{
"cells": [
{
"cell_type": "markdown",
"id": "1ac0b644",
"metadata": {},
"source": [
"# Week 8: Exercises"
]
},
{
"cell_type": "markdown",
"id": "50592686",
"metadata": {},
"source": [
"## Exercises -- Long Day"
]
},
{
"cell_type": "markdown",
"id": "0a56f147",
"metadata": {},
"source": [
"**Note:**\n",
"The expression $|\\det{\\pmb{J}_{\\pmb{r}}(\\pmb{u})}|$ is in mathematical literature often referred to as \"the Jacobian\". This phrase is also often used for the Jacobian matrix, though. To avoid confusion, we will at DTU refer to $|\\det{\\pmb{J}_{\\pmb{r}}(\\pmb{u})}|$ as \"the Jacobian function\", although this is not a phrase you will see in literature outside of DTU. Alternatively, we might simply refer to it in full:\n",
"\"the absolute value of the determinant of the Jacobian matrix\"."
]
},
{
"cell_type": "markdown",
"id": "f8032e13",
"metadata": {},
"source": [
"### 1: Plane Integrals over Rectangles. By Hand"
]
},
{
"cell_type": "markdown",
"id": "bfa4dd3d",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "739aeb4c",
"metadata": {},
"source": [
"Consider the region $B=\\left\\lbrace (x,y) \\bigm| 0\\leq x\\leq 2 \\wedge -1\\leq y\\leq 0\\right\\rbrace$ in $\\mathbb{R}^2$. Calculate the plane integral \n",
"\n",
"\\begin{equation*}\n",
" \\int_B (x^2y^2+x) \\mathrm{d}\\pmb{x}\n",
"\\end{equation*}\n",
"\n",
"using the formula for double integrals over (axis-parallel) rectangles."
]
},
{
"cell_type": "markdown",
"id": "ec6bb8a5",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "b47ad400",
"metadata": {},
"source": [
"Let us calculate the plane integral from above one more time, but now in a manner that at first glance may appear more complicated. Use the change-of-variables theorem for integrals over $\\mathbb{R}^2$."
]
},
{
"cell_type": "markdown",
"id": "80e1d148",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "a9526cd3",
"metadata": {},
"source": [
"Calculate the plane integral \n",
"\n",
"\\begin{equation*}\n",
" \\int_B \\frac{y}{1+xy} \\;\\mathrm{d}\\pmb{x}, \\quad\\text{hvor}\\quad B=\\left\\lbrace (x,y) \\mid 0\\leq x\\leq 1 \\, \\wedge \\, 0\\leq y\\leq 1\\right\\rbrace.\n",
"\\end{equation*}"
]
},
{
"cell_type": "markdown",
"id": "86cf0bae",
"metadata": {},
"source": [
"### 2: Polar Coordinates. By Hand"
]
},
{
"cell_type": "markdown",
"id": "dbca83e7",
"metadata": {},
"source": [
"A function $f:\\mathbb{R}^2 \\to \\mathbb{R}$ is given by\n",
"\n",
"\\begin{equation*}\n",
" f(x,y)=x^2-y^2.\n",
"\\end{equation*}\n",
"\n",
"For a given point $\\pmb{x}=(x,y)$ in the plane, let $r = \\Vert \\pmb{x} \\Vert$ denote the distance from the point to the origin $(0,0)$. Also, let $\\theta$ denote the angle between the $x$-axis and the position vector to the point - for the sign of the angle $\\theta$, we define the positive angular orientation as counterclockwise. A set of points $B$ contains all points that fulfill (in polar coordinates),\n",
"\n",
"\\begin{equation*}\n",
" 0\\leq r \\leq a \\, \\text{ and } \\, -\\frac{\\pi}{4} \\leq \\theta \\leq \\frac{\\pi}{2},\n",
"\\end{equation*}\n",
"\n",
"\n",
"\n",
"\n",
"where $a$ is an arbitrary positive real number."
]
},
{
"cell_type": "markdown",
"id": "2d840afe",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "57251697",
"metadata": {},
"source": [
"Make a sketch of $B$, and determine the area of $B$, first using integration and then purely from elementary geometric considerations."
]
},
{
"cell_type": "markdown",
"id": "9a42e62f",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "96813857",
"metadata": {},
"source": [
"Determine the plane integral $\\int_B f(x,y) \\;\\mathrm{d}\\pmb{x}$."
]
},
{
"cell_type": "markdown",
"id": "f6366f74",
"metadata": {},
"source": [
"### 3: The Volume of a Parallelotope"
]
},
{
"cell_type": "markdown",
"id": "5d5b5441",
"metadata": {},
"source": [
"A parallelotope $P$ in $\\mathbb{R}^n$ \"spanned by\" the vectors $\\pmb{a}_1, \\pmb{a}_2, \\dots, \\pmb{a}_n$ is defined by:\n",
"\n",
"\\begin{equation*}\n",
" P = \\left\\{ \\pmb{y} \\in \\mathbb{R}^n \\mid \\, \\pmb{y} = A\\pmb{x}, \\quad \\text{where } x_i \\in [0,1] \\text{ for }$i=1,2,\\dots, n$ \\right\\},\n",
"\\end{equation*}\n",
"\n",
"where $A = [\\pmb{a}_1 | \\pmb{a}_2 | \\cdots | \\pmb{a}_n]$ is the $n \\times n$ matrix whose $i$'th column is $\\pmb{a}_i$. This set of points can in short-hand notation be written as $P=A([0,1]^n)$. \n",
"\n",
"It can be shown with tools *solely* from Mathematics 1a (in particular the characterization of the determinant) that the $n$-dimensional volume of $P$ is:\n",
"\n",
"\\begin{equation*}\n",
" \\mathrm{vol}_n(P) = |\\mathrm{det}(A)|.\n",
"\\end{equation*}\n",
"\n",
"(For the interested student, such a proof can be found here https://textbooks.math.gatech.edu/ila/determinants-volumes.html)\n",
"\n",
"In $\\mathbb{R}^2$, a parallelotope is the well-known pallelogram, and $\\mathrm{vol}_n(P)$ is then the area of $P$, while it in $\\mathbb{R}^3$ becomes a parallelepiped with a volume."
]
},
{
"cell_type": "markdown",
"id": "8735d37d",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "042cda7e",
"metadata": {},
"source": [
"Show that $\\mathrm{vol}_n(P) = |\\mathrm{det}(A)|$ using the change-of-variables theorem for integrals over $\\mathbb{R}^n$."
]
},
{
"cell_type": "markdown",
"id": "3e8d73c1",
"metadata": {},
"source": [
"*In the rest of this exercise we want to investigate the proposition $\\mathrm{vol}_n(P) = |\\mathrm{det}(A)|$ without use of integration techniques.*"
]
},
{
"cell_type": "markdown",
"id": "ded8b241",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "7beb785e",
"metadata": {},
"source": [
"Let $n=2$. Choose two linearly independent vectors $\\pmb{a}_1, \\pmb{a}_2$ in $\\mathbb{R}^2$. It might be smart to choose $\\pmb{a}_1 \\in \\mathrm{span}(\\pmb{e}_1)$. Calculate (using elementary geometric considerations) the area of the parallelogram \"spanned by\" the two vectors. Also calculate $|\\mathrm{det}(A)|$ and compare the results."
]
},
{
"cell_type": "markdown",
"id": "8df9523e",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "2b00e70e",
"metadata": {},
"source": [
"Let $n=2$, and now let $\\pmb{a}_1, \\pmb{a}_2$ be arbitrary but linearly independent vectors in $\\mathbb{R}^2$. Can you prove the formula $\\mathrm{area}(P) = |\\mathrm{det}(A)|$, where $P$ is the parallelogram \"spanned by\" the two vectors? You may assume (why?) that $\\pmb{a}_1 \\in \\mathrm{span}(\\pmb{e}_1)$, if this helps in your argumentation."
]
},
{
"cell_type": "markdown",
"id": "ad2f6edb",
"metadata": {},
"source": [
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "e71613a0",
"metadata": {},
"source": [
"Let $n=3$. Choose three linearly independent vectors $\\pmb{a}_1, \\pmb{a}_2, \\pmb{a}_3$ in $\\mathbb{R}^3$. It can be smart to choose $\\pmb{a}_1, \\pmb{a}_2 \\in \\mathrm{span}(\\pmb{e}_1, \\pmb{e}_2)$. Calculate (using elementary geometric considerations) the volume of the parallelepiped \"spanned by\" the three vectors. Also calculate $|\\mathrm{det}(A)|$ and compare the two results."
]
},
{
"cell_type": "markdown",
"id": "bc6783f0",
"metadata": {},
"source": [
"#### Question e (Extra, can Wait Until After the Exercises of the Day)"
]
},
{
"cell_type": "markdown",
"id": "06eb878c",
"metadata": {},
"source": [
"Let $n=3$, and now let $\\pmb{a}_1, \\pmb{a}_2, \\pmb{a}_3$ be arbitrary but linearly independent vectors in $\\mathbb{R}^3$. Can you prove the formula $\\mathrm{areal}(P) = |\\mathrm{det}(A)|$, where $P$ is the parallelepiped \"spanned by\" the three vectors? You may assume (why?) that $\\pmb{a}_1, \\pmb{a}_2 \\in \\mathrm{span}(\\pmb{e}_1, \\pmb{e}_2)$, if that helps your argumentation."
]
},
{
"cell_type": "markdown",
"id": "3996eeb6",
"metadata": {},
"source": [
"### 4: Plane Integral with Parametrization I. By Hand"
]
},
{
"cell_type": "markdown",
"id": "6714ed74",
"metadata": {},
"source": [
"In the $(x,y)$ plane we are given the point $P_0=(1,2)$ and the set of points \n",
"\n",
"\\begin{equation*}\n",
" C=\\left\\lbrace (x,y)\\Big\\vert \\frac 32\\leq y \\leq \\frac 52 \\wedge 0\\leq x\\leq \\frac 12 y^2\\right\\rbrace.\n",
"\\end{equation*}"
]
},
{
"cell_type": "markdown",
"id": "66ebe8da",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "663e0591",
"metadata": {},
"source": [
"Make a preliminary sketch of $C$ and provide a parameterization $\\pmb{r}(u,v)$ for $C$ with appropriate intervals for $u$ and $v$, i.e., specify $\\Gamma$ such that $\\pmb{r}(\\Gamma) = C$. Justify that the chosen parameterization is injective (if the chosen parameterization is not injective, you must find a new one)."
]
},
{
"cell_type": "markdown",
"id": "3dd868d3",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "1a3cf8b2",
"metadata": {},
"source": [
"Determine the two parameter values $u_0$ and $v_0$ such that $\\pmb{r}(u_0,v_0)=P_0$.\n",
"Make an illustration of $C$ (both a sketch by hand and a plot in Sympy are fine) where you from $P_0$ draw the tangent vectors $\\pmb{r}'_u(u_0,v_0)$ and $\\pmb{r}'_v(u_0,v_0)$. Determine the area of the parallelogram that is spanned by these tangentvectors, see according to Exercise [](exercise:volumen-af-et-parallellotop)."
]
},
{
"cell_type": "markdown",
"id": "525ffeb5",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "e216710d",
"metadata": {},
"source": [
"Determine the Jacobian determinen that corresponds to $\\pmb{r}(u,v)$, and argue that the two column vectors that constitute the Jacobian matrix are linearly independent for all $(u,v) \\in \\Gamma$. Calculate the Jacobian determinant at the point $(u_0,v_0)$."
]
},
{
"cell_type": "markdown",
"id": "54afafb4",
"metadata": {},
"source": [
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "afe5eedd",
"metadata": {},
"source": [
"Calculate the plane integral \n",
"\n",
"\\begin{equation*}\n",
" \\int_C \\frac{1}{y^2+x} \\mathrm{d}\\pmb{x}\n",
"\\end{equation*}\n",
"\n",
"using the change-of-variables theorem for integrals over $\\mathbb{R}^2$. You must argue that changing variables is a usable method for this case. Check your result with the theorem on axis-parallel regions."
]
},
{
"cell_type": "markdown",
"id": "5ef3672b",
"metadata": {},
"source": [
"### 5: The Plane Integral with Parametrization II"
]
},
{
"cell_type": "markdown",
"id": "71ffa425",
"metadata": {},
"source": [
"We want to determine the plane integral\n",
"\n",
"\\begin{equation*}\n",
" \\int_B 2xy\\,\\mathrm{d} \\pmb{x} \\quad\\text{hvor}\\quad B=\\pmb{r}([0,1]^2),\n",
"\\end{equation*}\n",
"\n",
"given by the parametrization\n",
"\n",
"\\begin{equation*}\n",
" \\pmb{r}(u,v)=(u,v(1-u)),\\;\\text{hvor}\\; u\\in\\left[ 0,1\\right]\\text{ and } v\\in\\left[ 0,1\\right].\n",
"\\end{equation*}\n",
"\n",
"Follow the below steps."
]
},
{
"cell_type": "markdown",
"id": "88c33a85",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "233fffbd",
"metadata": {},
"source": [
"Describe the region $B$ using inequalities (such as $x+5y\\ge 7$). Then sketch $B$."
]
},
{
"cell_type": "markdown",
"id": "2764986a",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "225cae32",
"metadata": {},
"source": [
"Determine the Jacobian determinant for the parametrization $\\pmb{r}(u,v)$. Is the Jacobian determinant different from zero on the interior of the parameter domain (this is a requirement for using the change-of-variables theorem)?"
]
},
{
"cell_type": "markdown",
"id": "1b5a4a35",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "bef87473",
"metadata": {},
"source": [
"Now determine the wanted integral."
]
},
{
"cell_type": "markdown",
"id": "ca579f07",
"metadata": {},
"source": [
"### 6: A Triple Integral"
]
},
{
"cell_type": "markdown",
"id": "313d89c5",
"metadata": {},
"source": [
"Calculate the triple integral \n",
"\n",
"\\begin{equation*}\n",
" \\displaystyle{\\int_1^2\\int_1^2\\int_1^2 \\frac{xy}{z} \\mathrm dx\\mathrm dy\\mathrm dz.}\\\n",
"\\end{equation*}"
]
},
{
"cell_type": "markdown",
"id": "136bcd57",
"metadata": {},
"source": [
"### 7: Partial Integration and Integration by Substitution in Two Variables"
]
},
{
"cell_type": "markdown",
"id": "a47fcac0",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "699acc9a",
"metadata": {},
"source": [
"Determine $\\displaystyle{\\int_0^{\\frac{\\pi}{2}}\\left(\\int_0^{\\frac{\\pi}{2}} u\\cos(u+v)\\mathrm{d}u\\right)\\mathrm{d}v.}$"
]
},
{
"cell_type": "markdown",
"id": "f2c8bba9",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "7c6e6854",
"metadata": {},
"source": [
"Determine $\\displaystyle{\\int_0^1\\left(\\int_0^1 \\frac{v}{(uv+1)^2}\\mathrm{d}u\\right)\\mathrm{d}v.}$"
]
},
{
"cell_type": "markdown",
"id": "d12abd68",
"metadata": {},
"source": [
"----"
]
},
{
"cell_type": "markdown",
"id": "4da688b7",
"metadata": {},
"source": [
"## Exercises -- Short Day"
]
},
{
"cell_type": "markdown",
"id": "41077c21",
"metadata": {},
"source": [
"### 1: Parametrized Spatial Region. By Hand."
]
},
{
"cell_type": "markdown",
"id": "548d406d",
"metadata": {},
"source": [
"A region $B$ in $(x,y,z)$ space is given by the parametric representation \n",
"\n",
"\\begin{equation*}\n",
" \\pmb{r}(u,v,w)=\\big(\\frac{1}{2}u^2-v^2,-uv,w\\big),\\quad u\\in \\left[ 0,2\\right],v\\in \\left[ 0,2\\right],w\\in \\left[ 0,2\\right].\n",
"\\end{equation*}"
]
},
{
"cell_type": "markdown",
"id": "4d5f5147",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "8cb7f137",
"metadata": {},
"source": [
"In $B$ we are given the point \n",
"\n",
"\\begin{equation*}\n",
" \\pmb{x}_0=\\pmb{r}(1,1,1).\n",
"\\end{equation*}\n",
"\n",
"Find $\\pmb{x}_0$. When placed at $\\pmb{x}_0$, the tangent vectors $\\pmb{r}_u'(1,1,1),\\pmb{r}_v'(1,1,1)$ and $\\pmb{r}_w'(1,1,1)$ span a parallelepiped $P$, see Exercise [](exercise:volumen-af-et-parallellotop). Determine the volume of this parallelepiped. It would be good training to illustrate this with Sympy."
]
},
{
"cell_type": "markdown",
"id": "e7a931ec",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "9f69caef",
"metadata": {},
"source": [
"Determine the absolute value of the Jacobian determinant corresponding to $\\pmb{r}$. Evaluate it at $\\pmb{x}_0$."
]
},
{
"cell_type": "markdown",
"id": "23efa5f7",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "e9af05d8",
"metadata": {},
"source": [
"Calculate the volume of $B$."
]
},
{
"cell_type": "markdown",
"id": "35d0f591",
"metadata": {},
"source": [
"### 2: Mass Distributions in the $(x,y)$ Plane"
]
},
{
"cell_type": "markdown",
"id": "030e179c",
"metadata": {},
"source": [
"Consider the sets of points in $\\mathbb{R}^2$ given by:\n",
"\n",
"\\begin{equation*}\n",
" B=\\left\\lbrace (x,y) \\in \\mathbb{R}^2 \\;\\Big\\vert \\; 1\\leq x\\leq 2 \\, \\wedge \\, 0\\leq y\\leq x^3\\right\\rbrace,\n",
"\\end{equation*}\n",
"\n",
"and consider (again)\n",
"\n",
"\\begin{equation*}\n",
" C=\\left\\lbrace (x,y) \\in \\mathbb{R}^2 \\;\\Big\\vert \\; \\frac 32\\leq y \\leq \\frac 52 \\wedge 0\\leq x\\leq \\frac 12 y^2\\right\\rbrace.\n",
"\\end{equation*} \n",
"\n",
"We will think of $f(x,y)$ as a function that expresses the mass density at the point $(x,y)$ (so, with units such as $\\mathrm{kg/m^2}$)."
]
},
{
"cell_type": "markdown",
"id": "3d52912a",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "23c69b1c",
"metadata": {},
"source": [
"Assume that the mass density is constant, $f(x,y)=1$ for $(x,y)\\in B$. Determine the mass and centre of mass of $B$."
]
},
{
"cell_type": "markdown",
"id": "a5248599",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "b04910ef",
"metadata": {},
"source": [
"Assume that the mass density is $f(x,y)=x^2$ for $(x,y)\\in B$. Determine the mass and the centre of mass of $B$."
]
},
{
"cell_type": "markdown",
"id": "8163ac99",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "662d6fcd",
"metadata": {},
"source": [
"Assume that the mass density is constant $f(x,y)=1$ for $(x,y)\\in C$. Determine the mass and the centre of mass of $C$."
]
},
{
"cell_type": "markdown",
"id": "87d699b8",
"metadata": {},
"source": [
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "7fbc32b8",
"metadata": {},
"source": [
"Assume that the mass density is $f(x,y)=x^2$ for $(x,y)\\in C$. Determine the mass and the centre of mass of $C$."
]
},
{
"cell_type": "markdown",
"id": "b2d61ab4",
"metadata": {},
"source": [
"### 3: Spherical Regions in 3D Space"
]
},
{
"cell_type": "markdown",
"id": "c2a6a9e2",
"metadata": {},
"source": [
"Consider the spatial region $\\pmb{r}(\\Gamma)$ given by \n",
"\n",
"\\begin{equation*}\n",
" \\pmb{r}(u,v,w)=\\big(u\\sin(v)\\cos(w),u\\sin(v)\\sin(w),u\\cos(v)\\big), \\quad (u,v,w) \\in \\Gamma, \n",
"\\end{equation*}\n",
"\n",
"where $\\Gamma = [a,b] \\times [c,d] \\times [e,f] \\subset [0, \\infty[ \\times [0,\\pi] \\times [0,2\\pi]$. Meaning, we are dealing with the following parameter values:\n",
"$u\\in [a,b],v\\in [c,d],w\\in [e,f]$."
]
},
{
"cell_type": "markdown",
"id": "20d5212a",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "e69d6556",
"metadata": {},
"source": [
"What do the parameters represent?"
]
},
{
"cell_type": "markdown",
"id": "98053f08",
"metadata": {},
"source": [
"#### Question b\n",
"\n",
"Let $A$ be the region that is determined by the choice: \n",
"\n",
"\\begin{equation*}\n",
" a=1,b=3,c=\\frac{\\pi}{4},d=\\frac{\\pi}{3},e=0,f=\\frac{3\\pi}{4},\n",
"\\end{equation*}\n",
"\n",
"and let $B$ be the region determined by the choice: \n",
"\n",
"\\begin{equation*}\n",
" a=2,b=4,c=\\frac{\\pi}{4},d=\\frac{\\pi}{2},e=-\\frac{\\pi}{4},f=\\frac{\\pi}{4}.\n",
"\\end{equation*}"
]
},
{
"cell_type": "markdown",
"id": "24af2dcd",
"metadata": {},
"source": [
"Describe in words each of the regions $A$, $B$ and $A\\cap B$, and calculate their volumes."
]
},
{
"cell_type": "markdown",
"id": "a6bf3f81",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "0dd55f06",
"metadata": {},
"source": [
"Let $\\symbols x=(x_1,x_2,x_3)$. Calculate all of the integrals \n",
"\n",
"\\begin{equation*}\n",
" \\int_A x_1 \\, \\mathrm{d}\\pmb{x}, \\quad \\int_Bx_1 \\, \\mathrm{d}\\pmb{x} \\quad \\text{og} \\quad \\int_{A\\cap B}x_1 \\, \\mathrm{d}\\pmb{x}.\n",
"\\end{equation*}"
]
},
{
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"### 4: An Indefinite Integral in the Plane\n",
"\n",
"Let $B$ be the unit square $[0,1]^2$. We will investigate the indefinite plane integral: \n",
"\n",
"\\begin{equation*}\n",
" I := \\int_B \\frac{1}{x_2-x_1-1} \\mathrm{d}\\pmb{x}.\n",
"\\end{equation*}\n",
"\n",
"The integrand $f(x_1,x_2)=\\frac{1}{x_2-x_1-1}$ is not Riemann integrable over $B$, since $f$ is not defined at the point $(x_1,x_2)=(0,1)$. We wish to find out whether it is still possible to give the integral a value by considering limits."
]
},
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"source": [
"#### Question a"
]
},
{
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"id": "1c34ce35",
"metadata": {},
"source": [
"Find those points in the $(x,y)$ plane where $f(x_1,x_2)$ is not defined. Find the range of $f$ as a function on $B \\setminus \\{(0,1)\\}$."
]
},
{
"cell_type": "markdown",
"id": "88001983",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "40200da8",
"metadata": {},
"source": [
"Let $B_a = [a,1] \\times [0,1]$ for a fixed $a \\in [0,1]$. Make a sketch of $B_a$ and create a parametrization of $B_a$. Determine the Jacobian determinant of the parametrization."
]
},
{
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"id": "bf399ebb",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "458b6aed",
"metadata": {},
"source": [
"Calculate the Riemann integral \n",
"\n",
"\\begin{equation*}\n",
" I_a := \\int_{B_a} \\frac{1}{x_2-x_1-1} \\mathrm{d}\\pmb{x} \n",
"\\end{equation*}\n",
"\n",
"for every $a \\in ]0,1]$."
]
},
{
"cell_type": "markdown",
"id": "c36a1b90",
"metadata": {},
"source": [
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "6f9a9004",
"metadata": {},
"source": [
"Calculate the limit of $I_a$ for $a \\to 0$. \n",
"\n",
"\n",
"\n",
"#### Question e\n",
"\n",
"Let $B_b = [0,1] \\times [0,b]$. Define $I_b := \\int_{B_b} \\frac{1}{x_2-x_1-1} \\mathrm{d}\\pmb{x} $. Find $\\lim_{b \\to 1} I_b$. Compare with the above."
]
}
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