Differential calculus dependent variables

Here, we will need some simple facts from matrix differential calculus. If X is a matrix variable and (3 is a parameter that X depends on, then... 

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. 

Differential calculus can be used to find maximum and minimum values of a function. A relative minimum or maximum value of a variable y which depends on x is found at a point where dy/dx = 0. 

Our problem is to take the uncertainties in x, X2, and so on, and calculate the uncertainty in y, the dependent variable. This is called the propagation of errors. If the errors are not too large, we can take an approach based on a differential calculus. The fundamental equation of differential calculus is Eq. (7.9),... 

St is made smaller and smaller without limit but we constantly find that dx/dt is used when Sx/St is intended. For convenience, D is sometimes used as a symbol for the operation in place of djdx. The notation we are using is due to Leibnitz 2 Newton, the discoverer of this calculus, superscribed a small dot over the dependent variable for the first differential coefficient, two dots for the second, thus x, x. .. 

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. 

There are many uses for differential calculus in physical chemistry however, before going into these, let us first review the mechanics of differentiation. The functional dependence of the variables of a system may appear in many different forms as first- or second-degree equations, as trigonometric functions, as logarithms or exponential functions. For this reason, consider the derivatives of these types of functions that are used extensively in physical chemistry. Also included in the list below are rules for differentiating sums, products, and quotients. In some cases, examples are given in order to illustrate the application to physicochemical equations. 

From basic calculus, it is known that a function of a single variable is analytic at a given interval if and only if it has well-defined derivatives, to any order, at any point in that interval. In the same way, a function of several variables is analytic in a region if at any point in this region, in addition to having well-defined derivatives for all variables to any order, the result of the differentiation with respect to any two different variables does not depend on the order of the differentiation. 

The curly d in this equation is called a partial differential, used in calculus when we have functions that depend on several variables. The notation... 

In a search problem, almost nothing is known in advance about how the criterion of effectiveness depends upon the operating variables, the only way to learn being to perform experiments. Here the obstacle to using the calculus is the complete lack of a function that can be differentiated. The objective of the search is to get as close as possible to the optimum after only a limited number of experiments. Box and Wilson, with their paper The Experimental Attainment of Optimum Conditions published in 1951, were the first to interest engineers in search problems (B4). 

The alert reader will notice that although the left-hand-side of this equation depends only on x, the right-hand-side depends only on t. So both sides must be equal to the same constant. Now you have two easy ordinary differential equations in one unknown each. Also you have an unidentified flying parameter, namely the constant that both sides of the equation must equal. In the grand tradition of calculus textbooks, let us call this constant C. So now we have two separate equations to deal with, each in only one variable. The first one is ... 

Functional variation. Given a function of one variable, y = f(x), one can think of two types of variations of y, one associated with x, the other with /. For a fixed functional dependence f(x), the ordinary differential dy measures how y changes as a result of a variation x — x + dx of the variable x. This is the variation studied in ordinary calculus. Similarly, for a fixed point x, the functional variation 5y measures how the value y at this point changes as a result of a variation in the functional form f(x). This is the variation studied in variational calculus. 

The kinematical unknowns of the problem depend only on the variable x (a, 6x, Ox) ) and will be established from the solution of global differential equations of motion, which will be formulated employing the calculus of variations. In order to facilitate the analysis, the following stress resultants are defined as... 

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