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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 309

For a function, pullback by the flow gives
$$
\mathcal L_Xf=\frac{d}{ds}\bigg|_0 f(\Phi_s(\,\cdot\,))=X(f).
$$
In flow-box coordinates $X=\partial_t$, write $Y=Y^\mu\partial_\mu$. Then
$$
\mathcal L_XY=(\partial_tY^\mu)\partial_\mu
=[\partial_t,Y]=[X,Y].
$$
Both sides are vector fields defined invariantly, so equality in these coordinates proves
$$
\boxed{\mathcal L_Xf=X(f),\qquad \mathcal L_XY=[X,Y]}
$$
in every coordinate system.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 1 / a / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 1 / a / iii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 1 / a / iii / Solution

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