Functions of Several Independent Variables

A function of several independent variables is similar to a function of a single independent variable except that you must specify a value for each of the independent 

We can choose any three of the variables as independent variables so long as at least of them is proportional to ihe size of ihe system. For example, we could also 

In physical chemistry, we sometimes work with mathematical formulas that represent various functions. For example, if the temperature of a gas is fairly high and its volume is large enough, the pressure of a gas is given to a good approximation by the ideal gas equation 

In addition to formulas, functions can be represented by graphs, by tables of values, or by infinite series. However, graphs, tables, and series become more complicated when used for a function of several variables than for functions of a single variable. 

Chapter 7 Calculus With Several Independent Variables 

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Thus far in this chapter, functions of only a single variable have been considered. However, a function may depend on several independent variables. For example, z — f(x,y), where x and y are independent variables. If one of these variables, say y, is held constant the function depends only on x. Then, the derivative can be found by application of the methods developed in this chapter. In this case the derivative is called the partial derivative of z with respect to jc, which is represented by dz/dx or Bf/Bx. The partial derivative with respect to y is analogous. The same principle can be applied to implicit functions of several independent variables by the method developed in Section 2.5. Clearly, the notion of partial derivatives can be extended to functions of any number of independent variables. However, it must be remembered that when differentiating with respect to a given independent variable, all others are held constant. 

Estimation of the Probability of an Event as a Function of Several Independent Variables". 

Functions of several independent variables occur frequently in physical chemistry, both in thermodynamics and in quantum mechanics. 

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... 

A partial differential equation ( PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space or distributed in space and time. (Definition taken from Wikipedia http //en.wikipedia.org/wiki/Partial differenti al equation)... 

Parametric models are more or less white box or first principle models. They consist of a set of equations that express a set of quantities as explicit functions of several independent variables, known as parameters . Parametric models need exact information about the inner stmcture and have a limited number of parameters. For instance, for the description of the dynamics, the order of the system should be known. Therefore, for these models, process knowledge is required. Examples are state space models and (pulse) transfer functions. Non-parametric models have many parameters and need little information about the inner stmcture. For instance, for the dynamics, only the relevant time horizon shoirld be known. By their stmcture, they are predictive by nature. These models are black box and can be constructed simply from experimental data. Examples are step and pulse response functions. 

A function of several independent variables gives the value of a dependent variable if you specify a value for each of the independent variables. The equilibrium thermodynamic properties of a fluid (gas or Uquid) system of one substance and one phase are functions of three independent variables. If we choose a set of values for the temperature, T, volume, V, and amount of the substance in moles, n, then the other thermodynamic properties, such as the pressure, P, and the... 

Differential equations that contain partial derivatives are called partial differential equations. These equations involve functions of several independent variables. 

Euler s theorem is a mathematical theorem that applies to homogeneous functions. A function of several independent variables, f ni,n2,ti3,. ..,nc), is said to be homogeneous of degree k if... 

The >i are the values of the observations, the Xix are coefficients and the ai are unknowns (the parameters). The observational model is a general linear model, because the dependent variable y, is described as a function of several independent variables x,. The function is a linear function (there are no terms of a degree higher than 1). The matrix method is used to solve the problem. This method is described in every good textbook... 

The next chapter will explore some examples of extending the techniques of this chapter to partial differential equations and systems of partial differential equations. This extends the numerical approaches developed in this chapter to functions of several independent variables. 

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