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Integral progress and stuff

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master
E. Almqvist 3 years ago
parent ab4ddd5aa4
commit 90a563ef6b
7 changed files with 56 additions and 19 deletions
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  1. 2
      ma5/uppg/main.aux
  2. 10
      ma5/uppg/main.fdb_latexmk
  3. 2
      ma5/uppg/main.fls
  4. 22
      ma5/uppg/main.log
  5. BIN
      ma5/uppg/main.pdf
  6. BIN
      ma5/uppg/main.synctex.gz
  7. 39
      ma5/uppg/main.tex

2
ma5/uppg/main.aux
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@ -24,4 +24,4 @@
\@writefile{toc}{\contentsline {section}{\numberline {2}Uppgiftlösningar}{3}{section.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}1}{3}{subsection.2.1}\protected@file@percent }
\newlabel{psi_gen}{{4}{3}{1}{equation.2.4}{}}
\gdef \@abspage@last{4}
\gdef \@abspage@last{5}

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@ -1,5 +1,5 @@
# Fdb version 3
["pdflatex"] 1648207301 "main.tex" "main.pdf" "main" 1648207302
["pdflatex"] 1648211808 "main.tex" "main.pdf" "main" 1648211809
"/usr/share/texmf-dist/fonts/map/fontname/texfonts.map" 1640600964 3524 cb3e574dea2d1052e39280babc910dc8 ""
"/usr/share/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1640600964 1004 54797486969f23fa377b128694d548df ""
"/usr/share/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1640600964 916 f87d7c45f9c908e672703b83b72241a3 ""
@ -9,11 +9,13 @@
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"/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1640600964 30251 6afa5cb1d0204815a708a080681d4674 ""
"/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1640600964 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
"/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1640600964 37912 77d683123f92148345f3fc36a38d9ab1 ""
@ -79,9 +81,9 @@
"/usr/share/texmf-dist/web2c/texmf.cnf" 1640600964 39911 2da6c67557ec033436fe5418a70a8a61 ""
"/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1647853768 4243317 6cd4c1d3b72728bfa7db8ef2e14c0430 ""
"/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1647853732 2920113 d8a499937a1239664546ceed552c7509 ""
"main.aux" 1648207302 1487 a4ec684567e5177a4979c66f29306855 "pdflatex"
"main.out" 1648207302 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex"
"main.tex" 1648207301 8477 1a6c71bcad5833f57176814b77d70aaf ""
"main.aux" 1648211809 1487 653c38bcf6a6df8e18201b53f41b3454 "pdflatex"
"main.out" 1648211809 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex"
"main.tex" 1648211807 9607 ed2c06c8caf7132ab4ead7927c510590 ""
(generated)
"main.aux"
"main.log"

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ma5/uppg/main.fls
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@ -521,10 +521,12 @@ INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm
INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmr12.tfm
INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm
INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm
INPUT main.aux
INPUT ./main.out
INPUT ./main.out
INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb
INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbxti10.pfb
INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb
INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
INPUT /usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb

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@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.21) 25 MAR 2022 12:21
This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.21) 25 MAR 2022 13:36
entering extended mode
restricted \write18 enabled.
file:line:error style messages enabled.
@ -378,23 +378,23 @@ LaTeX Font Info: Trying to load font information for U+msb on input line 68.
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
) [1
{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] (./main.aux)
{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] [5] (./main.aux)
Package rerunfilecheck Info: File `main.out' has not changed.
(rerunfilecheck) Checksum: 7F4A6E130B4A0841F3A3FE085E8F354A;561.
)
Here is how much of TeX's memory you used:
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407459 words of font info for 43 fonts, out of 8000000 for 9000
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407818 words of font info for 44 fonts, out of 8000000 for 9000
1141 hyphenation exceptions out of 8191
60i,7n,63p,809b,330s stack positions out of 5000i,500n,10000p,200000b,80000s
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Output written on main.pdf (4 pages, 172885 bytes).
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PDF statistics:
137 PDF objects out of 1000 (max. 8388607)
98 compressed objects within 1 object stream
17 named destinations out of 1000 (max. 500000)
147 PDF objects out of 1000 (max. 8388607)
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18 named destinations out of 1000 (max. 500000)
33 words of extra memory for PDF output out of 10000 (max. 10000000)

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39
ma5/uppg/main.tex
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@ -210,11 +210,44 @@ $$
\implies \int_0^L \shrodprob dx\ = D^2 \int_0^L \sin^2\left( \frac{\eta\pi}{L} x \right) dx\ = 1.0
$$
Vi behöver nu beräkna integralen och få fram dess uttryck. Vi använder oss därmed av \emph{u-substitution} och \emph{trigonometriska ettan}:
$$ %TODO: remove me
2 \sin^2 x = 1 - \cos x
Vi behöver nu beräkna integralen och få fram dess uttryck. Vi använder oss därmed av \emph{u-substitution} och \emph{trigonometriska ettan}. \emph{\textbf{Låt $u = \frac{\eta\pi}{L}x$}}:
$$
\int_0^L \sin^2\left( \frac{\eta\pi}{L} x \right) dx\ = \int_0^L \sin^2(u) dx\
$$
$$
\implies \frac{du}{dx} = \frac{\eta\pi}{L} \implies dx\ = \frac{L}{\eta\pi} du\
$$
$$
\implies \int_0^L \sin^2(u) dx\ = \frac{L}{\eta\pi} \int_0^L sin^2(u) du\
$$
Väljer att \emph{\textbf{skriva om och förenkla $sin^2\left(u\right)$}} och enlight den \emph{trigonometriska ettan} får vi:
$$
\cos(2u) = 1 - 2\sin^2 u, \quad \text{(dubbla vinkeln för cosinus)}, \quad +2\sin^2 u
$$
$$
\cos 2u + 2\sin^2 u = 1, \quad -\cos 2u
$$
$$
2\sin^2 u = 1 - \cos 2u, \quad /2
$$
$$
\implies \sin^2 u = \frac{1}{2} - \frac{1}{2}\cos 2u
$$
Vi kan nu stoppa in vår förenklade version av $sin^2(u)$ med något vi faktiskt kan integrera:
$$
\implies \int_0^L \shrodprob dx\
= D^2 \int_0^L \sin^2\left( \frac{\eta\pi}{L} x \right) dx\
= D^2 \frac{L}{\eta\pi} \int_0^L \left\{ \frac{1}{2} - \frac{1}{2}\cos(2u) \right\} du\
$$
$$
= D^2 \frac{L}{2\eta\pi} \int_0^L \left\{ 1 - \cos(2u) \right\} du\
= D^2 \frac{L}{2\eta\pi} \left( \int_0^L 1 du\ - \int_0^L \cos(2u) du\ \right)
$$
$$
= D^2 \frac{L}{2\eta\pi} \left[u - \int \cos(2u) du\ \right]_0^L
$$

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