diff --git a/ma5/uppg/main.fdb_latexmk b/ma5/uppg/main.fdb_latexmk index 64009da..160ddc7 100644 --- a/ma5/uppg/main.fdb_latexmk +++ b/ma5/uppg/main.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1648211808 "main.tex" "main.pdf" "main" 1648211809 +["pdflatex"] 1648213365 "main.tex" "main.pdf" "main" 1648213366 "/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 "" @@ -13,6 +13,7 @@ "/usr/share/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1640600964 1288 655e228510b4c2a1abe905c368440826 "" "/usr/share/texmf-dist/fonts/tfm/public/cm/cmr17.tfm" 1640600964 1292 296a67155bdbfc32aa9c636f21e91433 "" "/usr/share/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1640600964 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" + "/usr/share/texmf-dist/fonts/tfm/public/cm/cmti7.tfm" 1640600964 1492 86331993fe614793f5e7e755835c31c5 "" "/usr/share/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm" 1640600964 768 1321e9409b4137d6fb428ac9dc956269 "" "/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1640600964 32080 340ef9bf63678554ee606688e7b5339d "" "/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbxti10.pfb" 1640600964 36554 b67dc2cfa451409e100b3fcf5f506509 "" @@ -81,9 +82,9 @@ "/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" 1648211809 1487 653c38bcf6a6df8e18201b53f41b3454 "pdflatex" - "main.out" 1648211809 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex" - "main.tex" 1648211807 9607 ed2c06c8caf7132ab4ead7927c510590 "" + "main.aux" 1648213365 1487 653c38bcf6a6df8e18201b53f41b3454 "pdflatex" + "main.out" 1648213365 561 7f4a6e130b4a0841f3a3fe085e8f354a "pdflatex" + "main.tex" 1648213364 10029 189ca3e9f10596bb1446685a27f92be0 "" (generated) "main.aux" "main.log" diff --git a/ma5/uppg/main.fls b/ma5/uppg/main.fls index 072a7ab..2c917b9 100644 --- a/ma5/uppg/main.fls +++ b/ma5/uppg/main.fls @@ -522,6 +522,10 @@ 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 /usr/share/texmf-dist/fonts/tfm/public/cm/cmti7.tfm +INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm +INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmti7.tfm +INPUT /usr/share/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm INPUT main.aux INPUT ./main.out INPUT ./main.out diff --git a/ma5/uppg/main.log b/ma5/uppg/main.log index b7257fb..0aaa28c 100644 --- a/ma5/uppg/main.log +++ b/ma5/uppg/main.log @@ -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 13:36 +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 14:02 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -383,15 +383,15 @@ Package rerunfilecheck Info: File `main.out' has not changed. (rerunfilecheck) Checksum: 7F4A6E130B4A0841F3A3FE085E8F354A;561. ) Here is how much of TeX's memory you used: - 9098 strings out of 478353 - 140920 string characters out of 5854635 + 9103 strings out of 478353 + 140983 string characters out of 5854635 463010 words of memory out of 5000000 - 27118 multiletter control sequences out of 15000+600000 - 407818 words of font info for 44 fonts, out of 8000000 for 9000 + 27122 multiletter control sequences out of 15000+600000 + 409234 words of font info for 48 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 -Output written on main.pdf (5 pages, 188516 bytes). +Output written on main.pdf (5 pages, 189172 bytes). PDF statistics: 147 PDF objects out of 1000 (max. 8388607) 104 compressed objects within 2 object streams diff --git a/ma5/uppg/main.pdf b/ma5/uppg/main.pdf index 854f57c..235868b 100644 Binary files a/ma5/uppg/main.pdf and b/ma5/uppg/main.pdf differ diff --git a/ma5/uppg/main.synctex.gz b/ma5/uppg/main.synctex.gz index e8d7460..4efd9c9 100644 Binary files a/ma5/uppg/main.synctex.gz and b/ma5/uppg/main.synctex.gz differ diff --git a/ma5/uppg/main.tex b/ma5/uppg/main.tex index 4f2ad83..f16bce3 100644 --- a/ma5/uppg/main.tex +++ b/ma5/uppg/main.tex @@ -249,6 +249,24 @@ $$ = D^2 \frac{L}{2\eta\pi} \left[u - \int \cos(2u) du\ \right]_0^L $$ +$$ +\emph{\textbf{Låt $\mu = 2u$}} +$$ +$$ +\implies \int \cos(2u) du\ = \int \cos(\mu) du\ +$$ +$$ +\implies \frac{d\mu}{du} = 2 \implies du\ = \frac{1}{2} d\mu\ +$$ +$$ +\int \cos(2u) du\ = \frac{1}{2} \int \cos(\mu) d\mu\ = -\frac{1}{2} \sin(\mu) + C +$$ + +$$ +\implies \int_0^L \shrodprob dx\ += D^2 \frac{L}{2\eta\pi} \left[u - \int \cos(2u) du\ \right]_0^L += D^2 \frac{L}{2\eta\pi} \left[ u + \frac{1}{2}\sin \mu \right]_0^L +$$ \end{document}