Would Leibniz lie to you? (Three aspects of the affine connection) (Q1078162)

The paper presents a new viewpoint on the second derivative of a scalar function. One can consider it as a second-rank (covariant) tensor when put together with the first derivative as a new object \((d\phi/du,d^ 2\phi/du^ 2)\) or, when defined an affine second derivative with the affine connection -\(\lambda(u)\), \(\delta^ 2\phi/\delta u^ 2=d^ 2\phi /du^ 2-\lambda (u)d\phi/du\); \(du^ 2/(du)^ 2\) can then be considered as a special form of \(-\lambda(u).\)