we can write the differentials of and in terms of the differentials of and :
It turns out that the chain rule also tells us how to rewrite differential operators in terms of the variables. But these go in the other direction. That is, we can write the differential operators and in terms of the operators and .
First of all, let’s write down the differential of in terms of and and in terms of and :
and now we can rewrite and in terms of and .
Now by uniqueness we can read off the partial derivatives of in terms of and :
Finally, we pull all mention of out of our notation and just write out the differential operators.
Now we’re done rewriting, but for good form we should express these coefficients in terms of and .
It’s important to note that there’s really no difference between these last two steps. The first one uses the variables and while the second uses the variables and , but they express the exact same functions, given the original substitutions above.
More generally, let’s say we have a vector-valued function defining a substitution
Cauchy’s invariant rule tells us that this gives rise to a substitution for differentials.
We can play it a little loose and write this out in matrix notation:
Now if we have a function in terms of the variables, we can use the substitution above to write it as a function of the variables. We can write the differential of in terms of each
Next we use the substitutions of the differentials to rewrite the first form as
Then uniqueness allows us to match up the coefficients and write out the partial derivatives in terms of the variables
It is in this form that the chain rule is most often introduced, or the similar form
And now we can remove mention of from the formulæ and speak directly in terms of the operators
Again, we can play it a little loose and write this in matrix notation
This is very similar to the substitution for differentials written in matrix notation. The differences are that we transform from -derivations to -derivations instead of from -differentials to -differentials, and the two substitution matrices are the transposes of each other. Those who have been following closely (or who have some background in differential geometry) should start to see the importance of this latter fact, but for now we’ll consider this a statement about formulas and methods of calculation. We’ll come to the deeper geometric meaning when we come through again in a wider context.