Question on the transition map of tangent bundle

Question

This is a follow-up question on diffeomorphism of the bundle chart of a Tangent bundle

I don't understand how I can write $\frac{\partial}{\partial x^i} = \frac{\partial y^j}{\partial x^i}\frac{\partial}{\partial y^j}$. Shouldn't it be $\frac{\partial}{\partial x^i} = \sum\limits_{j}\frac{\partial y^j}{\partial x^i}\frac{\partial}{\partial y^j}$, if I were to use chain rule?

How can I see that $\frac {\partial v} {\partial x}$ and $\frac {\partial y} {\partial u}$ is zero?

I would really appreciate any help.

score 2 · Accepted Answer · edited Feb 22 '21 at 02:05

2

The sum over a repeated index is implied. This is known as the Einstein convention.

Note that the $j$ in $$ \frac{\partial}{\partial x^i} = \frac{\partial y^j}{\partial x^i}\frac{\partial}{\partial y^j}$$ is otherwise a "free variable."

edited Feb 22 '21 at 02:05

Kajelad

14,951

answered Feb 22 '21 at 01:55

peter a g

4,923

@Kajelad fyi I basically copied your answer here. I hope you don't mind. – peter a g Feb 22 '21 at 01:56
I don't mind at all; in hindsight It should have been an answer rather than a comment. – Kajelad Feb 22 '21 at 02:07

Question on the transition map of tangent bundle

1 Answers1