On page 116 of Choquet-Bruhat,

for a Lie group [itex]G[/itex]

[itex]

f:G \rightarrow G; x \mapsto x^{-1}

[/itex]

is differentiable. I know from the previous definitions that a function [itex] f [/itex] on a manifold is differentiable at [itex]x[/itex] if

[itex]

\psi \circ f \circ \phi^{-1}

[/itex]

is differentiable, where [itex](U, \phi)[/itex] and [itex](W, \psi)[/itex] are charts for neighborhoods of [itex] x [/itex] and [itex]y=f(x)=x^{-1}[/itex]. It's probably an indication of how weak my analysis is, but I don't see how to proceed from there. Can anyone point me in the right direction?

*Analysis, Manifolds, and Physics*, Lie groups are defined, and the first exercise after that asks you to prove thatfor a Lie group [itex]G[/itex]

[itex]

f:G \rightarrow G; x \mapsto x^{-1}

[/itex]

is differentiable. I know from the previous definitions that a function [itex] f [/itex] on a manifold is differentiable at [itex]x[/itex] if

[itex]

\psi \circ f \circ \phi^{-1}

[/itex]

is differentiable, where [itex](U, \phi)[/itex] and [itex](W, \psi)[/itex] are charts for neighborhoods of [itex] x [/itex] and [itex]y=f(x)=x^{-1}[/itex]. It's probably an indication of how weak my analysis is, but I don't see how to proceed from there. Can anyone point me in the right direction?

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