{
"cells": [
{
"cell_type": "markdown",
"id": "0f47af6d",
"metadata": {},
"source": [
"# Week 8: The Riemann Integral in $n$-D"
]
},
{
"cell_type": "markdown",
"id": "2b4338d2",
"metadata": {},
"source": [
"## Key Terms"
]
},
{
"cell_type": "markdown",
"id": "32d08bf8",
"metadata": {},
"source": [
"* The Riemann integral for scalar functions of $n$ variables\n",
"* The Riemann integral for vector functions\n",
"* Change-of-variables theorem (also known as the transformation theorem): Coordinate change in $\\mathbb{R}^n$\n",
"* The Jacobian determinant\n",
"* [Typical coordinates](https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations)...\n",
" - in $\\mathbb{R}^2$: Cartesian and polar coordinates\n",
" - in $\\mathbb{R}^3$: Cartesian, spherical, cylindrical/semi-polar coordinates"
]
},
{
"cell_type": "markdown",
"id": "99dd4df5",
"metadata": {},
"source": [
"## Preparation and Syllabus"
]
},
{
"cell_type": "markdown",
"id": "974fd7ba",
"metadata": {},
"source": [
"* Reading material: The rest of Chapter 6 \n",
"* Python [demo](../demos/demo08_integration_in_multiple_dimensions)\n",
""
]
},
{
"cell_type": "markdown",
"id": "3fe2c6cb",
"metadata": {},
"source": [
"___\n",
"\n",
"## Exercises -- Long Day"
]
},
{
"cell_type": "markdown",
"id": "ab2454ee",
"metadata": {},
"source": [
"### 1: Plane Integrals over Rectangles. By Hand"
]
},
{
"cell_type": "markdown",
"id": "f5bf32af",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "77408939",
"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 $\\mathbb{R}^2$. Compute 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": "4f903271",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "4ac996a0",
"metadata": {},
"source": [
"We want to compute the same plane integral once more, but now in what at first sight appears to be a more complicated manner, which is to use the change-of-variables theorem for integrals over $\\mathbb{R}^2$."
]
},
{
"cell_type": "markdown",
"id": "6a93784b",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "09fdda87",
"metadata": {},
"source": [
"Compute the plane integral \n",
"\n",
"\\begin{equation*}\n",
" \\int_B \\frac{y}{1+xy} \\;\\mathrm{d}\\pmb{x}, \\quad\\text{where}\\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": "142172e6",
"metadata": {},
"source": [
"### 2: Polar Coordinates. By Hand"
]
},
{
"cell_type": "markdown",
"id": "03ceb8e9",
"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, $r = \\Vert \\pmb{x} \\Vert$ denotes the distance from the point to the origin $(0,0)$. Similarly, $\\theta$ denotes the angle between the $x$ axis and the position vector to the point, considered with a sign with counterclockwise being the positive direction. A set of points $B$ is in polar coordinates described as the points for which \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": "63b34e75",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "5a74b26c",
"metadata": {},
"source": [
"Draw a sketch of $B$, and compute the area of $B$ both using integration as well as via elementary geometric considerations."
]
},
{
"cell_type": "markdown",
"id": "b4538cc8",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "c9f10905",
"metadata": {},
"source": [
"Compute the plane integral $\\int_B f(x,y) \\;\\mathrm{d}\\pmb{x}$."
]
},
{
"cell_type": "markdown",
"id": "f4d14a00",
"metadata": {},
"source": [
"### 3: Volume of a Parallelotope"
]
},
{
"cell_type": "markdown",
"id": "badb495e",
"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 be written as $P=A([0,1]^n)$. \n",
"\n",
"It can be shown via 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, a proof of this is found here: https://textbooks.math.gatech.edu/ila/determinants-volumes.html .)\n",
"\n",
"In $\\mathbb{R}^2$, a parallelotope is the well-known parallelogram, and $\\mathrm{vol}_n(P)$ is the area of $P$, while we in $\\mathbb{R}^3$ find the parallelepiped with the usual volume."
]
},
{
"cell_type": "markdown",
"id": "c3b87487",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "514a74f7",
"metadata": {},
"source": [
"Show $\\mathrm{vol}_n(P) = |\\mathrm{det}(A)|$ by use of the change-of-variables theorem for integrals over $\\mathbb{R}^n$."
]
},
{
"cell_type": "markdown",
"id": "9176e49c",
"metadata": {},
"source": [
"*In the rest of this exercise we wish to investigate the proposition $\\mathrm{vol}_n(P) = |\\mathrm{det}(A)|$ without use of integration techniques.*"
]
},
{
"cell_type": "markdown",
"id": "9dd6a80b",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "5f221829",
"metadata": {},
"source": [
"Let $n=2$. Choose two linearly independent vectors $\\pmb{a}_1, \\pmb{a}_2$ in $\\mathbb{R}^2$. For example, you could choose $\\pmb{a}_1 \\in \\mathrm{span}(\\pmb{e}_1)$. Compute (via simple geometric considerations) the area of the parallelogram \"spanned\" by the two vectors. Also compute $|\\mathrm{det}(A)|$ and compare the sizes."
]
},
{
"cell_type": "markdown",
"id": "83d32d9d",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "c7f78b46",
"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 that helps your argument."
]
},
{
"cell_type": "markdown",
"id": "2c6f273f",
"metadata": {},
"source": [
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "d9f8322e",
"metadata": {},
"source": [
"Let $n=3$. Choose three linearly independent vectors $\\pmb{a}_1, \\pmb{a}_2, \\pmb{a}_3$ in $\\mathbb{R}^3$. It might be beneficial to choose $\\pmb{a}_1, \\pmb{a}_2 \\in \\mathrm{span}(\\pmb{e}_1, \\pmb{e}_2)$. Compute (using simple geometric considerations) the volume of the parallelepiped that is \"spanned\" by the three vectors. Also compute $|\\mathrm{det}(A)|$ and compare the sizes."
]
},
{
"cell_type": "markdown",
"id": "4d3561af",
"metadata": {},
"source": [
"#### Question e (optional/extra)"
]
},
{
"cell_type": "markdown",
"id": "48a2c5de",
"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 $\\mathrm{area}(P) = |\\mathrm{det}(A)|$, where $P$ is the parallelepiped that is \"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 makes your argument easier."
]
},
{
"cell_type": "markdown",
"id": "b9c8903b",
"metadata": {},
"source": [
"### 4: Plane Integral over Parametrized Region I"
]
},
{
"cell_type": "markdown",
"id": "b5450bc0",
"metadata": {},
"source": [
"In the $(x,y)$ plane we are given the point $P_0=(1,2)$ as well as 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": "e22a1a44",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "0237e3bf",
"metadata": {},
"source": [
"Draw a temporary sketch of $C$ and provide a parametric representation $\\pmb{r}(u,v)$ of $C$ with fitting intervals for $u$ and $v$, meaning choose $\\Gamma$ such that $\\pmb{r}(\\Gamma)=C$. Argue that the chosen parametrization is injective (if the chosen parametrization is not injective, then find another one)."
]
},
{
"cell_type": "markdown",
"id": "c61dfebb",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "7c113e89",
"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 $C$ (e.g. using SymPy) where you place the tangent vectors $\\pmb{r}'_u(u_0,v_0)$ and $\\pmb{r}'_v(u_0,v_0)$ with their starting points at $P_0$. Determine the area of the parallelogram that is spanned by the tangent vectors (see Exercise [](exercise:volumen-af-et-parallellotop))."
]
},
{
"cell_type": "markdown",
"id": "388f6ce1",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "55a40c1a",
"metadata": {},
"source": [
"Determine the Jacobian determinant that corresponds to $\\pmb{r}(u,v)$ and argue that the two column vectors in the Jacobian matrix are linearly independent for all $(u,v) \\in \\Gamma$. Compute the Jacobian determinant at the point $(u_0,v_0)$."
]
},
{
"cell_type": "markdown",
"id": "160fd1d4",
"metadata": {},
"source": [
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "23ae0766",
"metadata": {},
"source": [
"Compute the plane integral: \n",
"\n",
"\\begin{equation*}\n",
" \\int_C \\frac{1}{y^2+x} \\mathrm{d}\\pmb{x}\n",
"\\end{equation*}\n",
"\n",
"via the change-of-variables theorem for integrals over $\\mathbb{R}^2$. You must argue for why the change-of-variables theorem can be used."
]
},
{
"cell_type": "markdown",
"id": "b367a563",
"metadata": {},
"source": [
"### 5: Plane Integral over Parametrized Region II"
]
},
{
"cell_type": "markdown",
"id": "23b50797",
"metadata": {},
"source": [
"We want to compute the plane integral\n",
"\n",
"\\begin{equation*}\n",
" \\int_B 2xy\\,\\mathrm{d} \\pmb{x} \\quad\\text{where}\\quad B=\\left\\lbrace (x,y) \\mid 0\\leq x \\, \\wedge \\, 0\\leq y, x+y\\leq 1\\right\\rbrace.\n",
"\\end{equation*}\n",
"\n",
"Follow the steps below."
]
},
{
"cell_type": "markdown",
"id": "b9e623c2",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "6da5bf39",
"metadata": {},
"source": [
"First, sketch the region $B$. Then determine a parametric representation of $B$."
]
},
{
"cell_type": "markdown",
"id": "27ff1f35",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "eb890edf",
"metadata": {},
"source": [
"Determine the Jacobian determinant that corresponds to this parametrization. Is the Jacobian determinant different from zero in the interior of the parameter region (which is a requirement from the change-of-variables theorem)?"
]
},
{
"cell_type": "markdown",
"id": "54d367e7",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "632d3514",
"metadata": {},
"source": [
"Now compute the wanted integral."
]
},
{
"cell_type": "markdown",
"id": "ec37e51d",
"metadata": {},
"source": [
"### 6: Integration by Parts and by Substitution in Two Variables"
]
},
{
"cell_type": "markdown",
"id": "d23de2e6",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "64ae63de",
"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": "4cce7d04",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "2142edf7",
"metadata": {},
"source": [
"Compute $\\displaystyle{\\int_0^1\\left(\\int_0^1 \\frac{v}{(uv+1)^2}\\mathrm{d}u\\right)\\mathrm{d}v.}$"
]
},
{
"cell_type": "markdown",
"id": "246321fc",
"metadata": {},
"source": [
"### 7: A Triple Integral"
]
},
{
"cell_type": "markdown",
"id": "3099a302",
"metadata": {},
"source": [
"Compute 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": "0e9927e4",
"metadata": {},
"source": [
"----"
]
},
{
"cell_type": "markdown",
"id": "bbdacda2",
"metadata": {},
"source": [
"## Exercises -- Short Day"
]
},
{
"cell_type": "markdown",
"id": "3f88149e",
"metadata": {},
"source": [
"### 1: Parametrized Spatial Region. By Hand."
]
},
{
"cell_type": "markdown",
"id": "e8e25f66",
"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": "f6b2d36b",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "f3412b6d",
"metadata": {},
"source": [
"Within $B$ we are given the point \n",
"\n",
"\\begin{equation*}\n",
" \\pmb{x}_0=\\pmb{r}(1,1,1).\n",
"\\end{equation*}\n",
"\n",
"Find $\\pmb{x}_0$. From the initial point $\\pmb{x}_0$, 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 $P$, as was described in Long-Day Exercise [](exercise:volumen-af-et-parallellotop). Compute the volume of this parallelepiped. Illustrate with SymPy."
]
},
{
"cell_type": "markdown",
"id": "28521a98",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "cbb39e05",
"metadata": {},
"source": [
"Compute the absolute value of the Jacobian determinant that corresponds to $\\pmb{r}$. Evaluate it at $\\pmb{x}_0$."
]
},
{
"cell_type": "markdown",
"id": "0fb2f815",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "803bc166",
"metadata": {},
"source": [
"Compute the volume of $B$."
]
},
{
"cell_type": "markdown",
"id": "bc9a370e",
"metadata": {},
"source": [
"### 2: Mass Distributions in the $(x,y)$ plane"
]
},
{
"cell_type": "markdown",
"id": "b2449298",
"metadata": {},
"source": [
"Consider the sets of points in $\\mathbb{R}^2$ given by:\n",
"\n",
"\\begin{equation*}\n",
" B=\\left\\lbrace (x,y)\\vert 1\\leq x\\leq 2 \\, \\wedge \\, 0\\leq y\\leq x^3\\right\\rbrace\n",
"\\end{equation*}\n",
"\n",
"and (again)\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*} \n",
"\n",
"We interpret $f(x,y)$ as the mass density (with units of $\\mathrm{kg/m^2}$) at the point $(x,y)$."
]
},
{
"cell_type": "markdown",
"id": "fe44c5f6",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "8a943a93",
"metadata": {},
"source": [
"Assume that the mass density is constant $f(x,y)=1$ for $(x,y)\\in B$. Determine the mass and the centre of mass of $B$."
]
},
{
"cell_type": "markdown",
"id": "29937cd2",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "9e3fb9ab",
"metadata": {},
"source": [
"Assume that the mass density is $f(x,y)=x^2$ for $(x,y)\\in B$. Compute the mass as well as the centre of mass of $B$."
]
},
{
"cell_type": "markdown",
"id": "0b0ee748",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "607d3caa",
"metadata": {},
"source": [
"Assume that the mass density is constant $f(x,y)=1$ for $(x,y)\\in C$. Compute the mass and the centre of mass of $C$."
]
},
{
"cell_type": "markdown",
"id": "5e7bbdc9",
"metadata": {},
"source": [
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "6a78cf1f",
"metadata": {},
"source": [
"Assume that the mass density is $f(x,y)=x^2$ for $(x,y)\\in C$. Compute the mass and the centre of mass of $C$."
]
},
{
"cell_type": "markdown",
"id": "410bebed",
"metadata": {},
"source": [
"### 3: Spherical Regions of 3D Space"
]
},
{
"cell_type": "markdown",
"id": "4d4de437",
"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 $\\Gamma = [a,b] \\times [c,d] \\times [e,f] \\subset [0, \\infty[ \\times [0,\\pi] \\times [0,2\\pi]$. In other words, we are considering the following parameter values:\n",
"$u\\in [a,b],v\\in [c,d],w\\in [e,f]$."
]
},
{
"cell_type": "markdown",
"id": "a006e09a",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "0983ef9c",
"metadata": {},
"source": [
"What do these parameters represent?"
]
},
{
"cell_type": "markdown",
"id": "50cdf620",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "a07a91d7",
"metadata": {},
"source": [
"Let $A$ be the region that we achieve with 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 $B$ the region achieved with 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*}\n",
"\n",
"Describe in words each of the regions $A$, $B$, and $A\\cap B$, and compute their volumes.\n",
"\n",
"\n",
"\n",
"#### Question c\n",
"Compute the integrals \n",
"\n",
"\\begin{equation*}\n",
" \\int_A x_1 \\, \\mathrm{d}\\pmb{x}\\,,\\,\\, \\int_Bx_1 \\, \\mathrm{d}\\pmb{x}\\,,\\,\\text{ and } \\int_{A\\cap B}x_1 \\, \\mathrm{d}\\pmb{x}.\n",
"\\end{equation*}"
]
},
{
"cell_type": "markdown",
"id": "227e572d",
"metadata": {},
"source": [
"### 4: An Indefinite Integral in the Plane\n",
"\n",
"Let $B$ be the unit square $[0,1]\\times[0,1]$. We want to 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 we can still find an integration value by assigning a limit value to the integral."
]
},
{
"cell_type": "markdown",
"id": "73424f86",
"metadata": {},
"source": [
"#### Question a"
]
},
{
"cell_type": "markdown",
"id": "cf8da292",
"metadata": {},
"source": [
"Find those points in the $(x,y)$ plane where $f(x_1,x_2)$ is not defined. Find the image of $f$ as a function on $B \\setminus \\{(0,1)\\}$."
]
},
{
"cell_type": "markdown",
"id": "e0ebc1a2",
"metadata": {},
"source": [
"#### Question b"
]
},
{
"cell_type": "markdown",
"id": "f92746ef",
"metadata": {},
"source": [
"Let $B_a = [a,1] \\times [0,1]$ for a fixed $a \\in [0,1]$. Draw a sketch of $B_a$, and create a parametrization of $B_a$. Compute the Jacobian determinant of your parametrization."
]
},
{
"cell_type": "markdown",
"id": "d8df3798",
"metadata": {},
"source": [
"#### Question c"
]
},
{
"cell_type": "markdown",
"id": "d92d1664",
"metadata": {},
"source": [
"Compute 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 each $a \\in ]0,1]$.\n",
"\n",
"#### Question d"
]
},
{
"cell_type": "markdown",
"id": "1cfa70fe",
"metadata": {},
"source": [
"Compute the limit value of $I_a$ for $a \\to 0$. \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} $. Compute $\\lim_{b \\to 1} I_b$ and compare with the above results."
]
}
],
"metadata": {
"jupytext": {
"formats": "md:myst"
},
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
}
},
"nbformat": 4,
"nbformat_minor": 5
}