Understanding Tangent Map Derivation in S.S. Chern's Ebook

[Sumanta]

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 .

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

Could anyone kindly explain.

Thx
Sumanta

 

[Fredrik]

It took me a while to get used to his notation, but now I see that what he's doing is this:

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

There's a bug that causes the wrong LaTeX images to appear in previews. The only workaround is to refresh and resend after each preview, and sometimes also after saving an edit. (Also note that a closing tex tag looks like this: [noparse][/tex][/noparse]).