Basis vectors definition

[Siberion]

Homework Statement

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.

Homework Equations

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.

 

[mighty2000]

Thank you for your question. You are correct in thinking that working with the Jacobian matrix will be helpful in proving the given expression for the normalized vectors. Let me guide you through the steps to solve this problem.

First, let's recall the definition of a vector in terms of its components and basis vectors:

\vec{A}=ƩA_i \hat{ε_i} = Ʃ\tilde{A_i} \vec{ε_i}

Where \hat{ε_i} and \vec{ε_i} are the basis vectors and A_i and \tilde{A_i} are the components of the vector A in the original and new basis, respectively.

Next, we can write the Jacobian matrix for the transformation of basis vectors as:

J=\begin{bmatrix} \frac{\partial{\hat{ε_1}}}{\partial{\tilde{ε_1}}}& \frac{\partial{\hat{ε_1}}}{\partial{\tilde{ε_2}}}&...&\frac{\partial{\hat{ε_1}}}{\partial{\tilde{ε_n}}}\\ \frac{\partial{\hat{ε_2}}}{\partial{\tilde{ε_1}}}& \frac{\partial{\hat{ε_2}}}{\partial{\tilde{ε_2}}}&...&\frac{\partial{\hat{ε_2}}}{\partial{\tilde{ε_n}}}\\ ...&...&...&...\\ \frac{\partial{\hat{ε_n}}}{\partial{\tilde{ε_1}}}& \frac{\partial{\hat{ε_n}}}{\partial{\tilde{ε_2}}}&...&\frac{\partial{\hat{ε_n}}}{\partial{\tilde{ε_n}}}\end{bmatrix}

Now, we can use the chain rule to express the basis vectors \hat{ε_i} in terms of the components \tilde{A_i} as follows:

\frac{\partial{\hat{ε_i}}}{\partial{\tilde{A_j}}}=\frac{\partial{\hat{ε_i}}}{\partial{\hat{ε_j}}}\frac{\partial{\hat{ε_j}}}{\partial{\tilde{A_j}}}=\frac{\partial{\hat{ε_i}}}{\partial{\hat{ε_j}}