From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Mon, 2 Sep 2019 13:46:59 +0900 Subject: Initial commit --- macros/latex/contrib/withesis/math.tex | 237 +++++++++++++++++++++++++++++++++ 1 file changed, 237 insertions(+) create mode 100644 macros/latex/contrib/withesis/math.tex (limited to 'macros/latex/contrib/withesis/math.tex') diff --git a/macros/latex/contrib/withesis/math.tex b/macros/latex/contrib/withesis/math.tex new file mode 100644 index 0000000000..ac00cfea42 --- /dev/null +++ b/macros/latex/contrib/withesis/math.tex @@ -0,0 +1,237 @@ +\chapter{Mathematics Examples} +This appendix provides an example of \LaTeX's typesetting +capabilities. Most of text was obtained from the University of +Wisconsin-Madison Math Department's example thesis file. + +\section{Matrices} +The equations for the {\em dq}-model of an induction machine in the +synchronous reference frame are +\begin{eqnarray} + \left[\begin{array}{c} v_{qs}^e\\v_{ds}^e\\v_{qr}^e\\v_{dr}^e \end{array}\right] + &=& \left[ \begin{array}{cccc} + r_s + x_s\frac{\rho}{\omega_b} & \frac{\omega_e}{\omega_b}x_s & x_m\frac{\rho}{\omega_b} & \frac{\omega_e}{\omega_b}x_m \\ + -\frac{\omega_e}{\omega_b}x_s & r_s + x_s\frac{\rho}{\omega_b} & -\frac{\omega_e}{\omega_b}x_m & x_m\frac{\rho}{\omega_b} \\ + x_m\frac{\rho}{\omega_b} & \frac{\omega_e -\omega_r}{\omega_b}x_m & r_r'+x_r'\frac{\rho}{\omega_b} & \frac{\omega_e - \omega_r}{\omega_b}x_r' \\ + -\frac{\omega_e -\omega_r}{\omega_b}x_m & x_m\frac{\rho}{\omega_b} & -\frac{\omega_e - \omega_r}{\omega_b}x_r' & r_r' + x_r'\frac{\rho}{\omega_b} + \end{array} \right] + \left[\begin{array}{c} i_{qs}^e\\i_{ds}^e\\i_{qr}^e\\i_{dr}^e\end{array} \right] \label{volteq}\\ + T_e&=&\frac{3}{2}\frac{P}{2}\frac{x_m}{\omega_b}\left(i_{qs}^ei_{dr}^e - i_{ds}^ei_{qr}^e\right) \label{torqueeq}\\ + T_e-T_l&=&\frac{2J\omega_b}{P}\frac{d}{dt}\left(\frac{\omega_r}{\omega_b}\right) \label{mecheq}. +\end{eqnarray} + +\section{Multi-line Equations} + +\LaTeX{} has a built-in equation array feature, however the +equation numbers must be on the same line as an equation. For example: +\begin{eqnarray} +\Delta u + \lambda e^u &= 0&u\in \Omega, \nonumber \\ +u&=0&u\in\partial\Omega. +\end{eqnarray} + +Alternatively, the number can be centered in the equation using the +following method. +% +% The equation-array feature in LaTeX is a bad idea. For centered +% numbers you should set your own equations and arrays as follows: +% +\def\dd{\displaystyle} +\begin{equation}\label{gelfand} +\begin{array}{rl} +\dd \Delta u + \lambda e^u = 0, & +\dd u\in \Omega,\\[8pt] % add 8pt extra vertical space. 1 line=23pt +\dd u=0, & \dd u\in\partial\Omega. +\end{array} +\end{equation} +The previous equation had a label. It may be referenced as +equation~(\ref{gelfand}). + + +\section{More Complicated Equations} +\section*{Rellich's identity}\label{rellich.section} +\setcounter{theorem}{0} +% +% + +Standard developments of Pohozaev's identity used an identity by +Rellich~\cite{rellich:der40}, reproduced here. + +\begin{lemma}[Rellich] +Given $L$ in divergence form and $a,d$ defined above, $u\in C^2 +(\Omega )$, we have +\begin{equation}\label{rellich} +\int_{\Omega}(-Lu)\nabla u\cdot (x-\overline{x})\, dx += (1-\frac{n}{2}) \int_{\Omega} a(\nabla u,\nabla u) \, dx +- +\frac{1}{2} \int_{\Omega} +d(\nabla u, \nabla u) \, dx +\end{equation} +$$ ++ +\frac{1}{2} \int_{\partial\Omega} a(\nabla u,\nabla u)(x-\overline{x}) +\cdot \nu \, dS +- +\int_{\partial\Omega} +a(\nabla u,\nu )\nabla u\cdot (x-\overline{x}) \, dS. +$$ +\end{lemma} +{\bf Proof:}\\ +There is no loss in generality to take $\overline{x} = 0$. First +rewrite $L$: +$$Lu = \frac{1}{2}\left[ \sum_{i}\sum_{j} +\frac{\partial}{\partial x_i} +\left( a_{ij} \frac{\partial u}{\partial x_j} \right) + +\sum_{i}\sum_{j} +\frac{\partial}{\partial x_i} +\left( a_{ij} \frac{\partial u}{\partial x_j} \right) +\right]$$ +Switching the order of summation on the second term and relabeling +subscripts, $j \rightarrow i$ and $i \rightarrow j$, then using the fact +that $a_{ij}(x)$ is a symmetric matrix, +gives the symmetric form needed to derive Rellich's identity. +\begin{equation} +Lu = \frac{1}{2} \sum_{i,j}\left[ +\frac{\partial}{\partial x_i} +\left( a_{ij} \frac{\partial u}{\partial x_j} \right) + +\frac{\partial}{\partial x_j} +\left( a_{ij} \frac{\partial u}{\partial x_i} \right) +\right]. +\end{equation} + +Multiplying $-Lu$ by $\frac{\partial u}{\partial x_k} x_k$ and integrating +over $\Omega$, yields +$$\int_{\Omega}(-Lu)\frac{\partial u}{\partial x_k} x_k \, dx= +-\frac{1}{2} \int_{\Omega} +\sum_{i,j}\left[ +\frac{\partial}{\partial x_i} +\left( a_{ij} \frac{\partial u}{\partial x_j} \right) + +\frac{\partial}{\partial x_j} +\left( a_{ij} \frac{\partial u}{\partial x_i} \right) +\right] +\frac{\partial u}{\partial x_k} x_k \, dx$$ +Integrating by parts (for integral theorems see~\cite[p. 20] +{zeidler:nfa88IIa}) +gives +$$= \frac{1}{2} \int_{\Omega} +\sum_{i,j} a_{ij} \left[ +\frac{\partial u}{\partial x_j} +\frac{\partial^2 u}{\partial x_k\partial x_i} + +\frac{\partial u}{\partial x_i} +\frac{\partial^2 u}{\partial x_k\partial x_j} +\right] x_k \, dx +$$ +$$ ++ +\frac{1}{2} \int_{\Omega} +\sum_{i,j} a_{ij} \left[ +\frac{\partial u}{\partial x_j} \delta_{ik} + +\frac{\partial u}{\partial x_i} \delta_{jk} +\right] \frac{\partial u}{\partial x_k} \, dx +$$ +$$- \frac{1}{2} \int_{\partial\Omega} +\sum_{i,j} a_{ij} \left[ +\frac{\partial u}{\partial x_j} \nu_{i} + +\frac{\partial u}{\partial x_i} \nu_{j} +\right] \frac{\partial u}{\partial x_k} x_k \, dx +$$ += $I_1 + I_2 + I_3$, where the unit normal vector is $\nu$. +One may rewrite $I_1$ as +$$I_1 = \frac{1}{2} \int_{\Omega} +\sum_{i,j} a_{ij} \frac{\partial}{\partial x_k}\left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) x_k \, dx +$$ +Integrating the first term by parts again yields +$$I_1 = -\frac{1}{2} \int_{\Omega} +\sum_{i,j} a_{ij} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) \, dx ++ +\frac{1}{2} \int_{\partial\Omega} +\sum_{i,j} a_{ij} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) x_k \nu_k \, dS +$$ +$$ +- +\frac{1}{2} \int_{\Omega} +\sum_{i,j} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) x_k \frac{\partial a_{ij}}{\partial x_k}\, dx. +$$ +Summing over $k$ gives +$$\int_{\Omega}(-Lu)(\nabla u\cdot x)\, dx = +-\frac{n}{2} \int_{\Omega} +\sum_{i,j} a_{ij} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) \, dx +$$ +$$ ++ +\frac{1}{2} \int_{\partial\Omega} +\sum_{i,j} a_{ij} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) (x\cdot \nu ) \, dS +- +\frac{1}{2} \int_{\Omega} +\sum_{i,j} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) (x\cdot \nabla a_{ij}) \, dx +$$ +$$ ++ +\frac{1}{2} \int_{\Omega} +\sum_{i,j,k} a_{ij} \left[ +\frac{\partial u}{\partial x_j} +\frac{\partial u}{\partial x_k} \delta_{ik} + +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_k} \delta_{jk} +\right] \, dx +$$ +$$- \frac{1}{2} \int_{\partial\Omega} +\sum_{i,j} a_{ij} \left[ +\frac{\partial u}{\partial x_j} \nu_{i} + +\frac{\partial u}{\partial x_i} \nu_{j} +\right] (\nabla u\cdot x) \, dS. +$$ +Combining the first and fourth term on the right-hand side +simplifies the expression +$$\int_{\Omega}(-Lu)(\nabla u\cdot x)\, dx += +(1-\frac{n}{2}) \int_{\Omega} +\sum_{i,j} a_{ij} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) \, dx +$$ +$$ ++ +\frac{1}{2} \int_{\partial\Omega} +\sum_{i,j} a_{ij} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) (x\cdot \nu ) \, dS +- +\frac{1}{2} \int_{\Omega} +\sum_{i,j} \left( +\frac{\partial u}{\partial x_i} +\frac{\partial u}{\partial x_j} +\right) (x\cdot \nabla a_{ij}) \, dx +$$ +$$ +- +\frac{1}{2} \int_{\partial\Omega} +\sum_{i,j} a_{ij} \left[ +\frac{\partial u}{\partial x_j} \nu_{i} + +\frac{\partial u}{\partial x_i} \nu_{j} +\right] (\nabla u\cdot x) \, dS. +$$ +Using the notation defined above, the result follows. + + -- cgit v1.2.3