Summary of Functions of Several Variables

Essential Concepts

  • The graph of a function of two variables is a surface in R3R3 and can be studied using level curves and vertical traces.
  • A set of level curves is called a contour map.

Key Equations

  • Vertical trace
    f(a,y)=xf(a,y)=x for x=ax=a or f(x,b)=zf(x,b)=z for y=by=b
  • Level surface of a function of three variables
    f(x,y,z)=cf(x,y,z)=c

Glossary

contour map
a plot of the various level curves of a given function f(x,y)f(x,y)
function of two variables
a function z=f(x,y)z=f(x,y) that maps each ordered pair (x,y)(x,y) in a subset DD of R2R2 to a unique real number zz
graph of a function of two variables
a set of ordered triples (x,y,z)(x,y,z) that satisfies the equation z=f(x,y)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)=cf(x,y)=c for some real number cc in the range of ff
level surface of a function of three variables
the set of points satisfying the equation f(x,y,z)=cf(x,y,z)=c for some real number cc in the range of ff
surface
the graph of a function of two variables, z=f(x,y)z=f(x,y)
vertical trace
the set of ordered triples (c,y,z)(c,y,z) that solves the equation f(c,y)=zf(c,y)=z for a given constant x=cx=c or the set of ordered triples (x,d,z)(x,d,z) that solves the equation f(x,d)=zf(x,d)=z for a given constant y=dy=d