Level curve

A level curve is the curve formed by the intersection of an n-dimensional function with an n-1-dimensional plane. For example, a two-dimensional level curve is formed by the intersection of a 3-dimensional function with a plane. Level curves are often used in mathematics to show on a two-dimensional graph how three-dimensional functions are changing. It can be easier to draw a three-dimensional curve in two dimensions by plotting its level curves, and then visualizing the extension of those curves into three dimensions.

For example, the set of level curves of a function f(x,y) consists of the curve of points (x,y) for each constant value of f(x,y). A set of level curves for different constants c is referred to as a contour plot.

The value of a function is identical at every point along any of its level curves. For example, the level curve of the paraboloid $Z(x,y)=x^{2}+y^{2}$ at Z=4 is the circle $x^{2}+y^{2}=4$. Therefore, the gradient of a function (which represents the rate of fastest change) is always perpendicular to its level curves because it is a vector that takes the direction of maximum increase in f.