Prove Tensor Analysis Relation: Γᵢₖᵣ = ∊ᵢ • ∂∊ⱼ/∂qᵏ

[Arham]

Hello. I'm learning tensor analysis. I have a problem. We know that

$$\Gamma^i_{jk}=\vec{\epsilon^i}\cdot\frac{\partial\vec{\epsilon_j}}{\partial q^k}$$

Please prove the relation

$$\frac{\partial\vec{\epsilon_j}}{\partial q^k}=\Gamma^m_{jk}\vec{\epsilon_m}$$

Thanks very much in advance

 

[belliott4488]

What'll you pay me?

 

[hanskuo]

Hello. I'm learning tensor analysis. I have a problem. We know that

$$\Gamma^i_{jk}=\vec{\epsilon^i}\cdot\frac{\partial\vec{\epsilon_j}}{\partial q^k}$$

Please prove the relation

$$\frac{\partial\vec{\epsilon_j}}{\partial q^k}=\Gamma^m_{jk}\vec{\epsilon_m}$$

Thanks very much in advance

$$\Gamma^m_{jk}\vec{\epsilon_m}\cdot\vec{\epsilon^i} =\Gamma^m_{jk}\delta^i{}_m=\Gamma^i_{jk}$$

 

[Arham]

Thanks hanskuo.

I knew this proof, but I thought that it is only correct for the inverse relation. I was wrong!

 

[hanskuo]

you are wellcome, Arham

Now I'm learning Differential Geometry,too.
do you begin to lerane covariant derivatives or not ?

 

[Arham]

I'm an undergraduate physics student, hanskuo. I am learning tensor analysis from George Arfken's book. As you know, this book has a brief introduction to Covariant Derivative; I have read it. But I should do more exercises and read more about it in future.

 

[hanskuo]

There are a lot of things interesting for covariant derivatives.
your original question likes this:
$$\nabla_{e_i}e_j=\Gamma^k{}_{ij}e_k$$