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It's obvious that the Lie derivative defined in terms of Lie brackets is anti-symmetric. But what is an intuitive way to visualize the anti-symmetry in the 'differentiating along a flow' definition?

Thanks!

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up vote 5 down vote accepted

Whichever definition you use that involves differentiating along a flow, it ought to have the property that $L_X X = 0$; in other words, for reasons that should be clear, the Lie derivative is alternating. In the presence of bilinearity, this is equivalent to antisymmetry.

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