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 fromtoIn general both new coordinates will be functions of the old. If we need to find the first derivatives of a functionexpressed in theplane in the new coordinate system. We use the chain rule:
From these we can write down the operatorsand
We can find the operators corresponding to the second derivatives by expanding the brackets: