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I am reading John M. Lee's book: Introduction to Smooth Manifolds ...
I am focused on Chapter 3: Tangent Vectors ...
I need some help in fully understanding Lee's conversation on computations with tangent vectors and pushforwards ... in particular I need help with a further aspect of Lee's exposition of pushforwards in coordinates concerning a map [itex]F: M \longrightarrow N[/itex] between smooth manifolds [itex]M[/itex] and [itex]N[/itex] ... ...
The relevant conversation in Lee is as follows:
In the above text, equation 3.7 reads as follows:
" ... ...
[itex]F_* \frac{ \partial }{ \partial x^i } |_p = F_* ( ( \phi^{-1} )_* \ \frac{ \partial }{ \partial x^i } |_{\phi(p)}) [/itex][itex]= ( \psi^{-1} )_* \ ( \tilde{F}_* \frac{ \partial }{ \partial x^i } |_{\phi(p)} )[/itex][itex]= ( \psi^{-1}_* ) ( \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p}) \frac{ \partial }{ \partial y^j }|_{ \tilde{F} ( \phi (p))} )[/itex][itex]= \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p} ) \frac{ \partial }{ \partial y^j }|_{ F (p) }[/itex]... ... ... ... ... 3.7
... ... ... "
I cannot see how Equation 3.7 is derived ... can someone please help ...
Specifically, my questions are as follows:Question 1
What is the explicit logic and justification for the step
[itex]F_* ( ( \phi^{-1} )_* \ \frac{ \partial }{ \partial x^i } |_{\phi(p)})[/itex] [itex]= ( \psi^{-1} )_* \ ( \tilde{F}_* \frac{ \partial }{ \partial x^i } |_{\phi(p)} )[/itex]
Question 2
What is the explicit logic and justification for the step[itex]= ( \psi^{-1} )_* \ ( \tilde{F}_* \frac{ \partial }{ \partial x^i } |_{\phi(p)} )[/itex][itex]= ( \psi^{-1}_* ) ( \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p}) \frac{ \partial }{ \partial y^j }|_{ \tilde{F} ( \phi (p))} )[/itex]
Question 3
What is the explicit logic and justification for the step
[itex] ( \psi^{-1}_* ) ( \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p}) \frac{ \partial }{ \partial y^j }|_{ \tilde{F} ( \phi (p))} )[/itex][itex]= \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p} ) \frac{ \partial }{ \partial y^j }|_{ F (p) }[/itex]
As you can see ... I am more than slightly confused by equation 3.7 ... hope someone can help ...Peter
===========================================================
To give readers the notation and context for the above I am providing the text of Lee's section on Computations in Coordinates (pages 69 -72) ... ... as follows:
I am focused on Chapter 3: Tangent Vectors ...
I need some help in fully understanding Lee's conversation on computations with tangent vectors and pushforwards ... in particular I need help with a further aspect of Lee's exposition of pushforwards in coordinates concerning a map [itex]F: M \longrightarrow N[/itex] between smooth manifolds [itex]M[/itex] and [itex]N[/itex] ... ...
The relevant conversation in Lee is as follows:
" ... ...
[itex]F_* \frac{ \partial }{ \partial x^i } |_p = F_* ( ( \phi^{-1} )_* \ \frac{ \partial }{ \partial x^i } |_{\phi(p)}) [/itex][itex]= ( \psi^{-1} )_* \ ( \tilde{F}_* \frac{ \partial }{ \partial x^i } |_{\phi(p)} )[/itex][itex]= ( \psi^{-1}_* ) ( \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p}) \frac{ \partial }{ \partial y^j }|_{ \tilde{F} ( \phi (p))} )[/itex][itex]= \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p} ) \frac{ \partial }{ \partial y^j }|_{ F (p) }[/itex]... ... ... ... ... 3.7
... ... ... "
I cannot see how Equation 3.7 is derived ... can someone please help ...
Specifically, my questions are as follows:Question 1
What is the explicit logic and justification for the step
[itex]F_* ( ( \phi^{-1} )_* \ \frac{ \partial }{ \partial x^i } |_{\phi(p)})[/itex] [itex]= ( \psi^{-1} )_* \ ( \tilde{F}_* \frac{ \partial }{ \partial x^i } |_{\phi(p)} )[/itex]
Question 2
What is the explicit logic and justification for the step[itex]= ( \psi^{-1} )_* \ ( \tilde{F}_* \frac{ \partial }{ \partial x^i } |_{\phi(p)} )[/itex][itex]= ( \psi^{-1}_* ) ( \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p}) \frac{ \partial }{ \partial y^j }|_{ \tilde{F} ( \phi (p))} )[/itex]
Question 3
What is the explicit logic and justification for the step
[itex] ( \psi^{-1}_* ) ( \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p}) \frac{ \partial }{ \partial y^j }|_{ \tilde{F} ( \phi (p))} )[/itex][itex]= \frac{ \partial \tilde{F}^j }{ \partial x^i } ( \tilde{p} ) \frac{ \partial }{ \partial y^j }|_{ F (p) }[/itex]
As you can see ... I am more than slightly confused by equation 3.7 ... hope someone can help ...Peter
===========================================================
To give readers the notation and context for the above I am providing the text of Lee's section on Computations in Coordinates (pages 69 -72) ... ... as follows: