Level curve

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In mathematics, a level curve is the curve formed by taking all the values of some functions <math>f(\vec{x})</math> 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.

These are also referred to as isocurves of the function. For functions of three variables, the terms level surface or isosurface are used.

The value of a function is identical at every point along any of its level curves. For example, the level curve of the paraboloid <math>Z(x,y)=x^2+y^2</math> at Z=4 is the circle <math>x^2+y^2=4</math>. 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.