The collection of all points (x,y) in the domain of a function f for which f (x,y) = c is called the level set of f at level c. The level set of f is empty if there is no point (x,y) in the domain of f for which f (x,y) = c. If (x(t),y(t)) is a curve in the domain of f such that f (x(t),y(t)) = c is constant, then the space curve (x(t),y(t),c) is called a level curve of f. The plane curve (x(t),y(t)) in the domain of f is called a contour.
Demos
Level Curves
Examples
Contours of Linear Functions
Contours of f (x,y) = x2 + y2
Exercises
Describe the contours of the function f(x,y) = \frac{1}{x^2+y^2}. What about the function f (x,y) = sin(x2 + y2)?
Describe the contours of the function f (x,y) = x2 – y2. The graph of this function is called a saddle.
Describe the contours of the function f (x,y) = x2 +Bxy + y2 for various values of B. For which B will there be a level set consisting of just one point?
Describe the contours of the function f (x,y) = x3 – 3xy2. (The graph of this function is called a monkey saddle.)
Describe the contours of the function f(x,y) = -x4 + 2x2 - y4 + 2y2.
Analyze Crater Lake shifted by an earthquake, with function f(x,y) = -(x2 + y2)2 + 2(x2 + y2) + mx for various values of m. For which m will the lake no longer hold water? Describe the critical levels, i.e. the level sets that contain critical points.
