level curve

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

$z=f(x,y)$ (1)

in ${ℝ}^3$ are the intersection curves of the surface and the planes  $z=\mathrm{constant}$. Thus the projections 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 paraboloid (http://planetmath.org/RuledSurface)  $z=xy$  are the rectangular hyperbolas  $xy=c$  (cf. this entry (http://planetmath.org/GraphOfEquationXyConstant)).

The gradient  $f_x^{\prime }(x,y)\overrightarrow{i}+f_y^{\prime }(x,y)\overrightarrow{j}$  of the function $f$ in any point of the surface (1) is perpendicular to the level curve (2), since the slope of the gradient is ${\displaystyle \frac{f_y^{\prime }}{f_x^{\prime }}}$ and the slope of the level curve is $-{\displaystyle \frac{f_x^{\prime }}{f_y^{\prime }}}$, whence the slopes are opposite inverses.

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 parallel 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