General Relativity For
Tellytubbys
Miscellaneous Mathematics
Sir Kevin Aylward B.Sc.,
Warden of the Kings Ale
Back
to the Contents section
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.
F=F(V(x))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOraiabg2da9iaacAeacaGGOaGaaiOvaiaacIcacaWG4bGaaiykaiaacMcaaaa@3D17@
Forming the derivative of
this gives
dF
dx
=
F(V(x+Δx)−F(V(x))
Δx
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaiOraaqaaiaadsgacaWG4baaaiabg2da9maalaaabaGaaiOraiaacIcacaGGwbGaaiikaiaadIhacqGHRaWkcqqHuoarcaWG4bGaaiykaiabgkHiTiaacAeacaGGOaGaaiOvaiaacIcacaWG4bGaaiykaiaacMcaaeaacqqHuoarcaWG4baaaaaa@4B41@
aslimΔx−>0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadohaciGGSbGaaiyAaiaac2gacqqHuoarcaWG4bGaeyOeI0IaeyOpa4JaaGimaaaa@3FB4@
dF
dx
=
F(V(x+Δx)−F(V(x))
Δv
.
Δv
Δx
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaiOraaqaaiaadsgacaWG4baaaiabg2da9maalaaabaGaaiOraiaacIcacaGGwbGaaiikaiaadIhacqGHRaWkcqqHuoarcaWG4bGaaiykaiabgkHiTiaacAeacaGGOaGaaiOvaiaacIcacaWG4bGaaiykaiaacMcaaeaacqqHuoarcaWG2baaaiaac6cadaWcaaqaaiabfs5aejaadAhaaeaacqqHuoarcaWG4baaaaaa@50C5@
limΔx−>0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaacMgacaGGTbGaeuiLdqKaamiEaiabgkHiTiabg6da+iaaicdaaaa@3DD6@
dF
dx
=
∂F
∂V
∂V
∂x
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaiOraaqaaiaadsgacaWG4baaaiabg2da9maalaaabaGaeyOaIyRaaiOraaqaaiabgkGi2kaabAfaaaWaaSaaaeaacqGHciITcaqGwbaabaGaeyOaIyRaamiEaaaaaaa@43D4@
Now suppose that the
function is a function of more then one function
F=F(V(x),U(x))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOraiabg2da9iaacAeacaGGOaGaaiOvaiaacIcacaWG4bGaaiykaiaacYcacaqGvbGaaiikaiaadIhacaGGPaGaaiykaaaa@40F5@
With a bit of piddling
about, using the standard derivative of product rules, the partial derivative
can be found to be
dF
dx
=
∂F
∂V
∂V
∂x
+
∂F
∂U
∂U
∂x
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaiOraaqaaiaadsgacaWG4baaaiabg2da9maalaaabaGaeyOaIyRaaiOraaqaaiabgkGi2kaabAfaaaWaaSaaaeaacqGHciITcaqGwbaabaGaeyOaIyRaamiEaaaacqGHRaWkdaWcaaqaaiabgkGi2kaacAeaaeaacqGHciITcaqGvbaaamaalaaabaGaeyOaIyRaaeyvaaqaaiabgkGi2kaadIhaaaaaaa@4DE5@
Or in more general terms,
for an arbitrary number of functions, F can be written as:
F=F(
X
α
(x))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOraiabg2da9iaacAeacaGGOaGaaiiwamaaCaaaleqabaGaeqySdegaaOGaaiikaiaadIhacaGGPaGaaiykaaaa@3EEF@
then
dF
dx
=
∂F
∂
X
1
∂
X
1
∂x
+
∂F
∂
X
2
∂
X
2
∂x
∂F
∂
X
3
∂
X
3
∂x
+...
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5FA4@
This can be written more
compactly as:
dF
dx
=
∂F
∂
X
α
∂
X
α
∂x
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaiOraaqaaiaadsgacaWG4baaaiabg2da9maalaaabaGaeyOaIyRaaiOraaqaaiabgkGi2kaabIfadaahaaWcbeqaaiabeg7aHbaaaaGcdaWcaaqaaiabgkGi2kaabIfadaahaaWcbeqaaiabeg7aHbaaaOqaaiabgkGi2kaadIhaaaaaaa@4784@
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
F=F(
X
α
(
x
1
,
x
2
,
x
3
,...))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOraiabg2da9iaacAeacaGGOaGaaiiwamaaCaaaleqabaGaeqySdegaaOGaaiikaiaadIhadaahaaWcbeqaaiaaigdaaaGccaGGSaGaamiEamaaCaaaleqabaGaaGOmaaaakiaacYcacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaaiilaiaac6cacaGGUaGaaiOlaiaacMcacaGGPaaaaa@47E8@
or
F=F(
X
α
(
x
β
))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOraiabg2da9iaacAeacaGGOaGaaiiwamaaCaaaleqabaGaeqySdegaaOGaaiikaiaadIhadaahaaWcbeqaaiabek7aIbaakiaacMcacaGGPaaaaa@40C7@
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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2022
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