James V. Herod*
version of 1 February 1996.
Here is a slightly different way of thinking about the characteristic equations (2.2) and their companion (2.3). First recall the multidimensional chain rule from your course on vector calculus, and suppose that a function z depends on time through moving coordinates x(t) and y(t). We write:
z(t) = u(x(t), y(t))
and differentiate with respect to time. The result is
z'(t) = x'(t) u_x + y'(t) u_y
(the partial derivatives u_x and u_y depend on x(t) and y(t)).
The point is that the first two terms of equation (2.1) will look like z'(t) if we agree that
x'(t) = p
y'(t) = q.
These equations are just the characteristic equations (2.2). Equation (2.3) comes about if we substitute for the first two terms in (2.2) and rewrite the other stuff as a function of time.
Throughout this section, u(x,y) and z will have the same values, but u(x,y) depends on the original variables of the problem while z will depend on another set of variables related to the characteristic curves, t and an eta which is used as a label to distinguish one characteristic curve from another.