Key Equations

• Vertical trace
  $f(a,y)=x$ for $x=a$ or $f(x,b)=z$ for $y=b$
• Level surface of a function of three variables
  $f(x,y,z)=c$

Glossary

contour map

a plot of the various level curves of a given function $f(x,y)$

function of two variables

a function $z=f(x,y)$ that maps each ordered pair $(x,y)$ in a subset $D$ of $\mathbb{R}^{2}$ to a unique real number $z$

graph of a function of two variables

a set of ordered triples $(x,y,z)$ that satisfies the equation $z=f(x,y)$ plotted in three-dimensional Cartesian space

level curve of a function of two variables

the set of points satisfying the equation $f(x,y)=c$ for some real number $c$ in the range of $f$

level surface of a function of three variables

the set of points satisfying the equation $f(x,y,z)=c$ for some real number $c$ in the range of $f$

surface

the graph of a function of two variables, $z=f(x,y)$

vertical trace

the set of ordered triples $(c,y,z)$ that solves the equation $f(c,y)=z$ for a given constant $x=c$ or the set of ordered triples $(x,d,z)$ that solves the equation $f(x,d)=z$ for a given constant $y=d$