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+#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,
+        )
+      ],
+  )
+]