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## Main Question or Discussion Point

Greetings,

I have just started studying manifolds, and have come across the idea that the basis vectors can be expressed as:

I tried to convince myself of this in 2D Cartesian coordinates using a pretty non-rigorous derivation (the idea being to get a simple intuitive grasp rather than an abstract mathematical one) starting with the position vector, as follows:

d

Expand with the product rule, everything other than [itex]\partial[/itex]x/[itex]\partial[/itex]x

d

My first question: I would conclude that

My second question: Is there an equivalent expression to

Thanks for any help you can give.

-HJ Farnsworth

I have just started studying manifolds, and have come across the idea that the basis vectors can be expressed as:

**e**_{[itex]\mu[/itex]}= [itex]\partial[/itex]/[itex]\partial[/itex]x^{[itex]\mu[/itex]}.I tried to convince myself of this in 2D Cartesian coordinates using a pretty non-rigorous derivation (the idea being to get a simple intuitive grasp rather than an abstract mathematical one) starting with the position vector, as follows:

**r**(x,y) = x**e**_{1}+ y**e**_{2}d

**r**= [itex]\partial[/itex]**r**/[itex]\partial[/itex]x dx + [itex]\partial[/itex]**r**/[itex]\partial[/itex]y dy = [itex]\partial[/itex](x**e**_{1}+y**e**_{2})/[itex]\partial[/itex]x dx + [itex]\partial[/itex](x**e**_{1}+y**e**_{2})/[itex]\partial[/itex]y dyExpand with the product rule, everything other than [itex]\partial[/itex]x/[itex]\partial[/itex]x

**e**_{1}dx and [itex]\partial[/itex]y/[itex]\partial[/itex]y**e**_{2}dy (each partial combination is obviously 1) goes to 0 since the Cartesian basis vectors are constant and dx and dy are independent, so that:d

**r**= [itex]\partial[/itex]**r**/[itex]\partial[/itex]x dx + [itex]\partial[/itex]**r**/[itex]\partial[/itex]y dy =**e**_{1}dx +**e**_{2}dyMy first question: I would conclude that

**e**_{[itex]\mu[/itex]}= [itex]\partial[/itex]**r**/[itex]\partial[/itex]x^{[itex]\mu[/itex]}, rather than**e**_{[itex]\mu[/itex]}= [itex]\partial[/itex]/[itex]\partial[/itex]x^{[itex]\mu[/itex]}. What did I miss here?My second question: Is there an equivalent expression to

**e**_{[itex]\mu[/itex]}= [itex]\partial[/itex]/[itex]\partial[/itex]x^{[itex]\mu[/itex]}for basis one-forms? If so, could anyone please provide a quasi-derivation similar to mine above, except correct in the way mine was wrong?Thanks for any help you can give.

-HJ Farnsworth