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Line 1: − 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. +
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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.
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.