Calculus How To

Level Curve of a Function: Definition

Share on

Calculus Handbook

Feel like "cheating" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.

Analytic Geometry >

A level curve of a function f(x, y) is a cross section of a three-dimensional figure, projected onto an x-y plane.

Level curves tell us something about heights on a landscape; they are used to create three-dimensional surfaces from two-dimensional contours. For example, the following image shows several level curves and what these curves look like when they are “lifted” and plotted at their respective heights [1]:

level curves of a function

L: Several level curves. Right: Level curves plotted at their respective heights.

We can find a level curve in the plane with the formula f(x, y) = c for some fixed number c [2]. For graphs of three variable functions w = f(x, y, z), the level curves are f(x, y, z) = k[3].

Level Curve of a Function and Contour Maps

contour map with level curves

A contour map of the United States showing level curves.

Level curves are identical to contour lines on a map; points on the same curve or line have a constant altitude (i.e., equal height) on the map with respect to the function. A level curve is the set of all points of one cross section, but if we take several cross sections of a three-dimensional shape, we create a contour map.

If f(x, y) represents altitude at point (x, y), then each contour can be described by f(x, y) = k, where k is a constant. They are created by finding the intersections of function values (or planes at a certain heights) with the graph of the function. The intersections are then plotted in the plane.

contour and level curve

A level curve (right) created by the intersection of a plane and a cone, traced onto the x-y plane.

Note: If a level curve of a function intersects itself at a nonzero angle at a point (x,y) then (x,y) is a critical point [4].


[1] Multivariable and Vector Functions [PDF]. Retrieved January 25, 2022 from:
[2] Croke, C. Functions of several variables.
[3] Functions of two or more variables. Retrieved January 25, 2022 from:
[4] Knill, O. Mathematics Maths 21a Summer 2004 Multivariable Calculus. Retrieved January 25, 2022 from:

Stephanie Glen. "Level Curve of a Function: Definition" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published.