For these variations to be (exact, i.e. not quasi-) gauge symmetries of the Lagrangian, it is necessary that δ δ λ λ L = 0 for all possible choices of the functions λ λ a ( x ) . If the variations are quasi-symmetries, it is then necessary that the current also depends linearly and differentially on the arbitrary functions, i.e. then δ δ λ λ L = d i K λ λ i , where K λ λ i = K a i λ λ a + K a i , j λ λ j a + K a i , j 1 j 2 λ λ j 1 j 2 a … … For simplicity, we will assume that all gauge symmetries are exact symmetries, but the general case is handled similarly.