Functions of Multiple Variables

Multivariate Calculus Applies to Functions of Multiple Variables 

Here we review the tools from multivariate calculus that we need to describe processes in which multiple degrees of freedom change together. We need these methods to solve two main problems to find the extrema of multivariate functions, and to integrate them. Here we introduce the mathematics. 

A function y = fix) assigns one number to y for each value of x. The function fix) may be y = 3x , or y = log(x - 14), etc. Now consider multivariate functions. Multivariate functions assign one number to each set of values of 

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In conjunction with the use of isoparametric elements it is necessary to express the derivatives of nodal functions in terms of local coordinates. This is a straightforward procedure for elements with C continuity and can be described as follows Using the chain rule for differentiation of functions of multiple variables, the derivative of a function in terms of local variables ij) can be expressed as... 

Physical intuition is needed in order to justify the fundamental relationships in step (a). Once the physical problem is converted into a mathematical one (step [b]), physical intuition is no longer needed and the gear must shift to mastering the how. At this point, a good handle of calculus becomes indispensable, in fact, a prerequisite for the successful completion of this material. Especially important is familiarity with functions of multiple variables, partial derivatives and path integrations. 

In dealing with functions of multiple variables we encounter various partial derivatives. For a function of two variables, F(x, y), the partial derivative... 

Equations of state deal with many variables. The total derivative of a function of multiple variables, F x, y,z,...), is defined as... 

When the radial variation of temperature must be taken into account, the problem assumes an entirely different character. Each of the equations is now a partial differential equation, and both radial and axial profiles must be calculated a mesh or network of radial and axial lines is set up, and the temperature and composition are calculated for each intersection. A great deal of work has been done on the formulation of difference equations for solving the related diffusion or heat-conduction equations most of this has been directed towards the case in which there is only one dependent variable and in which the source is a linear function of that variable. Although the results obtained for one dependent variable are only partially applicable to the multiple-variable problem,... 

A multiple integral has as its integrand function a function of several variables, all of which are integrated. 

A traditional or one-dimensional integral corresponds to the area under the curve between the imposed limit, as illustrated in Figure 1.11. Multiple integrals are simply extensions of these ideas to more dimensions. We shall illustrate the principles using a function of two variables,/(r,i/). The double integral... 

If the object has an exactly sinusoidal variation of absorption, thickness or refractive index in one dimension, diffracted beams appear only when d sin (/) = A (i.e. m = 1). This is important because of Fourier s Theorem, which states that any (single valued) function of a variable x can be expanded as a sum of sines and cosines of multiples of x. Thus any phase or intensity variation in the sample can be considered as a sum of sinusoidal variations of different wavelength, each giving a certain intensity at a single characteristic angle 0. The intensity at a point in the diffraction pattern corresponds to the strength of a variation of some sample property... 

There are two principal types of integrals of functions of several variables, the line integral and the multiple integral. 

Although other descriptions are possible, the mathematical concept that matches more closely the intuitive notion of smoothness is the frequency content of the function. Smooth functions are sluggish and coarse and characterized by very gradual changes on the value of the output as we scan the input space. This, in a Fourier analysis of the function, corresponds to high content of low frequencies. Furthermore, we expect the frequency content of the approximating function to vary with the position in the input space. Many functions contain high-frequency features dispersed in the input space that are very important to capture. The tool used to describe the function will have to support local features of multiple resolutions (variable frequencies) within the input space. 

A table of correlations between the variables from the instrumental set and variables from the sensory set may reveal some strong one-to-one relations. However, with a battery of sensory attributes on the one hand and a set of instrumental variables on the other hand it is better to adopt a multivariate approach, i.e. to look at many variables at the same time taking their intercorrelations into account. An intermediate approach is to develop separate multiple regression models for each sensory attribute as a linear function of the physical/chemical predictor variables. 

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