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?

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. 

