How Do You Take the Covariant Derivative of a Tensor Twice?

[Mr-R]

Doing some problems in D'INVERNO GR textbook and I am stuck on taking the covariant derivation of a tensor twice. Please see the attached picture and please do inform me if something is not clear

 

[bloby]

Hello.
Where do e comes from??

Take $A^i=B^i_jC^j$.
j is a dummy index, there is a summation over j.
i can be 1, 2 or 3. This equation tells that the equality is true for all three values of i.
$A^k=B^k_jC^j$ is exactly the same as $A^i=B^i_jC^j$, you must not keep track of indices from one equation to another.

 

[Mr-R]

Heya bloby,

For e, I just chose a new tensor to represent the covariant derivative of the original tensor. Then what should I have named it? T^{a}_{b} ?

Thanks

 

[bloby]

Rather $T^a_d$ the same indices than LHS. The indices are related to basis element. They must be consistent within an equation, like $v^i=\frac{dx^i}{dt}$, not $v^i=\frac{dx^j}{dt}$. The 3rd and 4th line of the thumbnail are the same(after corrections) with renamed indices .

 

[Mr-R]

Rather $T^a_d$ the same indices than LHS. The indices are related to basis element. They must be consistent within an equation, like $v^i=\frac{dx^i}{dt}$, not $v^i=\frac{dx^j}{dt}$. The 3rd and 4th line of the thumbnail are the same(after corrections) with renamed indices .

Much Appreciated bloby

Thanks