With regard to applications of the calculus of variations in physics we extend in this chapter the mathematical tools for field theories, especially with respect to the inverse problem in physics. The only difference from mechanics is the general dimension \(n>1\) of the base X of the fibred manifold \((Y,\,\pi ,\,X)\) and the resulting, somewhat more demanding, technical calculations. We focus on the practical use of the presented concepts which will be also explained with the help of some physical examples. For a more detailed presentation of concepts and proofs from the mathematical point of view, namely for 1-st and 2-nd prolongations of fibred manifolds, the reader can also follow Chap. 2 , with the fact that the general (coordinate free) properties of the presented mappings are the same in field theory as in mechanics. Even though for applications in physical field theories in Chap. 9 only 1-st and 2-nd prolongations of relevant geometrical objects (fibred manifolds, vector fields, differential forms) are needed we present here the general coordinate free definitions for correctness and completeness. They are then illustrated by examples of 1-st and 2-nd order.