Functions of several variables

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. 

If P is a function of several variables (e.g., when h deseribes more than one partiele in a eomposite system), and if F is a property that depends on a subset of these variables (e.g., when F is a property of one of the partieles in the eomposite system), then the expansion F=Zj ( )j> <( )j F> is viewed as relating only to F s dependenee on the subset of variables related to F. In this ease, the integrals <(l)k F> are earried out over only these variables thus the probabilities Pk = P depend parametrieally on the remaining variables. 

Andraos, J. On the Propagation of Statistical Errors for a function of Several Variables, /. Chem. Educ. 1996, 73, 150-154. 

Further investigation revealed that the sieving rate constant was a function of several variables ... 

A dichotomy arises in attempting to minimize function (h). You can either (1) minimize the cost function (h) directly or (2) find the roots of Equation (i). Which is the best procedure In general it is easier to minimize C directly by a numerical method rather than take the derivative of C, equate it to zero, and solve the resulting nonlinear equation. This guideline also applies to functions of several variables. 

Powell, M. J. D. An Efficient Method for Finding the Minimum of a Function of Several Variables Without Calculating Derivatives. Comput J 7 155-162 (1964). 

Equations 4.5 and 4.6 are examples of partial differential equations because they contain partial derivatives, i.e., and The d symbol indicates that [C] and T" are functions of several variables. In this case, the variables are time (0 and depth (z), respectively. To evaluate a partial derivative, all but one of the variables must be held constant in the case of depth (z) is held constant and only time (f) is considered a variable. 

The standard procedure for finding the minimum of a function of several variables is to take the partial derivative with respect to each variable and set the result equal to zero. Taking the derivative with respect to each unknown coefficient and setting the result equal to zero gives (summation interval suppressed)... 

REH I HINIHIZAT10N OF A FUNCTION OF SEVERAL VARIABLES 3404 REH < NELDER-HEAD HETHOD ... 

Column Performance. The only eluant found to elute Au-195m in any reasonable yield was NaCN, consequently the materials were investigated for variation in elution efficiency as a function of several variables, namely, concentration of cyanide, the volume of cyanide, and time/"specific activity" of Hg-195m on column. These parameters were of particular importance because of the over-riding requirement that the specific activity of Au-195m in solution should be greater than 20 mCi/ml. The parameter used to compare results was the % elution efficiency where ... 

Fleming Functions of Several Variables. Second edition. 

We expand the potential energy in a Taylor series about the equilibrium positions, which correspond to < , = q2 = = q3N = 0. The potential energy is a function of several variables, and the Taylor-series expansion is (Taylor and Mann, pp. 222-223)... 

The studies reported in this paper have focused on more complete elucidation of the nature of the interaction between the hydronium ion and active carbon. Both rate and extent of reaction have been studied as a function of several variables to obtain data which ultimately should contribute to a meaningful interpretation of pH effects on adsorption of organic solutes by active carbon. 

The Souders and Brown constant CSB is the C-factor [Eq. (14-77)] at the entrainment flood point. Most modern entrainment flooding correlations retain the Souders and Brown equation (14-80) as the basis, but depart from the notion that CSB is a constant. Instead, they express CSb as a weak function of several variables, which differ from one correlation to another. Depending on the correlation, CSB and us,flood are based on either the net area or on the bubbling area AB. 

Necessary conditions for a function of several variables to be a minimum are that the partial derivatives are equal to zero, viz. 

Metal Detectors. Metal detectors respond to changes in electrical conductivity caused by the presence of metallic objects, both ferrous and non-ferrous. At the same time, metal detectors are relatively insensitive to changes in soil moisture or groundwater conductivity. The magnitude of response from a metal detector is a function of several variables. 

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