This is a fundamental result that we shall apply in this chapter. We take a simple example of Equation (1.11) to show how things work. To this end, consider the Laplace equation in Cartesian geometry:
We now wish to transform this equation into an equation in a circular region defined by the polar coordinates:
The derivative in r is given by:
and you can check that the derivative with respect to
hence:
and:
Combining these results allows us to write Laplace's equation in polar coordinates as follows:
Thus, the original heat equation in Cartesian coordinates is transformed to a PDE of convection-diffusion type in polar coordinates.
We can find a solution to this problem using the Separation of Variables method, for example.
1.5 FUNCTIONS AND IMPLICIT FORMS
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: