# Basis vectors definition

1. Aug 25, 2013

### Siberion

1. The problem statement, all variables and given/known data

A vector is a geometrical object which doesn't depend on the basis we use to represent it, only its components will change. We can express this by $\vec{A}=ƩA_i \hat{ε_i} = Ʃ\tilde{A_i} \vec{ε_i}$, where it has been emphasized that the basis ε is not necessarily orthonormal. Prove that the normalized vectors $\hat{ε_i}$ are given by:

$$\hat{ε_i}=\frac{\frac{\partial{\vec{A}}}{\partial{\tilde{A_i}}}}{|\frac{\partial{\vec{A}}}{\partial{\tilde{A_i}}}|}$$

Use this to find expressions for the normalized vectors $\hat{r}.\hat{\theta},\hat{\phi}$ of spherical coordinates.

2. Relevant equations

3. The attempt at a solution

I honestly don't have any idea of where should I start. This could be very easy but I'm kind of lost at the moment. Should I try working with the Jacobian matrix somehow?

Thanks for your help, I'll try to get an answer in the meantime and will edit the post if anything comes to my mind.