where is constant and is a function of and . We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form where is a characteristic line. First, we find by the chain rule. Now, if we set and we get which is the left hand side of the PDE we started with. Thus So, along the characteristic line , the original PDE becomes the ODE . That is to say that along the characteristics, the solution is constant. Thus, where and lie on the same characteristic. Therefore, to determine the general solution, it is enough to find the characteristics by solving the characteristic system of ODEs: , letting we know , , letting we know , , letting we know . In this case, the characteristic lines are straight lines with slope , and the value of remains constant along any characteristic line. Characteristics of linear differential operators Let X be a differentiable manifold and P a linear differential operator of order k. In a local coordinate system xi, in which α denotes a multi-index. The principal symbol of P, denoted σP, is the function on the cotangent bundle T∗X defined in these local coordinates by where the ξi are the fiber coordinates on the cotangent bundle induced by the coordinate differentials dxi. Although this is defined using a particular coordinate system, the transformation law relating the ξi and the xi ensures that σP is a well-defined function on the cotangent bundle. The function σP is homogeneous of degree k in the ξ variable. The zeros of σP, away from the zero section of T∗X, are the characteristics of P. A hypersurface of X defined by the equation F(x) = c is called a characteristic hypersurface at x if Invariantly, a characteristic hypersurface is a hypersurface whose conormal bundle is in the characteristic set of P. Qualitative analysis of characteristics Characteristics are also a powerful tool for gaining qualitative insight into a PDE. One can use the crossings of the characteristics to find shock waves for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to along itself. Thus, when two characteristics cross, the function becomes multi-valued resulting in a non-physical solution. Physically, this contradiction is removed by the formation of a shock wave, a tangential discontinuity or a weak discontinuity and can