The Tensor Analysis of Partial Differential Equations

After the comprehensive works of Bortolotti on partial differential equations of the second order from the differential geometer's point of view, and the equally comprehensive memoir of Kawaguchi and Hombu on systems of higher order, the present note serves only to show that slightly different results can be obtained by keeping to the point of view that I have used in my former papers. The method, in particular, is to handle such systems as obey the following postulates and for the special transformation groups under which the postulates hold: (1) the system of equations transforms according to the tensor law; (2) the equations of variation of the given system are also tensorial when the variation itself is a vector; (3) there exists at least one operator which is vectorial in character and corresponds to total differentiation with respect to one of the independent variables.