The analytic method of representing functional relationships

The analytic method of representing functional relationships

type of methods

Every law of nature, connecting certain 
phenomena with others, establishes a functional relationship between magnitudes. There are many ways of representing a functional relationship, but the most important are the three following: 

(1) the analytic,

(2) the tabular,

(3) the graphical or geometrical method. 

We say that a functional relationship between magnitudes, or more simply, a function is represented, analytically, if these magnitudes are connected with each other by equations. These equations contain the magnitudes, subject to the various mathematical operations: addition, subtraction, division, taking logarithms etc. We always arrive at an analytic representation of a function on studying a problem theoretically: which means that, having established basic premises, we make use of mathematical analysis and obtain results in the form of mathematical formula. For instance, in celestial mechanics, all the possible motions, positions and interactions of the heavenly bodies are deduced from a single basic law, that of universal gravitation. If we have a direct expression for the function (i.e. the dependent variable) in terms of mathematical operations on the independent variables, we say that the function is given explicitly. The expression for the volume v of a gas at constant temperature in terms of the pressure is an example of an explicit function

(of one independent variable) :

Similarly, the expression for the area S of a triangle in terms of the sides:

is an example of an explicit function of three independent variables.

Another example may be given of an explicit function of one independent variable:

It is often inconvenient or impossible to write down the formula expressing a function in terms of the independent variables. We can write briefly instead:

This notation means that y is a function of the independent variable x, and f symbolizes the dependence of y on x. Of course another letter can be used in place of f. If we are considering several different functions of x, several different letters must be used to express symbolically the dependence on x:

This notation is not only used when the function is given analytically, but is used in the general case of functional dependence. Use is made of an analogous abbreviated notation for functions of several independent variables:

Here, v is a function of the variables x,y,z. We obtain particular values of functions by giving the independent variables particular values and carrying out the operations indicated by the symbols f, F . . . For example, the particular value of the function (1) for x = 1/2 is:

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek