Tangent map

Hi,

I am trying to understand the concept of tangent map and following the ebook of S S Chern.
I am a bit confused about the derivation of the tangent map acting on the basis

I tried for sometime to type out the equation but it appears I am having problems with the display and not sure what is being refreshed. Hence I am providing the link

http://www.worldscibooks.com/etextbook/3812/3812_chap1_2.pdf [Broken].

The derivation of equation 2.38 from the second equality to the third is not clear to me.

Could anyone kindly explain.

Thx
Sumanta

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Fredrik
Staff Emeritus
\begin{align*} &\sum_{j=1}^m\bigg\langle\frac{\partial}{\partial u^i},du^j\bigg\rangle\bigg(\frac{\partial F^\alpha}{\partial u^j}\bigg)_p =\sum_{j=1}^m\delta^j_i\bigg(\frac{\partial F^\alpha}{\partial u^j}\bigg)_p =\bigg(\frac{\partial F^\alpha}{\partial u^i}\bigg)_p =\sum_{\beta=1}^n\bigg(\frac{\partial F^\beta}{\partial u^i}\bigg)_p\delta^\alpha_\beta\\ &=\sum_{\beta=1}^n\bigg(\frac{\partial F^\beta}{\partial u^i}\bigg)_p\bigg\langle\frac{\partial}{\partial v^\beta},dv^\alpha\bigg\rangle =\bigg\langle\sum_{\beta=1}^n\bigg(\frac{\partial F^\beta}{\partial u^i}\bigg)_p\frac{\partial}{\partial v^\beta},dv^\alpha\bigg\rangle \end{align*}