Thursday, 19 July 2018

Commentary 2.2 Chain rule in multiple dimensions

The good old chain rule can be written as 
On page 62 we have Rm with coordinates xa, Rn with coordinates yb and Rl with coordinates zc. And functions fb which maps xa onto yb and gc which maps yb  onto zc 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.

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