Commentary 2.2 Chain rule in multiple dimensions

The good old chain rule can be written as 

On page 62 we have R^m with coordinates x^a, Rⁿ with coordinates y^b and R^l with coordinates z^c. And functions f^b which maps x^a onto y^b and g^c which maps y^b  onto z^c as shown below.

Picture copied from book. Thanks.

At equation (2.12) Carroll states that the chain rule relates the partial derivative of a composition to the partial derivatives of its individual maps, so:

I'm not quite sure where the c index should go, but the left hand side is the same as 

so (2.12) looks like (2) apart from the summation over b. I was a bit surprised by this and wondered about the summation.

In fact the right hand side of (2.12) is a chain rule generalised for functions of multiple variables. I eventually found it discussed on Wikipedia under the Chain rule - Higher dimensions. By a horrible change of variables I could show that Carroll and Wikipedia were talking the same language.

On the way I thoroughly refreshed my mind on differentiation of polynomials, showing that a sample set of polynomials could be differentiated by the chain rule or by simple expansion - giving the same result. Read here if you like.