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. | + | 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 there-dimensional curve in two dimensions by plotting its level curves, and then visualizing the extension of those curves into three dimensions. |
| − | 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. | + | 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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| + | 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. | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
| + | [[Category:vector analysis]] | ||
| + | [[Category:calculus]] | ||
| + | [[category:mathematics]] | ||
Revision as of 02:45, January 1, 2010
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 there-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.
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.
