Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 27

Some problems use functions of two variables that are written in the implicit form:

In this case we have an implicit relationship between the variables x and y. We assume that y is a function of x. The basic result for the differentiation of this implicit function is:

(1.12a)

or:

(1.12b)

We now use this result by posing the following problem. Consider the transformation:

and suppose we wish to transform back:

To this end, we examine the following differentials:

(1.13)

Let us assume that we wish to find dx and dy, given that all other quantities are known. Some arithmetic applied to Equation (1.13) (two equations in two unknowns!) results in:

where J is the Jacobian determinant defined by:

We can thus conclude the following result.

Theorem 1.1 The functions and exist if:

are continuous at (a, b) and if the Jacobian determinant is non-zero at (a, b).

Let us take the example:

You can check that the Jacobian is given by:

Solving for x and y gives:

You need to be comfortable with partial derivatives. A good reference is Widder (1989).