Riemann Curvature Tensor - 1

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This video looks at one method for deriving the Riemann Curvature tensor using covariant differentiation along different directions on a manifold.

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These videos are transparent with all the details given. It's nice that you chose a manifold with two directions. This can generalize to manifolds to n directions. However this is a nice way to produce the Riemann curvature tensor. Being concise, precise, exact and direct is the goal of what mathematics should be.

I'm already subscribed. Keep adding those rich details in your explanations. At least few are brave enough to discuss the details in full glory.

kummer

Hey Robert, I don't understand the notation of the covariant derivative. Nabla_alpha does mean we differentiate with respect to the fixed coordinate alpha, right? What does it mean without a subscript? You defined this version as e^alpha*del/del(x^alpha) which can be written as e^alpha*del_alpha, right? But that would mean we SUM over the basis vectors times derivatives in all different directions. Where does my understanding fail?

LUXi

Hi. Very. Nice. I have a question. Why. Take the. Tensor product?

georgeveropoulos

I appreciated, but I don't know why, at line 3 in slide 7, 10 terms out of 14 of them in the block are disappeared and only 4 terms are remained.

gsgp

Thank you for this video. However I am confused at 4.31 where you have treated '1' as a dummy variable and swapped mu for 1 in the second term compared to the line above. I do not see how that is an allowed procedure because surely one now has the direction of the covector restricted to the e^1 direction whereas in the line above it is a sum over all possible e^mu covectors.

I suspect my question demonstrates a fundamental misunderstanding on my part and so I hope you can explain.

Thanks,

Peter

petershotts

Hello, Sir.
I didn't understand how did you do the difference between de two tensors, because I thought that the tensor product was non-commutative.
Thanks for your attention

tomascampo

At 1:40, We see that the differential operator has a contravariant basis factor. Can you explain why this is contravariant instead of covariant basis? Please.

drlangattxdotnet

Dear sir, could you explain how to transform the 2nd term of line 4 on slide 6 to that of line 5? Or help to provide a source that offer the explanation? Many thanks and appreciate it.

cwc

Sorry, I figured out the reason of what I posted in the previous comment.

gsgp

Riemann Curvature Tensor - 1

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