100 lines
12 KiB
TeX
100 lines
12 KiB
TeX
\documentclass[twoside, 11pt, a4paper]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath,amsfonts,amssymb,graphicx,parskip}
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\usepackage{mathpazo}
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\usepackage{color}
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\usepackage[margin=1in]{geometry}
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%\usepackage{fourierx} % eller lmodern
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% for debug utskrift
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\newcommand{\debug}[1]{\texttt{#1}}
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% for ingen utskrift av kommentarer/debug
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%\newcommand{\debug}[1]{}
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\DeclareMathOperator{\erf}{erf}
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\DeclareMathOperator{\erfc}{erfc}
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\DeclareMathOperator{\eps}{\epsilon}
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\newcommand{\dee}{\mathrm{d}}
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\begin{document}
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Jacobian:
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\[
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\begin{split}
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\frac{\partial u}{\partial x} &= \frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x} \\
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\frac{\partial u}{\partial y} &= \frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial y}
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\end{split}
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\]
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Hessian:
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\[
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\begin{split}
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\frac{\partial^2 u}{\partial x^2} &= \frac{\partial}{\partial\xi}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\xi}{\partial x} \\
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&+ \frac{\partial}{\partial\eta}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\eta}{\partial x} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x}+\frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial x\partial\xi} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x}+\frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial x\partial\xi}\right)\frac{\partial\xi}{\partial x} \\
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&+\left(\frac{\partial^2 u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x} + \frac{\partial u}{\partial \xi}\frac{\partial^2\xi}{\partial\xi\partial\eta} + \frac{\partial^2 u}{\partial\eta^2}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial x\partial\eta}\right)\frac{\partial\eta}{\partial x} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x}+\frac{\partial u}{\partial\xi}\frac{\partial\xi}{\partial x} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x}\right)\frac{\partial\xi}{\partial x} \\
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&+\left(\frac{\partial^2 u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x} + \frac{\partial^2 u}{\partial\eta^2}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial\eta}{\partial x}\right)\frac{\partial\eta}{\partial x} \\
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&= \frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}
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\end{split}
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\]
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\[
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\begin{split}
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\frac{\partial^2 u}{\partial y^2} &= \frac{\partial}{\partial\xi}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial y}\right)\frac{\partial\xi}{\partial y} \\
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&+ \frac{\partial}{\partial\eta}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial y}\right)\frac{\partial\eta}{\partial y} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial y} + \frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial\xi\partial y} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial y}+\frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\xi\partial y}\right)\frac{\partial\xi}{\partial y} \\
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&+\left(\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y} + \frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial \eta\partial y} + \frac{\partial^2 u}{\partial\eta^2}\frac{\partial\eta}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\eta\partial y}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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\newpage
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\[
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\begin{split}
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\frac{\partial^2 u}{\partial x\partial y} &= \frac{\partial}{\partial\xi}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\xi}{\partial y} \\
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&+ \frac{\partial}{\partial\eta}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\eta}{\partial y} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x} + \frac{\partial u}{\partial \xi}\frac{\partial^2\xi}{\partial\xi\partial x} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\xi\partial x}\right)\frac{\partial\xi}{\partial y} \\
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&+ \left(\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}+\frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial\eta\partial x} + \frac{\partial^2u}{\partial\eta^2}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\eta\partial x}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x}\frac{\partial\xi}{\partial y} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x}\frac{\partial\xi}{\partial y}
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+ \frac{\partial^2u}{\partial\eta^2}\frac{\partial\xi}{\partial x}\frac{\partial \eta}{\partial y} + \frac{\partial^2 u}{\partial \xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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Jerk matrix:
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\[
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\begin{split}
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\frac{\partial^3u}{\partial x^3} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\xi}{\partial x} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\eta}{\partial x} \\
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&= \frac{\partial^3u}{\partial\xi^3}\left(\frac{\partial\xi}{\partial x}\right)^3 + \frac{\partial^3u}{\partial \xi\partial\eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2\frac{\partial\xi}{\partial x} + 2\frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\eta}{\partial x} \\
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&+\frac{\partial^3u}{\partial\eta^3}\left(\frac{\partial\eta}{\partial x}\right)^3 + \frac{\partial^3u}{\partial \xi^2\partial\eta}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\eta}{\partial x} + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial x}\left(\frac{\partial\eta}{\partial x}\right)^2
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\end{split}
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\]
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\[
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\begin{split}
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\frac{\partial^3u}{\partial y^3} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\xi}{\partial y} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^3u}{\partial\xi^3}\left(\frac{\partial\xi}{\partial y}\right)^3 + \frac{\partial u^3}{\partial\xi\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2\frac{\partial\xi}{\partial y} + 2 \frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial y}\right)^2\frac{\partial\eta}{\partial y} \\
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&+ \frac{\partial^3 u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial y}\right)^2\frac{\partial\eta}{\partial y} + \frac{\partial^3 u}{\partial\eta^3}\left(\frac{\partial \xi}{\partial y}\right)^3 + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial y}\left(\frac{\partial\eta}{\partial y}\right)^2
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\end{split}
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\]
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\newpage
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\[
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\begin{split}
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\frac{\partial^3u}{\partial x^2\partial y} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\xi}{\partial y} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^3u}{\partial\xi^3}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\xi}{\partial y} + \frac{\partial^3u}{\partial\xi\partial\eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2\frac{\partial\xi}{\partial y} + 2\frac{\partial^3u}{\partial\xi^2\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial x} \\
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&+ \frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\eta}{\partial y}+\frac{\partial^3u}{\partial\eta^3}\left(\frac{\partial\eta}{\partial x}\right)^2\frac{\partial\eta}{\partial y} + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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\[
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\begin{split}
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\frac{\partial^3u}{\partial x\partial y^2} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\xi}{\partial x} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\eta}{\partial x} \\
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&= \frac{\partial^3u}{\partial\xi^3}\frac{\partial\xi}{\partial x}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial \xi}{\partial x}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^3u}{\partial\xi^2\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y} \\
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&+ \frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial y}\right)^2\frac{\partial\eta}{\partial x} + \frac{\partial^3u}{\partial\eta^3}\frac{\partial\eta}{\partial x}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial x}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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\end{document}
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