- #1

HJ Farnsworth

- 128

- 1

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 dy

Expand 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}dy

My 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