Transformations of Basis Vectors on Manifolds

[ehrenfest]

Homework Statement

I am trying to show that
$$\vec{e'}_a = \frac{\partial x^b}{\partial x'^a} \vec{e}_b$$

where the e's are bases on a manifold and the primes mean a change of coordinates
I can get that $$\frac{\partial x^a}{ \partial x'^b} dx'^b \vec{e}_a = dx'^a \vec{e'}_a$$ from the invariance of ds but what should I do next?

Homework Equations

The Attempt at a Solution

 

[George Jones]

Homework Statement

I am trying to show that
$$\vec{e'}_a = \frac{\partial x^b}{\partial x'^a} \vec{e}_b$$

where the e's are bases on a manifold and the primes mean a change of coordinates

You're dealing with coodinates bases, or else your expression above isn't true. Thus

$$\vec{e}_b = \frac{\partial}{\partial x^b}$$

$$\vec{e'}_a = \frac{\partial}{\partial x'^a}.$$

 

[ehrenfest]

You're dealing with coodinates bases, or else your expression above isn't true. Thus

$$\vec{e}_b = \frac{\partial}{\partial x^b}$$

$$\vec{e'}_a = \frac{\partial}{\partial x'^a}.$$

Yes, I am dealing with coordinate bases.

How can you set a basis vector equal to a partial derivative operator?