\\\\(
\nonumber
\newcommand{\bevisslut}{$\blacksquare$}
\newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}}
\newcommand{\transp}{\hspace{-.6mm}^{\top}}
\newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace}
\newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}}
\newcommand{\eqnl}{}
\newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}}
\newcommand{\vekind}[2]{{_\mathrm{#1}\mathbf{#2}}}
\newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}}
\newcommand{\diver}[2]{{\mathrm{div}\mathbf{#1} (#2)}}
\newcommand{\rot}[1]{{\mathbf{rot}\mathbf{(#1)}}}
\newcommand{\am}{\mathrm{am}}
\newcommand{\gm}{\mathrm{gm}}
\newcommand{\E}{\mathrm{E}}
\newcommand{\Span}{\mathrm{span}}
\newcommand{\mU}{\mathbf{U}}
\newcommand{\mA}{\mathbf{A}}
\newcommand{\mB}{\mathbf{B}}
\newcommand{\mC}{\mathbf{C}}
\newcommand{\mD}{\mathbf{D}}
\newcommand{\mE}{\mathbf{E}}
\newcommand{\mF}{\mathbf{F}}
\newcommand{\mK}{\mathbf{K}}
\newcommand{\mI}{\mathbf{I}}
\newcommand{\mM}{\mathbf{M}}
\newcommand{\mN}{\mathbf{N}}
\newcommand{\mQ}{\mathbf{Q}}
\newcommand{\mT}{\mathbf{T}}
\newcommand{\mV}{\mathbf{V}}
\newcommand{\mW}{\mathbf{W}}
\newcommand{\mX}{\mathbf{X}}
\newcommand{\ma}{\mathbf{a}}
\newcommand{\mb}{\mathbf{b}}
\newcommand{\mc}{\mathbf{c}}
\newcommand{\md}{\mathbf{d}}
\newcommand{\me}{\mathbf{e}}
\newcommand{\mn}{\mathbf{n}}
\newcommand{\mr}{\mathbf{r}}
\newcommand{\mv}{\mathbf{v}}
\newcommand{\mw}{\mathbf{w}}
\newcommand{\mx}{\mathbf{x}}
\newcommand{\mxb}{\mathbf{x_{bet}}}
\newcommand{\my}{\mathbf{y}}
\newcommand{\mz}{\mathbf{z}}
\newcommand{\reel}{\mathbb{R}}
\newcommand{\mL}{\bm{\Lambda}}
\newcommand{\mnul}{\mathbf{0}}
\newcommand{\trap}[1]{\mathrm{trap}(#1)}
\newcommand{\Det}{\operatorname{Det}}
\newcommand{\adj}{\operatorname{adj}}
\newcommand{\Ar}{\operatorname{Areal}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\Rum}{\operatorname{Rum}}
\newcommand{\diag}{\operatorname{\bf{diag}}}
\newcommand{\bidiag}{\operatorname{\bf{bidiag}}}
\newcommand{\spanVec}[1]{\mathrm{span}{#1}}
\newcommand{\Div}{\operatorname{Div}}
\newcommand{\Rot}{\operatorname{\mathbf{Rot}}}
\newcommand{\Jac}{\operatorname{Jacobi}}
\newcommand{\Tan}{\operatorname{Tan}}
\newcommand{\Ort}{\operatorname{Ort}}
\newcommand{\Flux}{\operatorname{Flux}}
\newcommand{\Cmass}{\operatorname{Cm}}
\newcommand{\Imom}{\operatorname{Im}}
\newcommand{\Pmom}{\operatorname{Pm}}
\newcommand{\IS}{\operatorname{I}}
\newcommand{\IIS}{\operatorname{II}}
\newcommand{\IIIS}{\operatorname{III}}
\newcommand{\Le}{\operatorname{L}}
\newcommand{\app}{\operatorname{app}}
\newcommand{\M}{\operatorname{M}}
\newcommand{\re}{\mathrm{Re}}
\newcommand{\im}{\mathrm{Im}}
\newcommand{\compl}{\mathbb{C}}
\newcommand{\e}{\mathrm{e}}
\\\\)
Week 2, Short Day: Conic Sections
A quadratic equation with two variables represents a conic section whose main forms are ellipses, hyperbolas and parabolas. If you have a quadratic equation with two variables, how can you then decide which type of conic section it represents, and how is this positioned in the coordinate system? This requires that you master the techniques completing the square and reduction of a quadratic form, which we will introduce and train today.
Today’s Key Concepts
The quadratic equation with two variables. Quadratic form. Completing the square. Conic section. Ellipsis, hyperbola and parabola. Rotated coordinate system. Rotation matrix.
Preparation and Syllabus
Today we continue with subjects from eNote 15 Symmetric Matrices, Sections 15.8 and 15.9. In addition, we will open eNote 22 Quadratic Equations in Two and Three Variables about conic sections. See in particular Section 22.1.
Maple Ressources
The student
Maple package contains a very useful command for today’s topic:
completesquare
will complete the square for an equation.
Today’s Maple demo 24_ConicSections.
Activity Program
- 13:00 – 14.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
- 14.00 – 16.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
- 16.00 – 17.00: $\,$ Weekly test
Group Exercises
- The Equations for the Circle and the Sphere in Standard Form
- The Standard Equation for the Three Typical Conic Sections
- Identification of a Conic Section
- Determination of a Conic Section
Weekly Test
For all Weekly Tests, the following applies:
- The test is an on-location test, meaning it can only be accessed in the study area.
- No electronic aids are allowed (except for your own notes on e.g. a tablet).
- The test can be accessed in the Möbius quiz system via a link on DTU Learn in the module for 01006 (in the top menu click “Möbius”).
- Your solutions to the test questions must be typed into Möbius without in-between calculations or steps. The result is automatically evaluated by Möbius.
- To ensure a smooth experience use the Firefox or Chrome browser, and disable any add-blocker.
- Use a DTU network.
- You may discuss the test questions with fellow students in your study group, but you have your own version of the test with scrambled numbers that you yourself must solve and enter into Möbius.
- During the final hour on Fridays you have one attempt. Passing this attempt will grant you 1 bonus point. From Friday at 18:00 until Wednesday at 18:00 the test is reopened for repeated attempts. Passing during this phase will grant you ½ bonus point.