# level curve

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

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

$f(x,y)=c$ | (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 $\frac{{f}_{y}^{\prime}}{{f}_{x}^{\prime}}$ 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 |