= Curvature of an integral curve of a vector field
{title2=$\kappa=|F\times(F\cdot\nabla)F|/|F|^3$}
For a <continuously differentiable> nonzero <vector field> $F$, the <curvature of a space curve> following its direction is
$$
\kappa=\frac{|F\times(F\cdot\nabla)F|}{|F|^3}.
$$
Take the <unit tangent vector> $t=\pm F/|F|$ and differentiate with respect to <arc length>. The derivative of $|F|^{-1}$ is parallel to $F$ and disappears from $t\times t'$. Since $t\cdot t'=0$, we have $|t\times t'|=|t'|=\kappa$, proving the formula. It remains valid with zero curvature, even where the usual <Frenet frame> is undefined.
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