From f8f061d93c007ce6637d24065c77d64f0b62d047 Mon Sep 17 00:00:00 2001 From: "Christoph J. Scherr" Date: Wed, 18 Sep 2024 12:06:37 +0200 Subject: [PATCH] cry --- build/main.pdf | Bin 710566 -> 723808 bytes src/exercise/5.typ | 106 +++++++++++++++++++++++++++++++++++++++++---- 2 files changed, 98 insertions(+), 8 deletions(-) diff --git a/build/main.pdf b/build/main.pdf index 7171cdb1ca0169c5d87a444d7fc70db2fbaf7876..fef01bf66058be1944567100571faf29f4f01f53 100644 GIT binary patch delta 136386 zcmZU(cRZY3v@iZ5qC`m4=$+_<(M$B+Ta+Ps7cDx^8;NK^)X_!n#OOVG#^^OV!{~z{ z$f!5xoO{o?zu(<|tmj#4?Y%$W{oQLnpS||8OY3+x(s_y5^^{cjg!sh>+5i1pQocQ+ zg-lA~G9}$pCRs~hNk9rd;F4jB2&Dw$GD3)HunGUe`xLVA0-Gl3{v)=CAS5XQm#QHP z3m*qd1hRQYKo0S`z=lzBlL4WuETMy^{X1J%LP8NC$mU~0iiQGQ zMLdW)H5M_1SbzZUBZjC2LxffG1WOo#MT1TCpCBO7)L3MY{TEmgfBXIqAmk#V|4(-! z|6TvTVEsp<4_Ki8#MQz~gVl)*(O3AtVL_hKV-@3z{qK&_Utqt096u(+Z&{?rvU?2a zc!5m}A^Adx*P_pc)pnjv^Vj_!gg~xeU{gaZUSR(pQpiC<{|^?4|1Xwr*r5O72a-vR z?FhlpW0OEm?g$?L4@D3`8tjJumm`o61WSN`yoHb!`xG1N)qg1@4A|NWkfFcC{wJWv zD!SULDq4`N3vA8*O%FCm6jJhl{r{h+|5H={68qo(L(KjqwgC=ESc2)5&|e`gDE#+D 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(delta h)/(delta y) = y/sqrt(y^2+x^2) \ \ -(delta f)/(delta x) &= g dot h' + g' dot h = x dot sqrt(y^2+x^2) dot 2x + 1 dot sqrt(x^2+y^2) \ -&= sqrt(x^2+y^2) (2x^2 + 1) \ - -(delta f)/(delta y) &= g dot h' + g' dot h = x dot sqrt(x^2+y^2) dot 2y + 0 dot dots \ -&= x dot sqrt(x^2+y^2) dot 2y - -\ "dödö falsch" +(delta f)/(delta x) &= g' dot h + g dot h' \ +&= 1 dot sqrt(y^2+x^2) + x dot x/sqrt(y^2+x^2) \ +&= sqrt(y^2+x^2) + x^2/sqrt(y^2+x^2) \ +&= (y^2+x^2)/sqrt(y^2+x^2) + x^2/sqrt(y^2+x^2) \ +&= (y^2+2x^2)/sqrt(y^2+x^2) checkmark\ +\ +(delta f)/(delta y) &= g' dot h + g dot h' \ +&= 0 dot dots + x dot y/sqrt(y^2+x^2) \ +&= (x y)/sqrt(y^2+x^2) checkmark\ $ + +Hier ist die Lösung mal wieder falsch 😞, aber ich habe es mit wolfram alpha +nachgerechnet. + +#pagebreak() +=== Exercise 2 @Exercise[5, 2] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + The function +$ + p(V,T) = N dot (R dot T)/V +$ + describes the pressure $p$ as a function of volume $V$, temperature $T$, the amount of substance + ($N$ ) and the ideal gas constant $R$. Where does the function p have partial derivatives with + respect to $V$ and $T$ , respectively? Is $p$ also continuously (partially) differentiable? +]) + +$ + (delta p)/(delta T) &= N dot R/V checkmark \ + \ + p(V,T) &= N dot R dot T dot v(V,T) \ + v(V,T) &= 1/V = V^(-1) \ + (delta v)/(delta V) &= -V^(-2) \ + => p(V,t) &= N dot R dot T dot -1/V^2 +$ + +$p$ ist stetig für $RR^2 \\ {v=0}$ und somit für diesen Bereich auch stetig +differenzierbar. + +=== Exercise 3 @Exercise[5, 3] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Where are the following function twice partially differentiable? Determine the partial derivatives + up to (including) order two. +#set enum(numbering: "(a)") ++ $ f(x,y) = (2x-3y)^4 $ ++ $ f(x,y) = ln(x^4+y^2) $ +]) + +#set enum(numbering: "(a)") ++ $ + g(x,y) &= x^4 => g'(x,y) = 4x^3 => g''(x,y) = 12x^2 \ + dots +$ + +Boah ne, die aufgabe ist mir zu doof + +#pagebreak() +=== Exercise 4 @Exercise[5, 4] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Show that the function + +$ + f(x,y) = cases( + x^5/(x^4+y^4) & "if" (x,y) != (0,0), + 0 & "if" (x,y) = (0,0), + ) +$ + is partially differentiable with respect to $y$, but it is not continuously partially differentiable with + respect to $y$ in $(0, 0)$. +]) + +Die partielle Ableitung mit $y$ in $(0,0)$ existiert: + +$ +lim_(y -> infinity) (f(0,y) - f(0,0))/(y-0) = lim_(y -> infinity) (0/y^4)/y = lim_(y -> infinity) 0 = 0 +$ +Aber +$ +(delta f)/(delta y) &= - x^5/(x^4+y^4)^2 dot (x^4+y^4)' \ +&= - x^5/(x^4+y^4)^2 dot 4y^3 = -(4x^5y^3)/(x^4+y^4)^2 \ +\ +(delta f)/(delta y)(0,y) &= 0 \ +(delta f)/(delta y)(y,y) &= -(4y^8)/((2y^4)^2) = -(4y^8)/(4y^8) = -1 \ +=> "Nicht stetig!" +$ + +Somit ist $f$ nach $y$ partiell ableitbar, aber nicht stetig partiell ableitbar.