this transformation to the dependency diagram, we obtain the diagrams in Figure \[fig:lorenz2\].

![Functional dependencies among state variables and their derivatives in the Lorenz system, according to variable names (left) and according to the spaces of values involved (right) []{data-label="fig:lorenz2"}](lorenz2){width="3in"}

There are a number of pleasing features about a functional dependency diagram like the ones in Figure \[fig:lorenz2\]. The most obvious – and most trivial – is that the arrows (on the right frame) are actual functions, and could be labeled as such. The arrows out of the spaces corresponding to tuples of variables are projections, while the others are determined by one of the equations and by the definition of the derivative. Everything about is captured in the diagram on the right, in that the equations can be recovered from the diagram. The in-degree of a variable in Figure \[fig:lorenz2\] specifies the number of functional equations that constrain its value. This means that the independent variables listed in Figure \[fig:lorenz2\] are those with no arrows pointing into them. It is easy to see that these are the pairs $(x,y)$ and the triples $(x,y,z)$. This does *not* mean that there are no constraints on these independent variables, just that there 