Differential of a Functional

The concept of a differentia] enables the study of the change in a functional by rendering it linear and continuous over a sufficiently small change in the associated function. The notion follows the definition of the differential of a function explained in Section 9.8 (p. 271). 

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Define the differential of a function of more than one variable... 

This can be derived rigorously by using 3.35 to express the differentials of a function with respect to x and y in terms of those with respect to r and - We obtain... 

The differential of a function Ay/Ax is simply the slope of a graph of y versus x. If the function cannot be expressed analytically, the value of Ay/Ax at any value of x can be found by drawing a tangent to the curve and finding the slope of the tangent, as illustrated in Figure 21.11. 

While the operations of taking the nth power of a function (/ (x)) or differentiation of a function (f " (x)) are well known and famihar in all physics contexts, the composition operation (/l l(x)) is not usually encountered in standard physics problems. This, however, does not mean that it is less signiflcant. Iterations of simple maps, not necessarily related to or motivated by physical considerations, combined with computer graphics has led to an explosion in attempts to see how the process of iteration can generate (and store) images of remarkable complexity (see, e.g., Mandelbrot (1983), Peitgen and Richter (1986), Barnsley (1988), Peitgen... 

The process of finding the derivative or slope of a function is the basis of differential calculus. Since you will be dealing with spreadsheet data, you will be concerned not with the algebraic differentiation of a function, but with obtaining the derivative of a data set or the derivative of a worksheet formula by numeric methods. 

A second advantage of derivative methods is increased numerical accuracy. This is perhaps less important in the present era of highly accurate computers. Nevertheless, the numerical evaluation of higher energy derivatives is still a difficult problem. Indeed, Hartree (1968) observes that the differentiation of a function specified only by a table of values... is a notoriously unsatisfactory process, partieularly if higher derivatives than the first are required (citation taken from Gerratt and Mills (1968)). 

On the other hand, dQ and dW are inexact differentials because they cannot be obtained by differentiation of a function of the system. 

The differential of a function of several variables (an exact differential) has one term for each variable, consisting of a partial derivative times the differential of the independent variable. This differential form delivers the value of an... 

The differential of a function is called an exact differential. There can also be differential forms that are not differentials of any function. A general differential form or pfaffianform can be written... 

If this is the differential of a function, then M and N must be the appropriate derivatives of that function. Pfaffian forms exist in which M and N are not the appropriate partial derivatives of the same function. In this case du is called an inexact differential. It is an infinitesimal quantity that can be calculated from specified values of dx and dy, but it is not equal to the change in any function of x and y resulting from these changes. 

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... 

In this integral, the variables X2 and X3 in M must be replaced by functions of x corresponding to the curve on which the integral is performed, with similar replacements in N and P. The line integral of the differential of a function depends only on the end points of the curve, which the line integral of an inexact differential depends on the path of the curve as well as on the end points. 

To find the differential of a function of two independent variables. This can be best done in the following manner, partly... 

The differentiation of a function of three independent variables may be left as an exercise to the reader. Neglecting quantities of a higher order if u be the volume of a rectangular parallelopiped1 having the three dimen-sions x, y, e, independently variable, then u = xyz, and... 

The reason many differential equations are so difficult to solve is due to the fact that they have been formed by the elimination of constants as well as by the elision of some common factor from the primitive. Such an equation, therefore, does not actually represent the complete or total differential of the original equation or primitive. The equation is then said to be inexact. On the other hand, an exact differential equation is one that has been obtained by the differentiation of a function of x and y and performing no other operation involving x and y. 

All this means is that the differential of a function represents the change in the value of the function when the variable suffers an infinitesimal change. The student learned this the first day he attacked the calculus. The ratio dy y is called the proportional, relative, or fractional error, that is to say, the ratio of the error involved in the whole process to the total quantity sought while 100% y is called the percentage error. The degree of accuracy of a measurement is determined by the magnitude of the proportional error. 

To find the variation—not the differential—of a function. Let y be the given function. Write y + By in place of y, and subtract the new function from the old, and there you have it. We at once recognize the formal analogy of the operation with the process of differentiation. Thus, if... 

So far as I know the verb to variate or to vary, meaning to find the variation of a function in the same way that to differentiate means to find the differential of a function, is not used. ... 

We derive the transform first using the simple algebraic approach of Boas (1966, p. 159). The total differential of a function / = f x, y) is written... 

In words, the integral of the differential of a function f(x,y) around a cyclical path equals the integral, over the enclosed area A, of the difference of the mixed derivatives shown. Use this theorem to argue that Eq. (9.13) holds when / is a thermodynamic state fimction. 

Differentiation of a function at a finite discontinuity produces a deltafunction. Consider, for example, the Heaviside unit step function ... 

This chapter introduces the fundamental concepts of optimal control. Beginning with a functional and its domain of associated functions, we learn about the need for them to be in linear or vector spaces and be quantified based on size measures or norms. With this background, we establish the differential of a functional and relax its definition to variation in order to include a broad spectrum of functionals. A number of examples are presented to illustrate how to obtain the variation of an objective functional in an optimal control problem. 

Equation (2.9) delivers a very important result. It shows that the Gateaux differential of a functional is equal to its directional derivative (Section 9.9.1, p. 272) along h, i. e., the change in the associated function y at yo. Thus, the value of the differential is the derivative of the functional with respect to the scalar multiplier to the function change, when evaluated at the zero value of the multiplier. 

In summary, we have three types of differentials of a functional ... 

The Frechet differential of a functional is also the Gateaux differential. In turn, the Gateaux differential of a functional is also the variation. Thus, for a functional, the existence of the Frechet differential implies the existence of the Gateaux differential. In turn, the existence of the Gateaux differential implies the existence of the variation. However, there is no guarantee that the reverse relations hold. For example, a functional may have the variation but not the Gateaux differential. Using conditional statements, these relations are... 

To show this equivalence, we first show that the Prechet differential of a functional is also its Gateaux differential. 

The properties of the total differential of a function of two variables, allow us to... 

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