When we transform the variables in an equation and wish to find the relationship between derivatives expressed in old and new coordinates, especially when there are more than one coordinates involved in both the old and new coordinate system, we need to use operator algebra.
Suppose we transform from
to
In general both new coordinates will be functions of the old. If we need to find the first derivatives of a function
expressed in the
plane in the new coordinate system. We use the chain rule:
and
From these we can write down the operators
and
and
We can find the operators corresponding to the second derivatives by expanding the brackets:
