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# level curve

The level curves (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 $z=xy$ are the rectangular hyperbolas $xy=c$ (cf. this entry).

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.

## Mathematics Subject Classification

53A05*no label found*53A04

*no label found*51M04

*no label found*

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