How does ∂aAb behave under coordinate transformations in special relativity? Work out ∂'aA'b
The Attempt at a Solution
I have been given back the solution sheet to this problem, but I don't understand it. This is what I have
I get the first line, (13). We have treated the partial as a covariant vector, and written out the usual transform for it. The part within the brackets is the normal transform of a contravariant vector A.
Looking at (14), the text says we have used the product rule here. I am familiar with this in normal calculus, where if k(x) = f(x)g(x) then k'(x) = f'(x)g(x) + f(x)g'(x). If I apply that here, I would expect to see ∂'aAb + ∂aA'b
Written out long hand, this would be similar to (14). However, the solution has an extra ∂Xd/∂X'a in the second term, and that weird ∂∂X'b thing in the first term.
For the second term in (14) we have every item in (13) just multiplied together in a different order; and the first term in (14) is the same, but with two of those items conflated to form the ∂∂X'b strangeness. Are they multiplied together in some sense, or is this now a second order partial derivative term? What on Earth is going on? If someone could post up a clear rule for how to approach this type of problem it would be great, as it will keep cropping up in my course and I am totally clueless how else to tackle it.
My tutor advised me to take a look at the wikipedia link for the chain rule, but I can't see how that is relevant to this problem, although I see it is different in higher dimensions to what I am used to. I have also searched to see if the product rule is different in higher dimensions, but it seems to work just the same for vectors as for normal functions.