alt="theta"/> is:
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:
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
