More Than One Variable

A distribution in more than one variable may be illustrated by the function describing the distribution of matter in a sphere with its center at the origin of a cartesian system. If the radius is c and density p, then the distribution of matter Pix y z) is a function of three variables with the definition that P Xjy,z) dx dy dz is the amount of matter in the element of volume bounded by the set of six cartesian planes through the points x,y,z) and (x + dx, y + dy, z + dz). The mass-distribution function P x,yyZ) is then equal to the density p, for all points satisfying the condition x + / + 2 c. If we are interested in the distribution of matter in only two dimensions, we can obtain a function P(x,y) from P(x,yjz) by integrating over all z  

The limits of this integration must be between z = zt(c2 — x — y ). On integration we find 

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If V is a function of more than one variable, then more complex criteria for determining maxima and minima are obtained. Generally, but not always, the second partial derivatives of the function with respect to all its variables are sufficient to determine the character of a stationary value of V. For such functions, the theory of quadratic forms as described by Langhaar [B-1] should be examined. 

Most chemical reactions are more complicated than this one, and the system potential energy is a function of more than one variable. Consider this reaction, which is a generalized group-transfer reaction ... 

When more than one variable is allowed to vary, then all possible combinations of their values will be included. For example, the following variable definitions will result in a total of 20 scan points ... 

Performance Relationships for Mixing Variables More Than One Variable Changing or Held Constant... 

The concept of homogeneity naturally extends to functions of more than one variable. For example, a generalized homogeneous function of two variables, f(x,y), can be written in the form... 

An analytical method is usually subject to more than one variable. The standard deviation for the method will therefore be a composite of individual standard deviations. So long as these variables are independent, the -standard deviations should combine as follows to give the over-all standard deviation s ... 

The discussion above may be generalized to more than one variable. In the general case, equation (3.18) is replaced by... 

The Fourier integral may be readily extended to functions of more than one variable. We now derive the result for a function /(x, y, z) of the three spatial variables x, y, z. If we consider /(x, y, z) as a function only of x, with y and z as parameters, then we have... 

Therefore we come to the examination of ANOVA of data depending on more than one variable. The basic operation of any ANOVA is the partitioning of the sums of squares. 

A multivariate ANOVA, however, has some properties different than the univariate ANOVA. In order to be multivariate, obviously there must be more than one variable involved. As we like to do, then, we consider the simplest possible case and the simplest case beyond univariate is obviously to have two variables. The ANOVA for the simplest multivariate case, that is, the partitioning of sums of squares of two variables (X and Y), proceeds as follows. From the definition of variance ... 

Thus, it would be natural to attempt to extend the QMOM approach to handle a bivariate NDF. Unfortunately, the PD algorithm needed to solve the weights and abscissas given the moments cannot be extended to more than one variable. Other methods for inverting Eq. (125) such as nonlinear equation solvers can be used (Wright et al., 2001 Rosner and Pykkonen, 2002), but in practice are computationally expensive and can suffer from problems due to ill-conditioning. 

Analytical methods are usually difficult to apply for nonlinear objective functions with more than one variable. For example, suppose that the nonlinear function fix) = /(jtj, jc2,. . . , xn) is to be minimized. The necessary conditions to be used are... 

Matlab does not include a routine of the kind of fzero for more than one variable. Only the function fsolve, which is part of the Optimisation Toolbox, can deal with systems of equations with several variables. Here we demonstrate the application of fsolve to the system of equations (3.70). 

We now consider how Steward s algorithm can help to ascertain whether or not the system of equations describing the process is determinate. It should be noted that if a system of equations having the same number of variables as equations incorporates a subset of equations that contains fewer variables than equations, a unique solution of the system equations is unlikely to exist. We have used the words is unlikely to rather than does not because there are some special classes of equations that specify more than one variable, and if such an equation is included in the system, the system may have a subset of equations with fewer variables than equations and still be determinate. For example, consider the system of Eq. (5) ... 

Chemometric techniques have been frequently used for optimization of analytical methods, as they are faster, more economical and effective and allow more than one variable to be optimized simultaneously. Among these, two level fractional factorial design (2 ) is used mainly for preliminary evaluation of the significance of the variables and its interactions [1]. 

Newton-Raphson can be fairly easily extended to iteration problems involving more than one variable. For example, suppose we have two functions/x(x, 2) and /2(xi, xj) that depend on two variables Xi and X2. We want to find the values of xi and Xj that satisfy the two equations... 

Multivariate A multivariate measurement is defined as one in which multiple measurements are made on a sample of interest. That is, more than one variable or response is measured for each sample. Using a sensor array to obtain multiple responses on a vapor sample is a multivariate measurement. 

The advantage of utilizing the standardized form of the variable is that quantities of different types can be included in the analysis including elemental concentrations, wind speed and direction, or particle size information. With the standardized variables, the analysis is examining the linear additivity of the variance rather than the additivity of the variable itself. The disadvantage is that the resolution is of the deviation from the mean value rather than the resolution of the variables themselves. There is, however, a method to be described later for performing the analysis so that equation 16 applies. Then, only variables that are linearly additive properties of the system can be included and other variables such as those noted above must be excluded. Equation 17 is the model for principal components analysis. The major difference between factor analysis and components analysis is the requirement that common factors have the significant values of a for more than one variable and an extra factor unique to the particular variable is added. The factor model can be rewritten as... 

Many word problems lend themselves to more than one equation with more than one variable. It s easier to write two separate equations, but it takes more work to solve them for the unknowns. And, in order for there to be a solution at all, you have to have at least as many equations as variables. 

Many engineering problems involve several parameters, that impede the elaboration of the pi space. Fortunately, in some cases, a closer look at a problem (or previous experience) facilitates reduction of the number of physical quantities in the relevance list. This is the case when some relevant variables affect the process by way of a so-called intermediate quantity. Assuming that this intermediate variable can be measured experimentally, it should be included in the problem relevance list, if this facilitates the removal of more than one variable from the list. 

Solutions of equations and those of extremum problems are closely related. A point is the root of the equations f(x) = only if it minimizes the function g = fTf. Oh the other hand every local extremum point of a differentiable function g satisfies the equations ag(x)/3x = 0. Though a root is not necessarily an extremum point of g, this transformation may be advantageous in one dimension. As will be discussed the situation is, however, completely different with more than one variable. 

In this section we deal with the problem of finding the minimum of a function of more than one variables. 

However, when more than one variable factor is Included in an experiment, plotting of the functional property studied against each of the variable factors may not be useful and may indeed be misleading. Plotting of data points when there are two or more variable factors is generally useful only when there are several values of a variable factor for each of one or more sets of fixed values of all other factors. Otherwise, separation of the influences of different factors on the functional property will usually be impossible on the basis of plots of the data. 

It is true that the above considerations and examples given in Fig. 21.6 must look somewhat far-fetched in the context of the dynamics of organic chemicals in the environment. This is no longer the case for models with more than one variable. Such models are often nonlinear and have multiple steady-states. In this respect, the purpose of Fig. 21.6 is primarily to open the door to a world of complexity which itself leads far beyond the scope of this book. The interested reader is referred to the corresponding literature (e.g., Arrowsmith and Place, 1992). 

The Limiting Form of Functions of More Than One Variable... 

Finding the limiting forms of functions of more than one variable. 

The ideal gas law has many uses in chemistry, some of which we shall meet later in this text. To begin to see how useful the law can be, recall that we have seen how to use the individual laws to make predictions when only one variable is changed, such as heating a fixed amount of gas at constant volume. The ideal gas law enables us to make predictions when more than one variable is changed. For example, when we pump up an actual bicycle tire, the temperature of the gas in the pump increases as we press in the piston, so the compression is not strictly isothermal as we assumed in Example 4.3. 

The transition must be shown to be reversible. If the protein concentration is low (less than 0.5 mg/ml), this can be shown to be true for RNase over a wide range of temperature but only at pH values below about 3. The transitions are frequently so sharp that accurate estimates of equilibrium constants can be made only over a narrow range of the variable. Thus comparisons of parameters usually involve alteration of more than one variable at a time (i.e., pH and temperature) in order to keep the measurements in an accessible range. Paying as much attention... 

The choice of a solvent for a particular reaction will usually depend on more than one variable. These include the liquid temperature range, the dielectric constant, and whether or not the solvent is reactive in the chemical reaction. The most important consideration in the latter context is often the presence of an easily transferred proton. Certain solvents, such as acetone, are considered aprotic but may transfer a proton under basic conditions. Thus the designations given below are general and approximate. They are intended to guide the reader to the more detailed information contained in the full tables. 

Many equations have more than one variable. To find the solution, solve for one variable in terms of the other(s). To do this, follow the rule regarding variables and numbers on opposite sides of the equal sign. Isolate only one variable. 

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