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.

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.