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arbitrary function assume axis boundary called canonical form Cauchy problem Chap characteristic curves characteristic directions characteristic equations coefficients complete integral cone const constant coordinates defined DEFINITION depend determine dimensions Dirichlet problem domain eigenvalues elliptic end point Euler's equation example family of characteristics first-order PDE formula Fourier transform given Green's function Green's identity Hamilton-Jacobi equation Hamilton's equations harmonic function Hence homogeneous hyperbolic equation independent variables infinity initial curve integral equation integral surface Lagrange bracket Laplace transform method Neumann problem nonhomogeneous nonlinear notation obtain ODEs one-parameter family parameter piston Poisson brackets Poisson's potential equation prescribed quasi-linear reduced region result Riemann function satisfy second-order linear solution solve space spacelike Suppose tangent teristics Theorem timelike ut(x values vanishes vector velocity wave equation write xy plane zero