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.