Difference between revisions of "Level curve"

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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.
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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.  
  
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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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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.

Revision as of 20:08, February 3, 2010

In mathematics, a level curve is the curve formed by taking all the values of some functions 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 at Z=4 is the circle . 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.

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