ABSTRACT
One relationship for determining Ir and Ix is given by Eq. (III.3 I). Another relationship is given by the total derivative ofJet, x):
dj = Ir dt +Ix dx
Equations (III.3 I ) and (III.32) can be written in matrix form as
(1Il.32)
(1Il.33)
As before, the partial derivatives Ir and Ix are uniquely determined unless the determinant of the coefficient matrix ofEq. (1Il.33) is zero. Setting that determinant equal to zero gives the characteristic equation, which is
a dx - b dt = 0
Solving Eq. (III.34) for dx/dt gives
(III.34)
(Bl.35)
Equation (III.35) is the differential equation for a family of paths in the solution domain along which Ir and Ix may be discontinuous, or multivalued. Since a and b are real functions, the characteristic paths always exist. Consequently, a single quasilinear firstorder PDE is always hyperbolic. The convection equation, Eq. (IIl.23), is an example of such a PDE.
