Level Sets and Contours

Calculus 3 · Axiom Academy

LESSON Level Sets and Contours Slice a surface with a horizontal plane, drop the intersection onto the floor, and you have drawn a contour. 1. A Level Curve Is a Slice, Dropped Think of a mountain. If you walk around it staying at exactly 1000 feet of elevation, you trace out a level curve. Walk at 1100 feet and you trace a different one. The mountain is the surface z = f(x,y) ; your path is where a horizontal plane cuts it, seen from above. That is the whole idea, and it is worth stating twice. The level curve f(x,y) = k is the horizontal plane z = k intersected with the surface z = f(x,y) , then dropped straight down onto the xy -plane. our running example — a circular bowl the level curve is a circle of radius Every level curve of this bowl is a circle centred at the origin — a true circle, not an oval — because x^2 + y^2 = k says exactly "distance from the origin is ." At k = 0 the level set collapses to the single point (0,0) , the bottom of the bowl. For it is empty: the bowl never goes below zero. One slice gives one curve. Step the cutting plane through several heights and every slice leaves a curve behind. Collect them all in the xy -plane and you have a contour map — a topographic map of the surface. For our bowl, take the five heights the lesson runs on:

This is the written version of the interactive lesson above. See the full Calculus 3 course.