Quotient Theorem (Tensors)

1. Apr 23, 2009

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?

2. Apr 23, 2009

tiny-tim

hmm … do you mean …

3. Apr 23, 2009

yep.

4. Apr 2, 2011

nhanle

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

5. Apr 2, 2011

tiny-tim

welcome to pf!

hi nhanle! welcome to pf!

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

6. Apr 4, 2011

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

7. Apr 4, 2011

tiny-tim

??

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

8. Apr 5, 2011

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?

9. Apr 5, 2011

tiny-tim

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

if so, that should show you how to do it

10. Apr 5, 2011

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 :(