Summary of Functions of Several Variables

Essential Concepts

  • The graph of a function of two variables is a surface in [latex]\mathbb{R}^{3}[/latex] and can be studied using level curves and vertical traces.
  • A set of level curves is called a contour map.

Key Equations

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

Glossary

contour map
a plot of the various level curves of a given function [latex]f(x,y)[/latex]
function of two variables
a function [latex]z=f(x,y)[/latex] that maps each ordered pair [latex](x,y)[/latex] in a subset [latex]D[/latex] of [latex]\mathbb{R}^{2}[/latex] to a unique real number [latex]z[/latex]
graph of a function of two variables
a set of ordered triples [latex](x,y,z)[/latex] that satisfies the equation [latex]z=f(x,y)[/latex] plotted in three-dimensional Cartesian space
level curve of a function of two variables
the set of points satisfying the equation [latex]f(x,y)=c[/latex] for some real number [latex]c[/latex] in the range of [latex]f[/latex]
level surface of a function of three variables
the set of points satisfying the equation [latex]f(x,y,z)=c[/latex] for some real number [latex]c[/latex] in the range of [latex]f[/latex]
surface
the graph of a function of two variables, [latex]z=f(x,y)[/latex]
vertical trace
the set of ordered triples [latex](c,y,z)[/latex] that solves the equation [latex]f(c,y)=z[/latex] for a given constant [latex]x=c[/latex] or the set of ordered triples [latex](x,d,z)[/latex] that solves the equation [latex]f(x,d)=z[/latex] for a given constant [latex]y=d[/latex]