# Level curve

In mathematics, a level curve is the curve formed by taking all the values of some functions $f(\vec{x})$ such that the function is some constant value c. 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.