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

Table of contents
- Solution ii

Solution

At a point where

X \neq = 0

, the flow-box theorem supplies a transverse hypersurface with coordinates

x^{i}

. Flow each point on it for parameter

t

and keep the

x^{i}

constant along the flow. In the resulting coordinates,

X = \frac{\partial}{\partial t} .

Because the coordinate basis is transported by this flow, the Lie derivative of any tensor is obtained by differentiating its coordinate components:

L_{X} T = \partial_{t} T^{μ_{1} \dots}_{ν_{1} \dots} the unchanged coordinate basis tensors .

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