level curve

The level curvesMathworldPlanetmath (in German Niveaukurve, in French ligne de niveau) of a surface

z=f(x,y) (1)

in 3 are the intersection curves of the surface and the planes  z=constant. Thus the projectionsMathworldPlanetmath of the level curves on the xy-plane have equations of the form

f(x,y)=c (2)

where c is a constant.

For example, the level curves of the hyperbolic paraboloidMathworldPlanetmath (http://planetmath.org/RuledSurface)  z=xy  are the rectangular hyperbolasMathworldPlanetmathxy=c  (cf. this entry (http://planetmath.org/GraphOfEquationXyConstant)).

The gradientfx(x,y)i+fy(x,y)j  of the function f in any point of the surface (1) is perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the level curve (2), since the slope of the gradient is fyfx and the slope of the level curve is -fxfy, whence the slopes are opposite inversesPlanetmathPlanetmath.

Analogically one can define the level surfaces (or contour surfaces)

F(x,y,z)=c (3)

for a function F of three variables x, y, z. The gradient of F in a point  (x,y,z)  is parallelMathworldPlanetmathPlanetmath to the surface normal of the level surface passing through this point.

Title level curve
Canonical name LevelCurve
Date of creation 2013-03-22 17:35:27
Last modified on 2013-03-22 17:35:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Definition
Classification msc 53A05
Classification msc 53A04
Classification msc 51M04
Synonym contour curve
Synonym isopleth
Related topic LevelSet
Related topic ConvexAngle
Defines level surface
Defines contour surface