# Covariant differentiation twice

1. Jul 9, 2014

### 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

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2. Jul 11, 2014

### 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.

3. Jul 11, 2014

### 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

4. Jul 11, 2014

### 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 .

5. Jul 11, 2014

### Mr-R

Much Appreciated bloby

Thanks