larger diagram, such as the one in the right frame of Figure \[fig:lorenz2\]. This new diagram is a bit more instructive, in that it is the derivatives that are determined by the values of the state variables. However, it still leaves unstated the relationship between the derivative of a state variable and the state variable itself. For instance, the derivative $dx/dt$ is determined independently both by the values of $x$ (alone) and by the values of $x$ and $y$ through the first equation of . It would be useful to encode all of this information into the diagram.

The way to perform this encoding is to reinterpret the meaning of the arrows in the dependency graphs. Instead of an arrow indicating that the variable on the head is determined in part by the variable on the tail, it is better to demand that arrows be actual functional relations. This stronger requirement is not satisfied by either of the diagrams in Figure \[fig:lorenz1\]. The problem is that in , the formula for $dx/dt$ depends *jointly* on the variables $x$ and $y$. Therefore, the functional dependence between $x$, $y$, and $dx/dt$ needs to be from *pairs* of values $(x,y)$. When we perform 