Consider how a vector ##\bf{v}##at ##x## of(adsbygoogle = window.adsbygoogle || []).push({}); ##M##, as a differential operator, acts on the pull-back of a function:

##{\bf{v}}(F^{*}f) = (F_{*}{\bf{v}})(f) = df(F_{*}{\bf{v}})##

I was wondering how to relate this to the following?

##\displaystyle{{\bf{v}}(F^{*}f)={\bf{v}}[f\{y(x)\}]=v^{i}\frac{\partial}{\partial x^{i}}[f\{y(x)\}]=v^{i}\bigg(\frac{\partial y^{j}}{\partial x^{i}}\bigg)\bigg(\frac{\partial f}{\partial y^{j}}\bigg)}##

I ask this question because ##\displaystyle{F_{*}{\bf{v}}=\bigg(\frac{\partial y^{j}}{\partial x^{i}}\bigg)v^{i}}##, with the implicit Einstein summation convention,

so that ##\displaystyle{(F_{*}{\bf{v}})(f)=v^{i}\bigg(\frac{\partial y^{j}}{\partial x^{i}}\bigg)f \neq v^{i}\bigg(\frac{\partial y^{j}}{\partial x^{i}}\bigg)\bigg(\frac{\partial f}{\partial y^{j}}\bigg)}##

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# A Action of a vector on the pull-back of a function

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