The tilt curves (in German die Neigungskurven) of a surface $$z = f(x,\,y)$$ are the curves on the surface which intersect orthogonally the level curves $f(x,\,y) = c$ of the surface. If the gravitation acts in direction of the negative $z$ -axis, then a drop of water on the surface aspires to slide along a tilt curve. For example, since the level curves of the sphere $z = \pm\sqrt{r^2-x^2-y^2}$ are the ``latitude circles'', the tilt curves of the sphere are the ``meridian circles''. The tilt curves of a helicoid are circular helices.

If the tilt curves are projected on the $xy$ -plane, the differential equation of those projection curves is

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Example. Let us find the tilt curves of the elliptic paraboloid $$z = \frac{x^2}{a^2}+\frac{y^2}{b^2}.$$ The level curves are the ellipses $\frac{x^2}{a^2}+\frac{y^2}{b^2} = c$ . Now we have $$f'_x(x,\,y) = \frac{\partial}{\partial x}\!\left(\frac{x^2}{a^2}\!+\!\frac{y^2}{b^2}\right) = \frac{2x}{a^2}, \quad f'_y(x,\,y) = \frac{\partial}{\partial y}\!\left(\frac{x^2}{a^2}\!+\!\frac{y^2}{b^2}\right) = \frac{2y}{b^2},$$ whence the differential equation of the tilt curves is $$\frac{dy}{dx} = \frac{a^2}{b^2}\!\cdot\!\frac{y}{x}.$$ The separation of variables and the integration yield $$\int\frac{dy}{y} = \frac{a^2}{b^2}\!\int\frac{dx}{x},$$ then $$\ln|y| = \frac{a^2}{b^2}\ln|x|+\ln|C| = \ln(|C||x|^{a^2/b^2}),$$ and finally

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Remark. Given an arbitrary family of parametre curves on a surface $$\vec{r} \,=\, (x(u,\,v),\;y(u,\,v),\;z(u,\,v))^\intercal$$ of $\mathbb{R}^3$ , e.g. in the form $$\frac{du}{dv} = f(u,\,v),$$ the family of its orthogonal curves on the surface has in the Gaussian coordinates $u,\,v$ the differential equation

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