From e55e364adc4f93d8a904cbb0f85ba4f6052a7f6d Mon Sep 17 00:00:00 2001 From: pkirwin Date: Mon, 1 Mar 2021 15:43:09 -0700 Subject: [PATCH] commit --- PS2/doc.fdb_latexmk | 19 +++++++++++-------- PS2/doc.fls | 15 +++++++++++++++ PS2/doc.log | 18 ++++++++++-------- PS2/doc.pdf | Bin 234173 -> 248293 bytes PS2/doc.synctex.gz | Bin 96971 -> 97998 bytes PS2/doc.tex | 25 +++++++++++++++++-------- 6 files changed, 53 insertions(+), 24 deletions(-) diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index c054bc6..82849c5 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,8 +1,8 @@ # Fdb version 3 -["pdflatex"] 1614637883 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614637886 +["pdflatex"] 1614638543 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614638561 "C:/Users/speed/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1614620319 11548 693fc704971bedc6e913b37b081cf40e "" "C:/Users/speed/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1612675286 9293900 83195ab665706b6f706823352908aa95 "" - "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614626913 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" + "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614638559 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993058908 1436 c7f957a372ef2fbe93c0982f96625e12 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1233951848 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -43,7 +43,9 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1247593067 32762 7fee39e011c23b3589931effd97b9702 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr9.pfb" 1247593066 33993 9b89b85fd2d9df0482bd47194d1d3bf3 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1247593066 32716 08e384dc442464e7285e891af9f45947 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1247593067 37944 359e864bd06cde3b1cf57bb20757fb06 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb" 1247593053 11493 4f5ed183a47d3197cf8cd322325db6de "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/iftex.sty" 1583523400 6501 4011d89d9621e0b0901138815ba5ff29 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/ifvtex.sty" 1583523400 1057 525c2192b5febbd8c1f662c9468335bb "" @@ -61,6 +63,7 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/base/article.cls" 1610120746 20145 aad8c3dd3bc36e260347b84002182bc2 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/base/omscmr.fd" 1610120746 2470 59e43b502a8bea3a12f4954945302e40 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/base/size10.clo" 1610120746 8449 a72d5d4e612221b46000c3d71724e0ef "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/cancel/cancel.sty" 1388370247 7592 dd751af313a16a0308545d5bfd7aaaa2 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/caption.sty" 1603736996 51746 e89c4da670ba533e6ab38e045ce6d1d9 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/caption3.sty" 1603736996 67929 d50ae850e1d9604065e8337a75a41029 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/subcaption.sty" 1603736996 6133 46bea384e36e9b74bd53a238fc2d0654 "" @@ -93,15 +96,15 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/setspace/setspace.sty" 1324294077 22913 a27d7908fc6f0385466454a966a316eb "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/titlesec/titlesec.sty" 1571240058 48596 2b6a95da931c07a430b1a61904aaa42d "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614637886 334 906353ae3fc5daad100ce8b921730ac7 "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614637882 10279 f262bef4de2ea276f6d0acd4c1d77d63 "" - "doc.aux" 1614637886 334 906353ae3fc5daad100ce8b921730ac7 "pdflatex" - "doc.tex" 1614637882 10279 f262bef4de2ea276f6d0acd4c1d77d63 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" + "doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "pdflatex" + "doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" (generated) - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" - "doc.log" "doc.pdf" + "doc.log" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc.aux" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" diff --git a/PS2/doc.fls b/PS2/doc.fls index fe1cd4c..b8cecb6 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -416,6 +416,17 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.s INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty @@ -534,5 +545,9 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\c INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr9.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb diff --git a/PS2/doc.log b/PS2/doc.log index d1e4495..2dd77cd 100644 --- a/PS2/doc.log +++ b/PS2/doc.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 15:31 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 15:42 entering extended mode **d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex (d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex @@ -230,6 +230,8 @@ Package: enumitem 2019/06/20 v3.9 Customized lists \enit@inbox=\box55 \enit@count@id=\count281 \enitdp@description=\count282 +) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/cancel\cancel.sty +Package: cancel 2013/04/12 v2.2 Cancel math terms ) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/listings\lstlang1.sty File: lstlang1.sty 2020/03/24 1.8d listings language file ) (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) @@ -332,17 +334,17 @@ Package pdftex.def Info: q1a_eigenvals.png used on input line 163. 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a/PS2/doc.tex b/PS2/doc.tex index 53c08b8..db73568 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -14,8 +14,8 @@ \usepackage{fancyhdr} \usepackage{titlesec} \usepackage{lipsum} - \usepackage{enumitem} +\usepackage{cancel} \titleformat{\subsection}[runin]{\normalfont \large \bfseries} {\thesubsection}{1em}{} @@ -171,29 +171,30 @@ legend({'Numerical','Analytical'},'Location','northwest'); The analytical solution is \begin{equation} - \Phi(x) = A sin \left( \frac{n \pi}{L} x \right) + \phi(x) = A sin \left( \frac{n \pi}{L} x \right) \label{eq:analytical} \end{equation} In order to normalise this equation it must conform to the following: \begin{equation} - \int_{0}^{L} \left| \Phi(x) \right| ^2 dx = 1 + \int_{0}^{L} \left| \phi(x) \right| ^2 dx = 1. \label{eq:normalise} \end{equation} - Using the following identity: + We use the following identity: \begin{equation} - foo + \int\sin^2{(\alpha x)}\:dx = \frac{1}{2} x -\frac{1}{4\alpha}\sin{(2\alpha x)}. \label{eq:integral_ident} \end{equation} - Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate to the above identity. + Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate (\ref{eq:analytical}) + to the above identity. Evaluating the integral gives us the following relationship \begin{equation*} - \frac{1}{A^2} = \frac{1}{2} L - \frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right) - \frac{1}{2} \cdot 0 + \frac{L}{4n \pi} sin(0) + \frac{1}{A^2} = \frac{1}{2} L - \cancel{\frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right)} - \cancel{\frac{1}{2} \cdot 0} + \cancel{\frac{L}{4n \pi} sin(0)} \end{equation*} From this, we find: @@ -224,7 +225,15 @@ legend({'Numerical','Analytical'},'Location','northwest'); \item %1c \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %1ci - \lipsum[1] + + From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that + $\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities + $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and + $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where + $\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \ell}{L}$. + + Fro + \item %1cii \lipsum[1] \item %1ciii