- #1
rbwang1225
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Homework Statement
Please show that the defining equation for the parallel transport of a contravariant vector along a curve [itex]\dot{\lambda}^a+\Gamma^a_{bc}\lambda^b\dot{x}^c=0[/itex] is coordinate-independent, given that the transformation formula for the christoffel symbol being ##\Gamma^{a'}_{b'c'}=(\Gamma^{d}_{ef}X^{a'}_d-X^{a'}_{ef})X^{e}_{b'}X^{f}_{c'}##.
The Attempt at a Solution
I have stuck by the following derivation ##X^{a'}_{ab}\dot x^b\lambda^a+X^{a'}_a\dot\lambda^a+(\Gamma^{d}_{ef}X^{a'}_d-X^{a'}_{ef})X^{e}_{b'}X^{f}_{c'}(X^{c'}_{ef}\dot x^fx^e+X^{c'}_e\dot x^e)##, where I can't simplify it to the unprimed equation [itex]\dot{\lambda}^a+\Gamma^a_{bc}\lambda^b\dot{x}^c=0[/itex].
Any advice will be appreciated!