Is the Quotient Theorem Applicable to 4th Rank Tensors?

[latentcorpse]

Prove that b_{ijkl}=\int_{r<a} dV x_i x_j \frac{\partial^2}{\partial_k \partial_l} (\frac{1}{r}) where r=|x| is a 4th rank tensor.

i've had a couple of bashes and got nowhere other than to establish that its quotient theorem.

can i just pick a tensor of rank 3 to multiply it with or something?

 

[tiny-tim]

hmm … do you mean …

Prove that b_ijkl = ∫_r<a dV x_i x_j ∂²(1/r)/∂_k∂_l, where r=|x|, is a 4th rank tensor.

 

[latentcorpse]

yep.

 

[nhanle]

I have no idea how to solve this too, can you give me some idea please?

 

[tiny-tim]

welcome to pf!

hi nhanle! welcome to pf!

ok, what is the test for something being a tensor?

 

[nhanle]

hi tiny-tim,
thank you for your reply. This is how vague the definition of tensor I am holding at the moment.
I am also confused about the Affine connection. Can you help me clarify this?

Thank you

 

[tiny-tim]

??

i'm not going to type out a lecture on tensors and connections

please go back to your book or your lecture notes, and read up about tensors

 

[nhanle]

those appears on my lecture notes and also my book (general relativity - M.P.Hobson, G. Efstathiou, A.N. Lasenby) with very vague definitions.

From my understanding, if one is to be a rank N-tensor, it should expect to have N derivative summations under coordinate transformation. Is that right?

 

[tiny-tim]

does your book show why the Christoffel symbols aren't tensors?

if so, that should show you how to do it ​

 

[nhanle]

it does but only with a few special case. So, I stumpled on this question "Prove that bijkl = ∫r<a dV xi xj ∂2(1/r)/∂k∂l, where r=|x|, is a 4th rank tensor."

How to transform the partial derivatives? Thank you for being so patient with me
I also have question about the affine connection https://www.physicsforums.com/showthread.php?t=189456 which was raised long ago but no one seems to be interested in answering :(