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General Relativity For Tellytubbys

Miscellaneous Mathematics

Sir Kevin Aylward B.Sc., Warden of the Kings Ale

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This section is a refresher to bits of stuff that, if you don't already know, you had better resign yourself to flipping hamburgers for one or two years yet, before you have any hope of inventing a new Warp Drive. Not all of the details will be filled in cos I cannot be bothered.

Derivative Chain Rule

Consider a function that is a function of a function i.e.

Forming the derivative of this gives

Now suppose that the function is a function of more then one function

With a bit of piddling about, using the standard derivative of product rules, the partial derivative can be found to be

Or in more general terms, for an arbitrary number of functions, F can be written as:

then

This can be written more compactly as:

Where it is now assumed that repeated index's in a product will be summed, even though no sigma sign is shown.

Finally, if F is a set of function of set of variables x, y, z, one can generally write

or

Then, it can be written

dF d x β = F X α X α x β MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaiOraaqaaiaadsgacaWG4bWaaWbaaSqabeaacqaHYoGyaaaaaOGaeyypa0ZaaSaaaeaacqGHciITcaGGgbaabaGaeyOaIyRaaeiwamaaCaaaleqabaGaeqySdegaaaaakmaalaaabaGaeyOaIyRaaeiwamaaCaaaleqabaGaeqySdegaaaGcbaGaeyOaIyRaamiEamaaCaaaleqabaGaeqOSdigaaaaaaaa@4B2A@

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