From 42b06eda82009ceefc56526fb4089f493d2abb97 Mon Sep 17 00:00:00 2001 From: Robert Date: Fri, 26 Mar 2021 18:18:09 +0100 Subject: [PATCH] Finished differential calculus --- chapters/real_analysis_2.tex | 1 + chapters/sections/diff_calc.tex | 478 ++++++++++++++++++++++++++++++++ script.pdf | Bin 595301 -> 640565 bytes script.tex | 13 +- 4 files changed, 490 insertions(+), 2 deletions(-) create mode 100644 chapters/sections/diff_calc.tex diff --git a/chapters/real_analysis_2.tex b/chapters/real_analysis_2.tex index 158febb..d815f57 100644 --- a/chapters/real_analysis_2.tex +++ b/chapters/real_analysis_2.tex @@ -5,4 +5,5 @@ \chapter{Real Analysis: Part II} \subfile{sections/lims_and_funcs.tex} + \subfile{sections/diff_calc.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/diff_calc.tex b/chapters/sections/diff_calc.tex new file mode 100644 index 0000000..2f285bd --- /dev/null +++ b/chapters/sections/diff_calc.tex @@ -0,0 +1,478 @@ +\documentclass[../../script.tex]{subfiles} +% !TEX root = ../../script.tex + +\begin{document} +\section{Differential Calculus} + +\begin{defi} + Let $I$ be an open interval ($(a, b)$, $a < b$, $a, b = \infty$ possible). Let $f: I \rightarrow \field$ and $x \in I$. + $f$ is called differentiable in $x$ if + \[ + f'(x) = \limes{h}{0} \underbrace{\frac{f(x + h) - f(x)}{h}}_{\text{Difference quotient}} + \] + exists. $f'(x)$ is called the differential quotient, or derivative of $f$ in $x$. $f$ is called differentiable if it is differentiable in every $x$. +\end{defi} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $f(x) = c$ with $c \in \field$ be a constant function + \[ + f'(x) = \limes{h}{0} \frac{c - c}{h} = 0 + \] + + \item For $n \in \natn$ consider $f: \realn \rightarrow \realn ~~x \mapsto x^n$ + \[ + f'(x) = \limes{h}{0} \frac{(x + h)^n - x^n}{h} = \limes{h}{0} \sum_{k=0}^n \binom{n}{k}h^{k-1}x^{k-1} = nx^{n-1} + \] + + \item Consider the exponential function + \[ + f'(x) = \limes{h}{0} \frac{\exp(x + h) - \exp(x)}{h} = \limes{h}{0} \exp(x) \frac{\exp(h) - 1}{h} = \exp(x) + \] + \end{enumerate} +\end{eg} + +\begin{thm} + Let $f: I \rightarrow \field$ be differentiable in $x$. Then $f$ is also continuous in $x$. +\end{thm} +\begin{proof} + Let $f$ be continuous in $x$. Then + \begin{equation} + \limes{h}{0} (f(x+h) - f(x)) = 0 + \end{equation} + Assume $f$ to be uncontinuous in $x$. This means that + \begin{equation} + \exists \epsilon > 0 ~\forall \delta > 0 ~\exists h \in (-\delta, \delta): ~~|f(x+h) - f(x)| \ge \epsilon + \end{equation} + In particular, for every $n$ there exists an $h_n \in \left( \frac{-1}{n}, \frac{1}{n} \right) \subset \set{0}$, such that + \begin{equation} + |f(x + h_n) - f(x)| \ge \epsilon + \end{equation} + $h_n$ is a null sequence and + \begin{equation} + \left| \frac{f(x + h_n) - f(x)}{h_n} \right| \ge \frac{\epsilon}{\frac{1}{n}} = n \cdot \epsilon \longrightarrow \infty + \end{equation} + So the above term doesn't converge, thus + \begin{equation} + \frac{f(x+h) - f(x)}{h} \longrightarrow \infty + \end{equation} + Therefore, $f$ isn't differentiable in $x$. +\end{proof} + +\begin{rem} + The inverse is not true. +\end{rem} + +\begin{thm} + Let $I$ be an open interval and $f, g: I \rightarrow \field$ differentiable in $x \in I$. Then $f+g$ and $f \cdot g$ are differentiable too, + and if $g(x) \ne 0$ then $f/g$ is also differentiable. + \begin{gather*} + (f + g)'(x) = f'(x) + g'(x) \\ + (f \cdot g)'(x) = f'(x)g(x) + f(x)g'(x) \\ + \left(\frac{1}{g}\right)'(x) = \frac{-g'(x)}{g(x)^2} + \end{gather*} +\end{thm} +\begin{proof} + \reader +\end{proof} + +\begin{thm}[Chain rule] + Let $I, J$ be open intervals, and let + \begin{align*} + g: J \longrightarrow I && f: i \longrightarrow \field + \end{align*} + $g$ and $f$ are to be differentiable in $x$ and $f(x)$ respectively. Then $f \circ g$ is differentiable in $x$ and + \[ + (f \circ g)' = g'(x) \cdot f'(g(x)) + \] +\end{thm} +\begin{proof} + Consider the following function + \begin{align} + \phi: J \longrightarrow \field && \phi(\xi) = \begin{cases} + \frac{f(g(x) + \xi) - f(g(x))}{\xi}, & \xi \ne 0 \\ + f'(g(x)), & \xi = 0 + \end{cases} + \end{align} + $\xi$ is continuous, since $f$ is continuous and + \begin{equation} + \limes{\xi}{0} \phi(\xi) = f'(g(x)) = \phi(0) + \end{equation} + $\forall \xi \in J$ the following holds + \begin{equation} + f(g(x) + \xi) - f(g(x)) = \phi(\xi) \cdot \xi + \end{equation} + With this we can now show that + \begin{equation} + \begin{split} + \frac{f(g(x+h)) - f(g(x))}{h} &= \frac{f(g(x) + (g(x + h) - g(x))) - f(g(x))}{h} \\ + &= \frac{\phi(g(x + h) - g(x))(g(x + h) - g(x))}{h} \\ + &= \underbrace{\phi(g(x + h) - g(x))}_{\conv{h \rightarrow 0}0} \cdot \underbrace{\frac{g(x + h) - g(x)}{h}}_{\conv{h \rightarrow 0}g'(x)} \\ + &\conv{h \rightarrow 0} g'(x) \cdot f'(g(x)) + \end{split} + \end{equation} +\end{proof} + +\begin{defi} + Let $I$ be an interval and $f: I \rightarrow \realn$. $x_0 \in I$ is called a global maximum if + \[ + f(x) \le f(x_0) ~~\forall x \in I + \] + $x_0 \in I$ is called a local maximum if + \[ + \exists \epsilon > 0: ~~f(x) \le f(x_0) ~~\forall x \in (x_0 - \epsilon, x_0 + \epsilon) + \] + An extremum is either maximum or minimum. +\end{defi} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $f: [-1, 1] \rightarrow \realn$, $f(x) = x^2$. + \begin{itemize} + \item $x_0 = 0$ is a local and global minimum + \item $x_0 = \pm 1$ is a local and global maximum + \end{itemize} + + \item Consider + \begin{align*} + f: \realn &\longrightarrow \realn \\ + x &\longmapsto \cos x + \frac{x}{2} + \end{align*} + $f$ has infinitely many local extrema, but no global ones! + + \begin{center} + \begin{tikzpicture}[domain=-12:12, scale=0.3] + \draw[very thin, step=3.0, color=gray] (-11, -11) grid (11, 11); + \draw[->] (-12.2,0) -- (12.2,0) node[right] {$x$}; + \draw[->] (0,-12.2) -- (0,12.2) node[above] {$y$}; + \draw[smooth, thick, color=blue] plot (\x,{cos(\x r) + \x / 2}) node[right] {$f(x)$}; + \end{tikzpicture} + \end{center} + + \item Consider + \begin{align*} + f: \realn &\longrightarrow \realn \\ + x &\longmapsto \begin{cases} + 1, & x \text{ rational} \\ + 0, & x \text{ irrational} + \end{cases} + \end{align*} + \begin{itemize} + \item $x_0$ rational is a global maximum + \item $x_0$ irrational is a global minimum + \end{itemize} + \end{enumerate} +\end{eg} + +\begin{thm} + Let $I$ be an open interval, and $f: I \realn \realn$ a function with a local extremum at $x_0 \in I$. Then + \[ + f \text{ differentiable in } x_0 \implies f'(x_0) = 0 + \] +\end{thm} +\begin{proof} + Assume $f'(x_0) \ne 0$ (w.l.o.g. $f'(x_0) > 0$, otherwise consider $-f$). Then + \begin{equation} + \exists \delta > 0: ~~\left| \frac{f(x_0 + h) - f(x)}{h} - f'(x_0) \right| < f'(x_0) ~~\forall h \in (-\delta, \delta) + \end{equation} + Especially + \begin{equation} + 0 < \frac{f(x_0 + h) - f(x_0)}{h} ~~\forall h \in (-\delta, \delta) + \end{equation} + For $h > 0$ this means $f(x_0 + h) > f(x_0)$. And for $h < 0$ this means that $f(x_0 + h) < f(x_0)$. Thus $x_0$ is not an extremum. +\end{proof} + +\begin{rem} + Let $f: I \rightarrow \realn$ be differentiable. To find the extrema of $f$, calculate $f'$ and find its roots. + However, the roots are to be insepcted more closely, as $f'(x_0) = 0$ is not a sufficient criterion (The function could have inflection points or behave badly at the boundaries of $I$). +\end{rem} + +\begin{thm}[Mean value theorem] + Let $a, b \in \realn$ with $a < b$, and let $f, g: [a, b] \rightarrow \realn$ be differentiable. Then $\exists \xi \in (a, b)$ such that + \[ + (f(b) - f(a))g'(\xi) = f'(\xi)(g(b) - g(a)) + \] + + \begin{center} + \begin{tikzpicture} + \begin{axis}[ytick={1, 10}, yticklabels={$f(a)$, $f(b)$}, xtick={1, 6.25, 10}, xticklabels={$a$,$\xi$,$b$}] + \addplot[thick, smooth, color=blue] coordinates { + (1, 1) + (10, 10) + } node[below right, pos=0.1] {$g(x)$}; + + \addplot[thick, smooth, color=red] coordinates { + (1, 1) + (2, 5) + (6, 6) + (7.8, 9.5) + (10, 10) + }node[above left, pos=0.3] {$f(x)$}; + + \addplot +[mark=none, dashed, gray] coordinates {(6.2, 0) (6.2, 10)}; + \end{axis} + \end{tikzpicture} + \end{center} +\end{thm} +\begin{proof} + Consider all + \begin{equation} + h(x) = (f(b) - f(a))g(x) - f(x)(g(b) - f(a)) + \end{equation} + $h$ is differentiable, which means $h$ is continuous on $[a, b]$: + \begin{equation} + h(a) = f(b)g(a) - f(a)g(b) = h(b) + \end{equation} + We need to show that $h'$ has a root in $[a, b]$. If $h$ is constant, this is trivial. So we assume $\exists x \in (a, b)$ such that $h(x) > h(a)$. + Since $h$ is continuous on $(a, b)$ there exists a global maximum $x_0 \in [a, b]$ with $x_0 \ne a$ and $x_0 \ne b$. + This implies that $h'(x_0) = 0$. If $h(x) < h(a)$ the same argument can be made. +\end{proof} + +\begin{rem} + This theorem is often written as + \[ + \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(\xi)}{g'(\xi)} + \] + And if $g(x) = x$ + \[ + \frac{f(b) - f(a)}{b - a} = f'(\xi) + \] +\end{rem} + +\begin{cor} + Let $I$ be an open interval and $f: I \rightarrow \realn$ differentiable. Then + \begin{enumerate}[(i)] + \item $f'(I) \subset [0, \infty) \iff \text{ monotonically increasing}$ + \item $f'(I) \subset (0, \infty) \implies \text{ strictly monotonically increasing}$ + \item $f'(I) \subset (-\infty, 0] \iff \text{ monotonically decreasing}$ + \item $f'(I) \subset (-\infty, 0) \implies \text{ striuctly monotonically decreasing}$ + \end{enumerate} +\end{cor} +\begin{proof} + We will only show the "$\implies$" direction for (i). Assume $f$ isn't monotonically increasing, then $\exists x, y \in I$ such that $x < y$ but $f(x) > f(y)$. + The mean value theorem thus states, $\exists \xi \in (x, y)$ such that + \begin{equation} + f'(\xi) = \frac{f(y) - f(x)}{y- x} < 0 + \end{equation} + All other statements are proven in the same fashion. +\end{proof} + +\begin{eg} + $f$ strictly monotonically increasing does NOT imply that $f'(I) \subset (0, \infty)$. Consider $f(x) = x^3$. +\end{eg} + +\begin{cor}[L'Hôpital's rule] + Let $a, b, x_0 \in \realn$, with $a < x_0 < b$ and let $f, g: (a, b) \rightarrow \realn$ be a differentiable function. We require $f(x_0) = g(x_0) = 0$. + If $g'(x) \ne 0 ~~\forall x \in I \setminus \set{x_0}$ and if + \[ + \limes{x}{x_0} \frac{f'(x)}{g'(x)} + \] + exists, then + \[ + \limes{x}{x_0} \frac{f(x)}{g(x)} = \limes{x}{x_0}\frac{f'(x)}{g'(x)} + \] +\end{cor} +\begin{proof} + Between two roots of $g$ there must be at least one root of $g'$. I.e. $g(x) \ne 0 ~~\forall x \in I \setminus \set{x_0}$. + This means, that + \begin{equation} + \forall x \in (a, x_0) ~\exists \xi_x: ~~\frac{f(x)}{g(x)} = \frac{f(x) - f(x_0)}{g(x) - g(x_0)} = \frac{f'(\xi_x)}{g'(\xi_x)} \implies \limes{x}{x_0}\frac{f'(x)}{g'(x)} + \end{equation} + Since $\xi_x \in (x, x_0)$ + \begin{equation} + \xi_x \conv{x \rightarrow x_0} x_0 + \end{equation} + For the limit from the left, this implies + \begin{equation} + \limes{x}{x_0} \frac{f(x)}{g(x)} = \limes{x}{x_0}\frac{f'(x)}{g'(x)} + \end{equation} + This argument can be made for the limit from the right as well. +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item For the computation of the limit it is enough to consider $f$ and $g$ on $(x_0 - \delta, x_0 + \delta)$ with $\delta > 0$. + \item L'Hôpital's rule also works for one-sided limits + \item Let $f, g: (a, b) \setminus \set{x_0} \rightarrow \realn$ be differentiable. Then it is enough to require + \[ + \limes{x}{x_0} f(x) = \limes{x}{x_0} g(x) = 0 + \] + \item L'Hôpital's rule doesn't generally apply to complex valued functions. + \item By substituring $\tilde{f}(x) = f\left(\frac{1}{x}\right)$ and $\tilde{g}(x) = g\left(\frac{1}{x}\right)$ we can also use + \[ + \limes{x}{\infty} \frac{\tilde{f}(x)}{\tilde{g}(x)} = \limes{x}{\infty} \frac{\tilde{f}'(x)}{\tilde{g}'(x)} + \] + \item The inverse + \[ + L = \limes{x}{x_0} \frac{f(x)}{g(x)} \implies \limes{x}{0} \frac{f'(x)}{g'(x)} = L + \] + is NOT true. + \end{enumerate} +\end{rem} + +\begin{eg} + Consider + \[ + \limes{x}{0} \frac{x^2}{1 - \cos x} = \mathquotes{\frac{0}{0}} + \] + The functions here are + \begin{align*} + f(x) = x^2 && g(x) = 1 - \cos x + \end{align*} + with the derivatives + \begin{align*} + f'(x) = 2x && g'(x) = \sin x + \end{align*} + However, the limit of the derivatives is still + \[ + \limes{x}{0} \frac{2x}{\sin x} = \mathquotes{\frac{0}{0}} + \] + We can derive the functions again + \begin{align*} + f''(x) = 2 && g''(x) = \cos x + \end{align*} + And thus + \[ + \limes{x}{0} \frac{2}{\cos x} = 2 \implies \limes{x}{0} \frac{x^2}{1 - \cos x} = 2 + \] +\end{eg} + +\begin{thm}[Derivative of inverse functions] + Let $I$ be an open inverval, and $f: I \rightarrow \realn$ differentiable with $f'(I) \subset (0, \infty)$. + Then $f$ has a differentiable inverse function $\inv{f}(x): f(I) \rightarrow \realn$ and for $y \in f(I)$ we have + \[ + \left(\inv{f}\right)'(y) = \frac{1}{f'\left(\inv{f}(y)\right)} + \] +\end{thm} +\begin{proof} + $f$ is strictly monotonically increasing, thus $\inv{f}$ exists and is continuous. Let $y \in f(I), ~x := \inv{f}(y)$ and + \begin{equation} + \xi(h) = \inv{f}(y + h) - \underbrace{\inv{f}(y)}_x + \end{equation} + Then + \begin{equation} + x + \xi(h) = \inv{f}(y + h) \implies f(x + \xi(h)) = y + h = f(x) + h + \end{equation} + Which in turn implies + \begin{equation} + f(x + \xi(h)) - f(x) = h + \end{equation} + Now we have + \begin{equation} + \begin{split} + \frac{\inv{f}(y + h) - \inv{f}(y)}{h} =& \frac{\xi(h)}{f(x + \xi(h)) - f(x)} \\ + =& \inv{\left( \frac{f(x + \xi(h)) - f(x)}{\xi(h)} \right)} \\ + \conv{h \rightarrow 0}& \inv{\left(f'(x)\right)} = \frac{1}{f'(\inv{f}(y))}> 0 + \end{split} + \end{equation} +\end{proof} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $n \in \natn$ and consider + \begin{align*} + f: (0, \infty) &\longrightarrow \realn \\ + x &\longmapsto x^n + \end{align*} + The derivative is $f'(x) = nx^{n-1}$. The inverse function is + \begin{align*} + g(y) = \sqrt[n]{y} && g'(y) = \frac{1}{f'(g(y))} = \frac{1}{n\left(\sqrt[n]{y}\right)^{n-1}} = \frac{1}{n} \cdot y^{\left(\frac{1}{n} - 1\right)} + \end{align*} + + \item The natural logarithm. Let $f(x) 0 \exp x$ and $g(y) = \ln y$. Then + \[ + (\ln y)' = \frac{1}{\exp(\ln(y))} = \frac{1}{y} + \] + + \item Let $f(x) = x^3$. Then + \[ + \inv{f}(y) = \begin{cases} + \sqrt[3]{y}, & y \ge 0 \\ + -\sqrt[3]{y}, & y < 0 + \end{cases} + \] + $\inv{f}$ is not differentiable in $y = 0$. + \end{enumerate} +\end{eg} + +\begin{defi} + Let $I$ be an open interval. $f: I \rightarrow \realn$ is said to be $(n+1)$-times differentiable if the $n$-th derivative of $f$ ($f^{(n)}$) is differentiable. + + $f$ is said to be infinitely differentiable (or smooth) if $f$ is $n$ times differentiable for all $n \in \natn$. + + $f$ is said to be $n$ times continuously differentiable if the $n$-th derivative $f^{(n)}$ is continuous. +\end{defi} + +\begin{defi} + Let $I$ be an open interval, and $f: I \rightarrow \realn$ $n$ times differentiable in $x \in I$. Then + \[ + T_n f(y) = \sum_{k=0}^n \frac{f^{(k)} (x)}{k!} (y - x)^k + \] + is called the Taylor polynomial of $n$-th degree at $x$ of $f$. +\end{defi} + +\begin{thm}[Taylor's theorem] + Let $I$ be an open interval and $f: I \rightarrow \realn$ an $(n+1)$-times differentiable function. + Let $x \in I$ and $h: I \rightarrow \realn$ differentiable. For every $y \in I$, there exists a $\xi$ between $x$ and $y$ such that + \[ + (f(y) - T_n f(y)) \cdot h'(\xi) = \frac{f^{(n+1)}(\xi)}{n!} (y - \xi)^n (h(y) - h(x)) + \] +\end{thm} +\begin{proof} + Let + \begin{equation} + \begin{split} + g: I &\longrightarrow \realn \\ + t &\longmapsto \sum_{k=0}^n \frac{f^{(k)}(t)}{k!} (y-t)^k + \end{split} + \end{equation} + Apply the mean value theorem to $g$ and $h$ to get + \begin{equation} + g'(\xi)(h(y) - h(x)) = (g(y) - g(x))h'(\xi) = (f(y) - T_nf(y))h'(\xi) + \end{equation} + and thus + \begin{equation} + \begin{split} + g'(t) &= \sum_{k=0}^n \underbrace{\left( \frac{f^{(k+1)}(t)}{k!} (y-t)^k - \frac{f^{(k)}(t)}{k!}k(y - t)^{k-1} \right)}_{\text{Telescoping series}} \\ + &= \frac{f^{n+1}(t)}{n!} (y-t)^n + \end{split} + \end{equation} + By inserting $\xi$ we receive the desired equation. +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item This is useful for when $h'(\xi) \ne 0$ + \item The choice of $h$ can yield different errors + \[ + R_{n+1}(y, x) := f(y) - T_nf(y) + \] + \item The Langrange error bound is for $h(t) = (y - t)^{n+1}$: + \[ + R_{n+1}(y, x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (y-x)^{n+1} + \] + \item This theorem makes no statement about Taylor series. + \end{enumerate} +\end{rem} + +\begin{cor} + Let $(a, b) \subset \realn$ and $f: (a, b) \rightarrow \realn$ a $n$-times continuously differentuable function with + \[ + 0 = f'(x) = f''(x) = \cdots = f^{(n-1)}(x) + \] + and $f^{(n)} \ne 0$. If $n$ is odd, then there is no local extremum in $x$. If $n$ is even then + \begin{align*} + f^{(n)}(x) > 0 &\implies x \text{ is a local maximum} \\ + f^{(n)}(x) < 0 &\implies x \text{ is a local minimum} + \end{align*} +\end{cor} +\begin{proof} + W.l.o.g. $f^{(n)} > 0$. We will use the Taylor series with Lagrange error bound. + According to prerequisites, $f^{(n)}$ is continuous, i.e. $\exists \epsilon > 0$ such that $f^{(n)}(\xi) > 0$ on $(x - \epsilon, x + \epsilon)$. + The Taylor formula tells us, that $\forall y \in (x - \epsilon, x + \epsilon) ~\exists \xi_y \in (x - \epsilon, x + \epsilon)$ such that + \begin{equation} + f(y) - T_{n-1}(f(y)) = f(y) - f(x) = \frac{f^{(n)}(\xi_y)}{n!} (y - x)^n + \end{equation} + For $n$ odd, $f(y) - f(x)$ assumes positive and negative values in every neighbourhood of $x$. If $n$ is even then $f(y) - f(x)$ cannot be negative, thus $x$ is a local minimum. +\end{proof} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 936cb0986cf2f762209a0bf2ca15159e325e3a16..7146ad2e9ee7249d320aabab55565f4f551272b5 100644 GIT binary patch delta 242932 zcmce$eP5PaAZ z{~3h;^u~WimN)66;oj7{nxgqGHdSLZ3aL!mUK-G%|A;s#YUgC`gbub9m{G4R8{200 zP7KT1xUp69hn=)*OLM9ER_@7Zbq+Q=e<7u4d9P!Qq9Rg&_n-!wEieMMvK)<0fvh`tskLv)R3lD$TjXnBx?}`;UgEyY+axkXt_eoBT z)(kz+aOI~_`wE-~c0IOZ@9J?Gan^Bp_Oomo)k|=Rn)@-h|H<_3ZWB6RAJqHJ*cwM~ zO{y9aa7gv2%a9vyoj)|&RXm&=X0yGld0fkvt*#tOSH*vS+`VPHo2Ksy))z%4b$XlK zN^J76S>`C0gmHC32cC>SzM0x;?I!F@6af2&3q;gp8wd1uk-**oPmEfp|l(}WMq<5t7ck3iHSo-441}8HA;F$H7fd_mieclM@gx7QWt8D z%`rktb*So2sB=bDy=FsbbJ|w7C8&;`4pfqxj;il&MfrMFrNU~~rXIOXqpn+8Q+{4H z(5^+TuaQouD0eSsXzf7Roe3AZTP00LXB^FKOqxkSqEOEa!pB}KhaZ4NxnV=g0NhVy zuM(lRKBO~x;bd-&esm?8R@X_jf%NYTXrYivwFY4f)JnBVD$Cs!$k!__nMH|9n~JA>89OVws@2n*)1PfW8@}n1*MO~wm;8^9xj(0w^j>@KA$Q!# 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\documentclass[11pt]{report} \usepackage{amsmath, amssymb, amstext, physics} -\usepackage{amsthm, stackrel, xifthen, mathtools} +\usepackage{amsthm, stackrel, xifthen, mathtools, graphicx} \usepackage[makeroom]{cancel} \usepackage{hyperref, cleveref} \usepackage{enumerate, subcaption} \usepackage[shortlabels]{enumitem} \usepackage{multicol} -\usepackage{tikz} +\usepackage{tikz, pgfplots} \usepackage{kbordermatrix} \usetikzlibrary{calc,trees,positioning,arrows,fit,shapes,angles} @@ -87,6 +87,15 @@ \newcommand{\leexpl}[1]{% \underset{\substack{\big\uparrow\\\mathrlap{\text{\hspace{-1.5em}#1}}}}{\le}} +\newsavebox{\mathbox}\newsavebox{\mathquote} +\makeatletter +\newcommand{\mathquotes}[1]{% \mathquotes{} + \savebox{\mathquote}{\text{``}}% Save quotes + \savebox{\mathbox}{$\displaystyle #1$}% Save + \raisebox{\dimexpr\ht\mathbox-\ht\mathquote\relax}{``}#1\raisebox{\dimexpr\ht\mathbox-\ht\mathquote\relax}{''} +} +\makeatother + \theoremstyle{definition} \newtheorem{defi}{Definition}[chapter]

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