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]:

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

**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.

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].

## References

[1] Multivariable and Vector Functions [PDF]. Retrieved January 25, 2022 from: https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?filename=1&article=1014&context=books&type=additional

[2] Croke, C. Functions of several variables.

[3] Functions of two or more variables. Retrieved January 25, 2022 from: https://lindagreen.web.unc.edu/wp-content/uploads/sites/5262/2020/08/courseNotes_math233_2019F_Part3_FnsOfSeveralVariables.pdf

[4] Knill, O. Mathematics Maths 21a Summer 2004 Multivariable Calculus. Retrieved January 25, 2022 from: https://people.math.harvard.edu/~knill/teaching/summer2004/quiz2.html

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