Equations, mathematical chain rule

We utilize the laboratory, which is not a separate course, in the process of introducing higher level mathematics. For instance, the first day of laboratory is given to mathematics exercises that review simple integrals and derivatives, and the chain rule. This is also where partial derivatives are introduced using the ideal gas law and the van der Waals equation as object lessons. It is here that we also introduce the triangle derivative rule for partial derivatives, Eqn 4. 

The proof can be done by mathematical induction. For convenience, denote the /th derivative by f. The fust derivative appears in Equation (9-34). Just by plugging in i=l, it is clear that f satisfies the relationship. Now, use the chain rule to differentiate f,... 

In order to simplify mathematical steps, the volumetric (AG i) and surface contributions (AGjjjjf) to AG are discussed separately. Eor the volumetric contribution (Equations 10.23 and 10.24), using the chain rule. 

Most boundary value problems occurring in mathematical physics involve second order differential equations (Tychonov and Samarski, 1964). To express such equations in ( q) coordinates, transformations similar to those in Equations 8-4 and 8-5 are therefore needed for fx f, and fyy. Throughout this book, f and F are considered to be sufficiently smooth, so that it is possible to interchange the order of differentiation between any two independent variables. By smooth, we mean that sudden discontinuities are not expeeted in physical solutions. Application of the chain rule to Equations 8-2 and 8-3 leads to... 

For a system with constant composition, the two properties that we choose to constrain the state of the system become the independent properties. We can write the differential change of any other property, the dependent property, in terms of these two properties, as illustrated by Equation (5.4). From a combined form of the first and second laws, we developed the fundamental property relations. We then used the rigor of mathematics to allow us to form this intricate web of thermodynamic relationships. Included in the web are the Maxwell relations, the chain rule, derivative inversion, the cyclic relation, and Equations (5.22) through (5.24). A set of useful relationships relating partial derivatives with T, P, s, and v is summarized in Figure 5.3. We use these relationships to solve first- and second-law problems similar to those in Chapters 2 and 3, but for real fluids. 

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