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level curve

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\begin{document}
The {\em level curves} (in German {\em Niveaukurve}, in French {\em ligne de niveau}) of a surface 
\begin{align}
z \;=\; f(x,\,y)
\end{align}
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
\begin{align}
f(x,\,y) \;=\; c
\end{align}
where $c$ is a constant.\\

For example, the level curves of the \PMlinkname{hyperbolic paraboloid}{RuledSurface} \,$z = xy$\, are the rectangular hyperbolas \;$xy = c$\, (cf. \PMlinkname{this entry}{GraphOfEquationXyConstant}).\\

The gradient \,$f'_x(x,\,y)\,\vec{i}\!+\!f'_y(x,\,y)\,\vec{j}$\, 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 $\displaystyle\frac{f'_y}{f'_x}$ and the slope of the level curve is $\displaystyle-\frac{f'_x}{f'_y}$, whence the slopes are opposite inverses.\\

Analogically one can define the {\em level surfaces} (or {\em contour surfaces})
\begin{align}
F(x,\,y,\,z) \;=\; c
\end{align}
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.




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