I'm familiar with the first of these equations, which is ## d \hat{T}/ds=\kappa \hat{N} ## in the notation I am familiar with where ## \hat{T} ## is a unit tangent vector along the path and ## \hat{N} ## is the unit vector perpendicular to it, with ## \kappa ## being the curvature. I googled it and found ## B ## in your second equation to be defined as a unit vector ## \hat{B}=\hat{T} \times \hat{N} ##, but I have not verified the accuracy of the second equation. In any case, I think I have something that might be useful. Along the curve x=x(t), y=y(t), and z=z(t), define ## R=(x^2+y^2+z^2)^{1/2} ##. If I computed it correctly, ## R \nabla R=\vec{R} ## and ## d \vec{R}/ds=\hat{T} ## so that ##d(R \nabla R)/ds=\hat{T} ## and from this you can take another d/ds to generate your first equation containing a gradient operation. Perhaps this is what you are looking for. Hopefully it is helpful.