Hi. For a diffeomorphism between 2 manifolds Φ : M → N with a tangent vector v in M I have the following equation for the push-forward of v(adsbygoogle = window.adsbygoogle || []).push({});

( Φ_{*}v)f = v( Φ^{*}f) where Φ^{*}is the pull-back. I understand this equation but i have also come across the following equation for the push-forward

(Φ_{*}v)f = (Φ^{-1})^{*}v (Φ*f ) . Surely these 2 equations are not the same. I'm confused.

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# I Confused about push-forwards

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