Implicit functions

Implicit functions

A function is called implicit when we have no direct analytic expression for it in terms of the independent variables, but only an equation relating its values to those of the independent variables. For instance, if a variable y is related to a variable x by the equation:

y is an implicit function of the independent variable x or on the other hand, x can be reckoned an implicit function of the independent variable y. An implicit function v of several independent variables x, y, z, . . . is defined in general by an equation:

We can only compute the value of this function by solving the equation with respect to v, thus putting v in the form of an explicit function of x, y, z, . . . :

In the above example, y is expressed in terms of x as:

However, it is by no means essential to solve the equation to obtain the various properties of the function v; often, an implicit function can be studied quite well from the equation that defines it, without attempting a solution. The volume v of a gas, for example, is an implicit function of the pressure p and temperature T, defined by the equation:

The angle c between sides a and b of a triangle of area S is an implicit function of a, b, S, defined by the equation:

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek