 901                      \(\dfrac{dy}{dx}>0\) &
 902                      \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3,  xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x)};  \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
 903                        \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0.1, xmax=4,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))};  \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
 904                          \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1.5,  xmax=1.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
 905                            \hline
 906                            \(\dfrac{dy}{dx}<0\) &
 907                            \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
 908                              \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0,  xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)};  \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
 909                                \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=1.5,  xmax=4.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
 910                                  \hline
 911                                  \(\dfrac{dy}{dx}=0\)&
 912                                  \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}&                       \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x)}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
 913                                    \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture}  \\Stationary inflection point}\\
 914                                      \hline
 915                    \end{tabularx}
 916                  \end{table*}
 917                  \begin{itemize}
 918                    \item
 919                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
 920                      \((a, f(a))\) is a local min (curve is concave up)
 921                    \item
 922                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
 923                      \((a, f(a))\) is local max (curve is concave down)
 924                    \item
 925                      if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
 926                      inflection
 927                    \item
 928                      if also \(f^\prime(a)=0\), then it is a stationary point of inflection
 929                  \end{itemize}
 930
 931                  \subsection*{Implicit Differentiation}
 932
 933                  \noindent Used for differentiating circles etc.
 934
 935                  If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
 936                  for all \(x\) and \(y\), then:
 937
 938                  \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
 939
 940                  \begin{cas}
 941                    Action \(\rightarrow\) Calculation \\
 942                      \hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)} \hfill(returns \(y^\prime= \dots\))
 943                  \end{cas}
 944
 945                  \subsection*{Slope fields}
 946
 947                  \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
 948                    \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
 949                      \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
 950                      \pgfplotsinvokeforeach{-4,...,4}{%
 951                        \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) --  ( {#1 +0.1}, {4  + diff(#1, 4) *0.1});
 952                        \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) --  ( {#1 +0.1}, {3  + diff(#1, 3) *0.1});
 953                        \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) --  ( {#1 +0.1}, {2  + diff(#1, 2) *0.1});
 954                        \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) --  ( {#1 +0.1}, {1  + diff(#1, 1) *0.1});
 955                        \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) --  ( {#1 +0.1}, {0  + diff(#1, 0) *0.1});
 956                        \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) --  ( {#1 +0.1}, {-1  + diff(#1, -1) *0.1});
 957                        \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) --  ( {#1 +0.1}, {-2  + diff(#1, -2) *0.1});
 958                        \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) --  ( {#1 +0.1}, {-3  + diff(#1, -3) *0.1});
 959                        \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) --  ( {#1 +0.1}, {-4  + diff(#1, -4) *0.1});
 960                      }
 961                    \end{axis}
 962                  \end{tikzpicture}
 963
 964                  \subsection*{Parametric equations}
 965
 966                  For each point on \(\left( f(t), g(t) \right)\):
 967
 968                  \begin{align*}
 969                    \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
 970                    \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
 971                    \text{Also...} \\
 972                    \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
 973                  \end{align*}
 974
 975                \subsection*{Integration}
 976
 977                \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 978
 979                  \subsubsection*{Definite integrals}
 980
 981                  \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
 982
 983                  \begin{itemize}
 984
 985                    \item
 986                      Signed area enclosed by\\
 987                      \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
 988                    \item
 989                      \emph{Integrand} is \(f\).
 990                  \end{itemize}
 991
 992                  \subsubsection*{Properties}
 993
 994                  \begin{align*}
 995                    \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\
 996                    \int^a_a f(x) \> dx &= 0 \\
 997                    \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\
 998                    \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\
 999                    \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\
1000                  \end{align*}
1001
1002                  \subsection*{Integration by substitution}
1003
1004                  \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
1005
1006                  \begin{warning}
1007                    \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
1008                  \end{warning}
1009                  \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1010                    \text{let } u&=x+4\\
1011                    \implies& {\frac{du}{dx}} = 1\\
1012                    \implies& x = u - 4\\
1013                    \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1014                    &\text{(solve as  normal integral)}
1015                  \end{align*}
1016
1017                  \subsubsection*{Definite integrals by substitution}
1018
1019                  For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1020                  \(b\) for \(f(u) \cdot du\).
1021
1022                  \subsubsection*{Trigonometric integration}
1023
1024                  \[\sin^m x \cos^n x \cdot dx\]
1025
1026                  \paragraph{\textbf{\(m\) is odd:}}
1027                  \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1028                  \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1029                  Substitute \(u=\cos x\)
1030
1031                  \paragraph{\textbf{\(n\) is odd:}}
1032                  \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1033                  \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1034                  Substitute \(u=\sin x\)
1035
1036                  \paragraph{\textbf{\(m\) and \(n\) are even:}}
1037                  use identities...
1038
1039                  \begin{itemize}
1040
1041                    \item
1042                      \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1043                    \item
1044                      \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1045                    \item
1046                      \(\sin 2x = 2 \sin x \cos x\)
1047                  \end{itemize}
1048
1049                  \subsection*{Partial fractions}
1050
1051                  To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
1052                  \begin{align*}
1053                    \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
1054                    \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
1055                    \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
1056                    \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
1057                  \end{align*}
1058
1059                  \subsubsection*{Repeated linear factors}
1060
1061                  \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
1062
1063                  \subsubsection*{Irreducible quadratic factors}
1064
1065                  \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
1066
1067                  \begin{cas}
1068                    Action \(\rightarrow\) Transformation:\\
1069                    \hspace{1em} \texttt{expand(..., x)}
1070
1071                    To reverse, use \texttt{combine(...)}
1072                  \end{cas}
1073
1074                  \subsection*{Graphing integrals on CAS}
1075
1076                  \begin{cas}
1077                    \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
1078                    Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
1079                  \end{cas}
1080
1081                  \subsection*{Applications of antidifferentiation}
1082
1083                  \begin{itemize}
1084
1085                    \item
1086                      \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1087                      stationary points on \(y=F(x)\)
1088                    \item
1089                      nature of stationary points is determined by sign of \(y=f(x)\) on
1090                      either side of its \(x\)-intercepts
1091                    \item
1092                      if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1093                      \(n+1\)
1094                  \end{itemize}
1095
1096                  To find stationary points of a function, substitute \(x\) value of given
1097                  point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1098                  original function.
1099
1100                  \subsection*{Solids of revolution}
1101
1102                  Approximate as sum of infinitesimally-thick cylinders
1103
1104                  \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
1105
1106                  \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
1107
1108                  \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
1109
1110                  \begin{align*}
1111                    V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
1112                    &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
1113                  \end{align*}
1114
1115                  \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
1116
1117                  \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1118                  \hfill where \(f(x) > g(x)\)
1119
1120                  \subsection*{Length of a curve}
1121
1122                  \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1123
1124                  \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1125
1126                  \begin{cas}
1127                    \begin{enumerate}[label=\alph*), leftmargin=5mm]
1128                      \item Evaluate formula
1129                      \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1130                    \end{enumerate}
1131                  \end{cas}
1132
1133                  \subsection*{Rates}
1134
1135                  \subsubsection*{Gradient at a point on parametric curve}
1136
1137                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1138
1139                  \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1140
1141                  \subsection*{Rational functions}
1142
1143                  \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1144
1145                  \subsubsection*{Addition of ordinates}
1146
1147                  \begin{itemize}
1148
1149                    \item
1150                      when two graphs have the same ordinate, \(y\)-coordinate is double the
1151                      ordinate
1152                    \item
1153                      when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1154                      (\(x\)-intercept)
1155                    \item
1156                      when one of the ordinates is 0, the resulting ordinate is equal to the
1157                      other ordinate
1158                  \end{itemize}
1159
1160                  \subsection*{Fundamental theorem of calculus}
1161
1162                  If \(f\) is continuous on \([a, b]\), then
1163
1164                  \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1165                  \hfill where \(F = \int f \> dx\)
1166                  
1167                  \subsection*{Differential equations}
1168
1169                  \noindent\textbf{Order} - highest power inside derivative\\
1170                  \textbf{Degree} - highest power of highest derivative\\
1171                  e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1172
1173                  \subsubsection*{Verifying solutions}
1174
1175                  Start with \(y=\dots\), and differentiate. Substitute into original
1176                  equation.
1177
1178                  \subsubsection*{Function of the dependent
1179                  variable}
1180
1181                  If \({\frac{dy}{dx}}=g(y)\), then
1182                  \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1183                  \(e^c\) as \(A\).
1184
1185
1186
1187                  \subsubsection*{Mixing problems}
1188
1189                  \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1190
1191                  \subsubsection*{Separation of variables}
1192
1193                  If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1194
1195                  \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1196
1197                  \subsubsection*{Euler's method for solving DEs}
1198
1199                  \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1200
1201                  \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1202
1203                  \include{calculus-rules}
1204
1205    \section{Kinematics \& Mechanics}
1206
1207      \subsection*{Constant acceleration}
1208
1209      \begin{itemize}
1210        \item \textbf{Position} - relative to origin
1211        \item \textbf{Displacement} - relative to starting point
1212      \end{itemize}
1213
1214      \subsubsection*{Velocity-time graphs}
1215
1216      \begin{itemize}
1217        \item Displacement: \textit{signed} area between graph and \(t\) axis
1218        \item Distance travelled: \textit{total} area between graph and \(t\) axis
1219      \end{itemize}
1220
1221      \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1222
1223        \begin{center}
1224          \renewcommand{\arraystretch}{1}
1225          \begin{tabular}{ l r }
1226              \hline & no \\ \hline
1227              \(v=u+at\) & \(x\) \\
1228              \(v^2 = u^2+2as\) & \(t\) \\
1229              \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1230              \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1231              \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1232            \end{tabular}
1233        \end{center}
1234
1235        \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1236        \begin{align*}
1237          \text{speed} &= |{\text{velocity}}| \\
1238          &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1239        \end{align*}
1240
1241        \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1242        \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1243
1244        \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1245        \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1246
1247      \subsection*{Vector functions}
1248
1249        \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1250
1251        \begin{itemize}
1252          \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1253          \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1254          \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1255        \end{itemize}
1256
1257      \subsection*{Vector calculus}
1258
1259      \subsubsection*{Derivative}
1260
1261        Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1262        \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1263
1264      \subfile{dynamics}
1265      \subfile{statistics}
1266  \end{multicols}
1267\end{document}