Differential calculus independent variables

SQ denotes the same element of heat hence the coefficients (c s, Z s, and 7 s) are not independent, but are related. The relations are obtained from the equations (1) and the rules for the change of the independent variable in the calculus. For the transformation of the differentials we have ... 

For n— 1 ( ordinary differential calculus), the dependent differential dz may be taken proportional to the differential dx of the independent variable,... 

Integration is used frequently in kinetics, thermodynamics, quantum mechanics and other areas of chemistry, where we build models based on changing quantities. Thus, if we know the rate of change of a property, y (the dependent variable), with respect to x (the independent variable), in the form of dy/dx, then integral calculus provides us with the tools for obtaining the form of y as a function of x. We see that integration reverses the effects of differentiation. 

This section reviews some calculus rules and procedures for solving differential equations like Equation 11.2-1. In what follows,. r is an independent variable, y(jr) is a dependent variable, and a is a constant. 

The calculus of functions of several independent variables is a natural extension of the calculus of functions of one independent variable. The partial derivative is the first important quantity. For example, a function of three independent variables has three partial derivatives. Each one is obtained by the same techniques as with ordinary derivatives, treating other independent variables temporarily as constants. The differential of a function of x, X2, and X3 is given by... 

The calculus of finite differences deals with the ohanges which take place in the value of a function when the independent variable suffers a finite change. Thus if n is increased a finite quantity h, the function x2 increases to (x + h)2, and there is an increment of (x + h)2 - x2 = 2xh + h2 in the given function. The independent variable of the differential calculus is only supposed to suffer infinitesimally small changes. I shall show in the next two sections some useful results which have been obtained in this subject meanwhile let us look at the notation we shall employ. 

The existence of states that are inaccessible to adiabatic processes was shown by Carath odory to be necessary and sufficient for the existence of an integrating factor that converts into an exact differential [2-4]. From the calculus we know that for differential equations in two independent variables, an integrating factor always exists in fact, an infinite number of integrating factors exist. Experimentally, we find that for pure one-phase substances, only two independent intensive properties are needed to identify a thermod)mamic state. So for the experimental situation we have described, we can write SQ gj, as a function of two variables and choose the integrating factor. The simplest choice is to identify the integrating factor as the positive absolute thermodynamic temperature X = T. Then (2.3.3) becomes... 

Differential Calculus with Several Independent Variables... 

The equilibrium macroscopic state of a one-phase simple system is specified by c + 2 independent variables, where c is the number of independent substances (components) in the system. If the system is closed, the amounts of the substances are fixed and only two variables can be varied independently. We take 17 to be a function of S and V for a simple closed system. An infinitesimal change in U that corresponds to a reversible process is given by the fundamental relation of differential calculus ... 

E is said to be a functional of the functional E = E[W] takes a value that depends on the functional form of W, not merely on particular numerical values of a set of independent variables (adjustable parameters). The analysis of the behaviour of E[W] with respect to variation of W is functional analysis and parallels the study of the behaviour of f(x) with respect to variation of x (i.e. analysis in the usual sense) thus, for example, it is possible to discuss variations in terms of functional derivatives SEISfP. We shall, however, make little use of such concepts, in spite of their formal value (see e.g. Feynman and Hibbs, 1965), since variation functions are most commonly defined in terms of a finite number of parameters, to which the ordinary rules of differential calculus can be applied. The other problems we meet can also be solved by elementary methods. 

X he most commonly encountered mathematical models in engineering and science are in the form of differential equations. The dynamics of physical systems that have one independent variable can be modeled by ordinary differential equations, whereas systems with two, or more, independent variables require the use of partial differential equations. Several types of ordinary differential equations, and a few partial differential equations, render themselves to analytical (closed-form) solutions. These methods have been developed thoroughly in differential calculus. However, the great majority of differential equations, especially the nonlinear ones and those that involve large sets of... 

---