Multivariate Functions

Integrating Multivariate Functions Sometimes Depends on the Path of Integration, and Sometimes It Doesn t 

Maxima or minima are not the only things that we need from multivariate functions. Sometimes w e need to integrate them. We have considered only differential changes df so far. We now focus on larger changes Af = df. Suppose 

You could first integrate over x holding y = y, then integrate over y. Or you could first integrate over y holding x = x,, then integrate over x. The sequence of x and y values you choose is called a pathway. Some possible 

Which pathway should you use to integrate a multivariate function Does it matter It depends. Some functions are state functions, which are independent of pathways, and some are path-dependent functions. 

EXAMPLE 5.9 State functions, xdy + ydx is a state function because it can be expressed as the differential of / = xy. 

---

Broomhead, D. S. Lowe, D. Multivariable functional interpolation and adaptive networks. Complex Syst. 1988, 2,312-355. 

For a multivariate function, the nature of convexity can best be evaluated by examining the eigenvalues of/(x) as shown in Table 4.1 We have omitted the indefinite case for H, that is when/(x) is neither convex or concave. 

Figure 4.16 illustrates the character of ffx) if the objective function is a function of a single variable. Usually we are concerned with finding the minimum or maximum of a multivariable function fix)- The problem can be interpreted geometrically as finding the point in an -dimension space at which the function has an extremum. Examine Figure 4.17 in which the contours of a function of two variables are displayed. 

One method of optimization for a function of a single variable is to set up as fine a grid as you wish for the values of x and calculate the function value for every point on the grid. An approximation to the optimum is the best value of /(x). Although this is not a very efficient method for finding the optimum, it can yield acceptable results. On the other hand, if we were to utilize this approach in optimizing a multivariable function of more than, say, five variables, the computer time is quite likely to become prohibitive, and the accuracy is usually not satisfactory. 

Notice that the methods presented in Sections 1.0.2 and 1.0.3 can be extended to estimate the parameters in multivariable functions that are linear in the parameters. 

A single-variable function is convex if the second derivative is strictly positive over the range of the dependent variable, as shown in Fig. 4(1). As show in Fig. 4(2), a function is concave when the second derivative is negative over the dependent variable range. For a multivariable function, the matrix of second derivatives (termed the Hessian, H(x)) is used to check the convexity (or concavity) of the function ... 

Derivative Methods.—The most well developed of the derivative methods are univariate in nature, that is, they approach the minimum of the multivariate function along a sequence of lines (directions) in the many-dimensional space, and the problem is then to determine an algorithm for the choice of these directions. Usually (but not always) it is required that the current direction be followed until a minimum of the function in that direction is found. One may say that these methods are based on a sequence of onedimensional searches. 

D. F. Shanno and K. H. Phua, ACM Trans. Math. Software, 6, 618 (1980). Remark on Algorithm 500 Minimization of Unconstrained Multivariate Functions. 

This interesting field was initiated by Bader [158]. Topological analysis provides the means for a concise description of multivariate functions. For functions that describe physical observables, the number and location of critical points, where the gradient vanishes, and their mutual relationship are often directly related to the properties of the system under study. The application of topological analysis to the one electron density is even more productive, furnishing rigorous quantum-mechanical definitions of and bonds in molecules. Cioslowski has extended this analysis to the study of the electron-electron interactions, based on the analysis of the intracule and extracula densities [159,160]. 

A MATLAB algorithm for optimization of an arbitrary multivariate function... 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

To respect the orientation of frames, we will make also x = y and X3 = x. By applying the chain rule for multivariate functions to the derivatives of Eqn. (4), we obtain ... 

Figure 2 depicts the struetirre of RBF neural networks. RBF networks were introduced into the neural network literature by Broomhead and Lowe (1988). The RBF network model imitates the locally tuned response observed in biological neurons. Neurons with a locally tuned response characteristic can be found in several parts of the nervous system, for example, in cells in the visual cortex sensitive to bars oriented in a certain direction or other visual featirres within a small region of the visual field. These locally tuned neurons show the response characteristics bounded to a small range of the input space. The theoretical basis of the RBF approach lies in the field of interpolation of multivariate functions. The objective of... 

For multivariate functions the Taylor series approximation to a function evaluated at (xq, yo) is given by... 

The difference [f(a) — f(Xfc)] of the two column vectors may be replaced by its equivalent as given by the mean value theorem of differential calculus for multivariable functions (Theorem A-7), since the continuity requirements of the theorem are satisfied by suppositions stated above. Then, each element / of [f(Xk) — f(a)] may be stated as follows... 

Next make use of the mean value theorem of differential calculus for multivariable functions (A-6), namely,... 

Theorem A-2-7 Mean value theorem of differential calculus for multivariable functions Let/(x, y, z) be continuous and have continuous first partial derivatives in a domain D. Furthermore, let (x0, y0> zo) and (x0 4 h, y0 + k, z0 + /) be points in D such that the line segment joining these points lies in D. Then... 

The point-slope formula and the example in the previous section illustrate the methodology to calculate the Legendre transform of a function of one independent variable. Consider the following multivariable function, y(xi,X2,X3,..., x ), where the associated slopes are... 

As a final example, it is not necessary to transform all independent variables of a multivariable function. If Legendre transformation of y is performed with respect to jci and X3, then... 

Dim F)= Dim w)= S. Also, P contains as elements, normalised positive definite multivariate functions of a position vector R of arbitrary dimension. That is, it can be formally written Jp(r)c/R = 1, being 1 the unity (hyper-)matri.x, as... 

Consider the following example. We want to compare molecular shapes, emphasizing each molecule s spatial extent (the property). To that purpose, we choose a van der Waals surface (a model) to evaluate the excluded space. In addition, we decide to disregard fixed nuclear positions and average the surface over the whole amplitude of ground state vibrations (a level of detail). Only after these choices have been made, can we arrive at a sensible choice for a shape descriptor. This shape descriptor can be a single number, a vector, a matrix, a one- or multivariable function, or a graph. 

This series converges for any stable system within the radius of convergence defined by the input ensemble. If the multivariate function F is not analytic but is continuous, then a finite-order Volterra model can be found that achieves any desirable degree of approximation (based on the Stone-Weierstrass theorem). 

The graphical method is based on the notion that the mathematical model of a discrete-time finite-order (stationary) dynamic system is, in general, a multivariate function /( ) of the appropriate lagged values of the input-output variables... 

In the thermodynamic description of multicomponent systems, a principal relation is the Gibbs-Duhem equation. Astarita [1] has shown that the Gibbs-Duhem equation is not merely a thermodynamic relation it is a general repercussion of the properties of homogeneous functions. Consider a multivariant function, such as... 

A functional dependent on a function is analogous to a multivariable function dependent on a vector of variables. For example, consider the functional... 

To extend the analogy further, consider a functional dependent on an integral of a continuous function. The latter has an infinite number of components over a non-zero range of integration. Thus, the functional is equivalent to a multivariable function dependent on the variable vector comprising those components. For example, the functional I in Equation (2.1) is equivalent to a multivariable function dependent on a vector of infinite components of /. 

A. Guzman. Derivatives and Integrals of Multivariable Functions. Birkhauser, Boston, MA, 2003. 

---