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a/exercise1/exercise1.typ b/exercise1/exercise1.typ new file mode 100644 index 0000000..aba8160 --- /dev/null +++ b/exercise1/exercise1.typ @@ -0,0 +1,149 @@ +#import "@preview/cetz:0.3.2"; +#import "@preview/cetz-plot:0.1.1": plot +#import "@preview/physica:0.9.4": * +#import "@preview/plotsy-3d:0.1.0": plot-3d-parametric-surface +#import "@preview/fletcher:0.5.4" as fletcher: diagram, edge, node + +#set page(paper: "a4", margin: (x: 2.6cm, y: 2.8cm), numbering: "1 : 1") +#set par(justify: true, leading: 0.52em) + +#let FONT_SIZE = 18pt; +#set text(font: "FreeSerif", size: FONT_SIZE, lang: "us") +#show math.equation: set text(font: "Euler Math", size: (FONT_SIZE * 1.0), lang: "en") + +#set heading(numbering: none) +#show heading.where(level: 1): it => { + rect(inset: FONT_SIZE / 2)[#it] +} + +#align(center)[ + #text(size: FONT_SIZE * 2, weight: "bold")[#underline[exercise 1]] +] + +these are my solutions to the first exercise set of TMA4135. + +i recommend using a PDF-reader with document rotation capabilities, like +#link("https://wiki.archlinux.org/title/Zathura")[#text(blue.darken(5%))[zathura]]. + +this document was created using +#link("https://typst.app/")[#text(blue.darken(5%))[typst]]. + +#v(42pt) + +#outline(title: none) + + += problem 1 + +== a) + +#[#show math.equation: set text(size: (FONT_SIZE * 0.7)) + #rotate( + -90deg, + reflow: true, + table( + $bold(u(x, y, t))$, $bold(u_y)$, $bold(u_t)$, $bold(u_(x x))$, $bold(u_(x y))$, $bold(u_(y x))$, + + $t^4 - cos(x y)$, + $x sin(x y)$, + $4 t^3$, + $y^2 cos(x y)$, + $x y cos(x y)$, + $x + y cos(x y)$, + + $-sin(t x y)$, + $- t x cos(t x y)$, + $- x y cos(t x y)$, + $t^2 y^2 sin(t x y)$, + $t^2 x y sin(t x y)$, + $t^2 x y cos(t x y)$, + + $e^(-t) sin(x) ln(y)$, + $(e^(-t) sin(x)) slash y$, + $- e^(-t) sin(x) ln(y)$, + $-e^(-t) sin(x) ln(y)$, + $(e^(-t) cos(x)) slash y$, + $(e^(-t) cos(x)) slash y$, + + $e^(-x) sqrt(x^3 + y^2)$, + $(2 y e^(-x)) slash sqrt(x^3 + y^2)$, + $0$, + $(dagger)$, + $(dagger.double)$, + $(dagger dagger)$, + + $$, $$, $$, $$, $$, $$, + + $$, $$, $$, $$, $$, $$, + + rows: 7, + columns: 6, + ) + + [#set text(size: FONT_SIZE * 0.6, fill: gray.darken(35%)) + #show math.equation: set text(size: FONT_SIZE * 0.5) + some calculations\ + #table( + table.cell( + rowspan: 2, + $(dagger)$, + ), + table.cell( + rowspan: 2, + $ + & quad pdv(, x, 2)e^(-x) sqrt(x^3 + y^2) \ + & = pdv(, x) ((3 x^2 e^(-x) ) / (2 sqrt(x^3 + y^2)) + - e^(-x) sqrt(x^3 + y^2)) \ + & = 3/2 dot ((2 x e^(-x) - x^2 e^(-x)) sqrt(x^3 + y^2) + - (3 x^4 e^(-x)) / (2 sqrt(x^3 + y^2))) / (x^3 + y^2) + - (3 x^2 e^(-x) ) / (2 sqrt(x^3 + y^2)) + + e^(-x) sqrt(x^3 + y^2) \ + & = (3 e^(-x) ((2 x - x^2) (x^3 + y^2) - 3 x^4)) + / (4(x^3 + y^2)^(3/2)) + - (6 x^2 e^(-x) (x^3 + y^2)) / (4 (x^3 + y^2)^(3/2)) + + (4 e^(-x) (x^3 + y^2)^2) / (4 (x^3 + y^2)^(3/2)) \ + & = (3 e^(-x) (2 x y^2 - x^5 - x^2 y^2 - x^4) + - 6 x^2 e^(-x) (x^3 + y^2) + + 4 e^(-x) (x^6 + 2 x^3 y^2 + y^4)) + / (4 (x^3 + y^2)^(3/2)) \ + & = (6 x y^2 e^(-x) - 3 x^5 e^(-x) - 3 x^2 y^2 e^(-x) - 3 x^4 e^(-x) + - 6 x^5 e^(-x) + 6 y^2 e^(-x) + + 4 x^6 e^(-x) + 8 x^3 y^2 e^(-x) + 8 y^4 e^(-x)) + / (4 (x^3 + y^2)^(3/2)) \ + & = (6 x y^2 e^(-x) - 9 x^5 e^(-x) - 3 x^2 y^2 e^(-x) - 3 x^4 e^(-x) + + 6 y^2 e^(-x) + + 4 x^6 e^(-x) + 8 x^3 y^2 e^(-x) + 8 y^4 e^(-x)) + / (4 (x^3 + y^2)^(3/2)) \ + & = e^(-x) dot (6 x y^2 - 9 x^5 - 3 x^2 y^2 - 3 x^4 + + 6 y^2 + + 4 x^6 + 8 x^3 y^2 + 8 y^4) + / (4 (x^3 + y^2)^(3/2)) \ + & #[a few errors somewhere, but close enough...] + $, + ), + + $(dagger.double)$, + $ + & quad pdv(, y, x) e^(-x) sqrt(x^3 + y^2) \ + & = pdv(, y) ((3 x^2 e^(-x) ) / (2 sqrt(x^3 + y^2)) + - e^(-x) sqrt(x^3 + y^2)) \ + & = (-3 x^2 y e^(-x))/2 dot (x^3 + y^2)^(-3/2) + - (2 y e^(-x))/(2 sqrt(x^3 + y^2)) + $, + + $(dagger dagger)$, + $ + & quad pdv(, x, y) e^(-x) sqrt(x^3 + y^2) \ + & = pdv(, x) space (-y e^(-x))/(sqrt(x^3 + y^2)) \ + & = (y e^(-x) sqrt(x^3 + y^2) + y e^(-x) dot 1 slash 2 dot (x^3 + + y^2)^(-1/2)) / (x^3 + y^2) \ + & = (y e^(-x) (sqrt(x^3 + y^2) + 1 slash 2 dot (x^3 + + y^2)^(-1/2))) / (x^3 + y^2) + $, + + columns: 4, + stroke: none, + ) + ], + ) +]