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.