Euler cross differentiation

McKay (2) showed the possibility of using the Euler cross differentiation equation for the thermodynamic treatment of a three-component system. 

The Euler criterion is therefore equivalent to the familiar mixed partials of a function are equal rule of calculus. This cross-differentiation rule is also the condition for the function z(x, y) to have well-defined (single-valued) first derivatives at each point, and thus to be graphable. 

In contrast to the forward (explicit) Euler method, which uses the slope at the left-hand side to step across the interval, the imphcit verison of the Euler method crosses the interval by using the slope at the right-hand side, as shown in Figure 6.8. The implicit formula does not give any direct approximation ofy +i, instead an iterative method, e.g. the Newton method, is added inside the loop, thus advancing the differential equation to solve fory +1. This obviously comes at the price of more computation, but allows stability... 

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