The good old chain rule can be written as

On page 62 we have

**R***with*^{m}*coordinates**x*,^{a}**R***with*^{n}*coordinates**y*and^{b}**R***with*^{l}*coordinates**z*. And functions^{c}*f*which maps^{b}*x*onto^{a}*y*and^{b}*g*which maps^{c}*y*onto^{b}*z*as shown below.^{c}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.

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