Skip to main content
Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
\newcommand{\va}{\vec{a}}
\newcommand{\vb}{\vec{b}}
\newcommand{\vc}{\vec{c}}
\newcommand{\vC}{\vec{C}}
\newcommand{\vd}{\vec{d}}
\newcommand{\ve}{\vec{e}}
\newcommand{\cursedihat}{\hat{\dot{i}}}
\newcommand{\vi}{\hat{\imath}}
\newcommand{\vj}{\hat{\jmath}}
\newcommand{\vk}{\hat{k}}
\newcommand{\vn}{\vec{n}}
\newcommand{\vm}{\vec{m}}
\newcommand{\vr}{\vec{r}}
\newcommand{\vs}{\vec{s}}
\newcommand{\vu}{\vec{u}}
\newcommand{\vv}{\vec{v}}
\newcommand{\vw}{\vec{w}}
\newcommand{\vx}{\vec{x}}
\newcommand{\vy}{\vec{y}}
\newcommand{\vz}{\vec{z}}
\newcommand{\vzero}{\vec{0}}
\newcommand{\vF}{\vec{F}}
\newcommand{\vG}{\vec{G}}
\newcommand{\vH}{\vec{H}}
\newcommand{\vR}{\vec{R}}
\newcommand{\vT}{\vec{T}}
\newcommand{\vN}{\vec{N}}
\newcommand{\vL}{\vec{L}}
\newcommand{\vB}{\vec{B}}
\newcommand{\vS}{\vec{S}}
\newcommand{\proj}{\text{proj}}
\newcommand{\comp}{\text{comp}}
\newcommand{\nin}{}
\newcommand{\vecmag}[1]{\left\lVert #1\right\rVert}
\newcommand{\grad}{\nabla}
\newcommand\restrict[1]{\raise-.5ex\hbox{$\Big|$}_{#1}}
\DeclareMathOperator{\curl}{curl}
\DeclareMathOperator{\divg}{div}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Activity 11.8.3 .
In this activity, we will look at the calculation and interpretation of the gradient for
\(f(x,y,z)=x+y-z\) and
\(g(x,y,z)=z-x^2-y^2\text{.}\)
(a) Calculate
\(\nabla f\) and
\(\nabla g\text{.}\)
(b) Draw the level surfaces of
\(f\) for the values
\(k=\{-2,-1,0,1,2\}\text{.}\) Write a few sentences about the shape of each of these level surfaces and describe how the level surfaces is changing in terms of the value of
\(k\text{.}\)
(c) Write a few sentences about how the direction and magnitude of
\(\nabla f\) is related to the level surfaces from the previous task.
(d) Draw the level surfaces of
\(g\) for the values
\(k=\{-2,-1,0,1,2\}\text{.}\) Write a few sentences about the shape of each of these level surfaces and describe how the level surfaces are changing in terms of the value of
\(k\text{.}\) Note here that each of the level surfaces can be expressed with
\(z\) as a function of
\(x\) and
\(y\text{.}\)
(e) Write a few sentences about how the direction and magnitude of
\(\nabla g\) is related to the level surfaces from the previous task.
