# Curvature of contour lines of a scalar field

How can I compute the curvature of the contour lines (equipotential lines) $\phi (\vec{r})=c$ for the scalar field $\phi (\vec{r})$ ? I expect the direction of the curvature vector to be along the gradient of the field, in analogy to the electric field vectors which are orthogonal to the equipotential lines. Wwhat about the magnitude?

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The formula for computing the curvature of a curve defined by an implicit equation can be found in my notes at http://u.math.biu.ac.il/~katzmik/egreglong.pdf on page 32. It is closely related to the Reiss relation in algebraic geometry. See references there.

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Is there a generalization of the Bateman-Reiss operator for a general N-Dimensional curvilinear system of coordinates ? – Tarek Apr 21 '13 at 17:31
@Tarek: Sure. However, it's not the coordinates matter, it's the metric and its Levi-Civita connection. For an oriented Riemannian surface $(S,g)$ with Levi-Civita connection $\nabla$ and a function $f:S\to \mathbb{R}$ with $df\not=0$, then the geodesic curvature of the level sets of $f$ is the function $$\kappa = \frac{\nabla^2f(\ J\nabla f,\ J\nabla f\ )}{|\nabla f|^3},$$ where $J:TS\to TS$ is the usual 'rotation by $90$ degrees' operator on the tangent spaces. – Robert Bryant Apr 21 '13 at 23:51
@Tarek: I realized, after I made my comment, that you had asked about the 'general $N$-dimensional curvilinear system of coordinates'. In that case, the answer is essentially the same, but now you are asking about the second fundamental form of the level sets. The formula in that case is just $$II(v,w) = \frac{\nabla^2f(v,w)}{|\nabla f|}$$ when $v,w\in T_xS$ satisfy $df(v)=df(w)=0$. – Robert Bryant Apr 22 '13 at 0:00
@Robert: thanks for your comments. Which reference would you cite for this? – Mikhail Katz Apr 22 '13 at 12:22
@katz: I don't really know, to tell the truth. I'm sure that this is very classical, but I wouldn't begin to know how to track down the first reference, which must be sometime in the 19th century (if not 18th). One place you might try is Ron Goldman's 2005 paper "Curvature formulas for implicit curves and surfaces" (available online). The introduction refers to a number of earlier sources for these implicit formulae (including Spivak's "Comprehensive Intro. to Diff. Geom."), the earliest being 1888 (Knoblauch), but I would be surprised if it couldn't be traced much further back than this. – Robert Bryant Apr 22 '13 at 19:11