Functions of more than one variable

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 ... 

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... 

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... 

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

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

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

Plots of the response as a function of more than one variable are called response surfaces. More complex models can be used in certain instances. This is described in Section 6.4.4. 

Many of the relations in thermodynamics are based on the formal laws of the theory of functions of more than one variable. Therefore, a solid knowledge of the rules of partial differentiation and something more is fundamental to understand thermodynamics. Frequently, thermodynamics is not interested or able to deal with the respective function itself, but rather with the first total derivative. The mathematics of thermodynamics is described in some monographs of thermodynamics biased to mathematics [3-5]. 

We noted at the start of the chapter that the price of an option is a function of the price of the underlying stock and its behaviour over the life of the option. Therefore, this option price is determined by the variables that describe the process followed by the asset price over a continuous period of time. The behaviour of asset prices follows a stochastic process, and so option pricing models must capture the behaviour of stochastic variables behind the movement of asset prices. To accurately describe financial market processes, a financial model will depend on more than one variable. Generally, a model is constructed where a function is itself a function of more than one variable. Ito s lemma, the principal instrument in continuous time finance theory, is used to differentiate such functions. This was developed by a mathematician, Ito (1951). Here we simply state the theorem, as a proof and derivation are outside the scope of the book. Interested readers may wish to consult Briys et al. (1998) and Hull (1997) for a background on Ito s lemma we also recommend Neftci (1996). Basic background on Ito s lemma is given in Appendices B and C. 

The omega notation is used to provide a lower bound, while the theta notation is used when the obtained bound is both a lower and an upper bound. The little oh notation is a very precise notation that does not find much use in the asymptotic analysis of algorithms. With these additional notations available, the solution to the recurrence for insertion and merge sort are, respectively, 0(n ) and 0(n logn). The definitions of O, 2, 0, and o are easily extended to include functions of more than one variable. For example, f(n,m) = 0(g(n, m)) if there exist positive constants c, uq and mo such that /(n, m) < cg(n, m) for all n> no and all m > mo. As in the case of the big oh notation, there are several functions g(n) for which /(n) = Q(g(n)). The g(n) is only a lower bound on f(n). The 0 notation is more precise that both the big oh and omega notations. The following theorem obtains a very useful result about the order of f(n) when f(n) is a polynomial in n. 

For functions of one variable, the derivative with respect to the variable equated to zero gives the equation from which the required value of the variable is obtained. With functions of more than one variable, it can be shown that equating the partial derivative with respect to each variable to zero will produce simultaneous equations from which the unknowns may be found. Applying this, the partial derivative with respect to m equated to zero gives ... 

The bar graph in Figure 40 describes a saddle-like trend from pyrene through to 2-naphthalene with a change in selectivity from o-glucose to D-galactose occurring in the middle of the series. To account for such a trend these results must be a function of more than one variable. 

In the above example, V, was considered to be a function of only a single variable t Such an equation, V, =/(f), can be represented by a series of points on a two-dimensional Cartesian coordinate system. Physicochemical systems, however, usually depend on more than one variable. Thus, it is necessary to extend the definition of function given above to include functions of more than one variable. For example, we find experimentally that the volume of a gas will vary with temperature according to Equation (2-2) only if the pressure of the gas is held constant. Thus, the volume of a gas is not only a function of temperature, but also is a function of pressure. Careful measurements in the laboratory will show that for most gases at or around room temperature and one atmosphere pressure the law relating the volume of a gas simultaneously to the temperature and the pressure of the gas is the well-known ideal gas law... 

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