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  • MAT232 Lecture 09

Level Curve and Surfaces

• If $f(x,y)$ is a function then a level curve for $f$ at level $c$ is the curve $f(x,y)=c$
  • The intersection of the surface $z=f(x,y)$ along the plane $z=c$
• If $f(x,y,z)$ is a function then a level surface for $f$ at level $c$ is the set of points in $R^3$ satisfying $f(x,y,z)=c$
• Example:
  • $x^2-y^2=c$
  • C is the leve
  • $c>0$ Hyperbola left and right
  • $c=0$ straight lines
  • $c<0$ Hyperbola up and down
• Example:
  • $f(x,y)=\left|x\right|+\left|y\right|$
  • At $f=3$
  • we get a line from $(0,3)\to (3,0)$
  • and another line from $(-3,0)\to (0,3)$
  • all possible combinations of $(\{0,3,-3\},\{0,3,-3\})$

    	left=-5; right=5;
    	top=5; bottom=-5;
    	---
    	3=y+-x|-3<=x<=0
    	x+y=3|0<=x<=3
    	-x-y=3|-3<=x<=0
    	x-y=3|0<=x<=3
    	(0,3)|label:(0,3)
    	(3,0)|label:(3,0)
    	(-3,0)|label:(-3,0)
    	(0,-3)|label:(0,-3)