\\\\( \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 1, Long Day: Functions of Two Variables

In the spring we will work with functions that depend on two (or more) variables. This week we investigate how we can transfer concepts like continuity and differentiability to these kinds of functions, and how we can draw their graphs. In order to describe the growth of functions we introduce concepts like level sets, partial derivatives, gradients and directional derivatives, many of which are generalisations of equivalent concepts from functions of one variable. Mountains and mountain climbing are popular metaphors within this rather visual subject.

Today’s Key Concepts

Partial derivative $f’_x,f’_y$. Continuity and differentiability of functions of two variables. Graph of functions of two variables. Level set. Gradient $\nabla f$. Restriction. Coordinate expression via simple parametrization. Tangent vector.

Preparation and Syllabus

We will fire off the Spring semester by covering eNote 19 Functions of Two Variables today.

Maple Ressources

Useful Maple commands for today’s work:

  • D differentiates an operator. You can differentiate and insert values simultaneously.
  • diff differentiates an expression. It can differentiate vectors as well.
  • plot3d plots a function of two variables (a surface) in 3D space.
  • contourplot and contourplot3d plot level curves of a function.
  • spacecurve plots a curve in 3D space.

Also, refamiliarize yourself with essential plotting commands such as

  • plot for plotting an expression.
  • implicitplot for plotting an equation.
  • arrow for plotting arrows in both 2D and 3D space.
  • pointplot for plotting points.

The following weeks will also train you how to plot parametric representations with these commands. Maple plotting will be an essential skill this semester, so don’t ignore the plotting exercises.

Today’s Maple demos are 19_GradientsLevelCurves about partial derivatives and gradients, level curves and directional derivatives and 20_CurveParametrization with a parametrization example in 2D space.

Activity Program

  • 10.00 – 12.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.00 – 17.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: $\,$ Your teachers are present in the study areas

Today’s Group Exercises:

  1. Partial Derivatives of First and Second Order. By Hand
  2. Level Curves and Gradients
  3. Visualizations
  4. Gradient Vector Field and Directional Derivatives
  5. Clarification about Differentiability
  6. More Visualizations

Tip: If you want a printable version of the exercises without hints or answers, go directly to your browsers print function when you have entered the exercises.