Changes

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.

 

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.

[[Category:Geometry]]

[[Category:Geometry]]