# level curve

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

 $\displaystyle z\;=\;f(x,\,y)$ (1)

in $\mathbb{R}^{3}$ 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

 $\displaystyle f(x,\,y)\;=\;c$ (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$f^{\prime}_{x}(x,\,y)\,\vec{i}\!+\!f^{\prime}_{y}(x,\,y)\,\vec{j}$  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 $\displaystyle\frac{f^{\prime}_{y}}{f^{\prime}_{x}}$ and the slope of the level curve is $\displaystyle-\frac{f^{\prime}_{x}}{f^{\prime}_{y}}$, whence the slopes are opposite inverses.

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

 $\displaystyle F(x,\,y,\,z)\;=\;c$ (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