text: The following, exact equation can be derived from an advection-diffusion equation (see below), with a diffusion term (D=0) of zero:
  In parallel with the definition of the Lyapunov exponent, we define the matrix 
  , as follows:
  It is easy to show that:
  If we define  as the squared lengths of the principal components of the tracer
  gradient matrix, , then:
  where the 's are arranged, as before, from largest to smallest.
  Therefore,
  growth in the error vector will cause a corresponding decrease in the tracer
  gradient and vice versa. This can be understood very simply and intuitively by
  considering two nearby points: since the difference in tracer concentration will
  be fixed, the only source of variation in the gradients between them will be their
  separation.