v in the y coordinates. Then in the y coordinates, the 1-jet of v is a new list of real numbers . Since it follows that So Expanding by a Taylor series, we have Note that the transformation law is second-order in the coordinate transition functions. Differential operators between vector bundles See also Jet group Jet bundle Lagrangian system References Krasil'shchik, I. S., Vinogradov, A. M., [et al.], Symmetries and conservation laws for differential equations of mathematical physics, American Mathematical Society, Providence, RI, 1999, . Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. , . Saunders, D. J., The Geometry of Jet Bundles, Cambridge University Press, 1989, Olver, P. J., Equivalence, Invariants and Symmetry, Cambridge University Press, 1995, Sardanashvily, G., Advanced Differential Geometry for Theoreticians: Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013, ; Differential geometry Smooth functions Singularity theory