Classical field theory with L(x, PSI, PSIsub(x), ...) lagrangian in N-dimensional x-space

The 1-st and 2-nd Noether's theorems are generalized for the case when Lagrangian in a N-dimensional space depends on higher derivatives of the field functions. In the course of the theorems generalization the invariance with respect to R parametric gradient transformation with parameters depending on psi and has been analysed. Expressions for corresponding currents are obtained. The relations following from conditions imposed by the demand of invariance of continuity equations as well as from some ones less tri l and obviouvia are derived. As isustration spectric transformation groups (translation group, Lorentz and isotopic rotation groups) are considered which corresponding dynamic invariances have been obtained