Exploring Tensor Calculus: A Brief Introduction

[trees and plants]

Hello.Questions: How tensor operations are done?Like addition, contraction,tensor product, lowering and raising indices. Why do we need lower and upper indices if we want and not only lower? Is a tensor a multilinear mapping?Or a generalisation of a vector and a matrix? Could a tensor be generalised in some ways?What about derivatives or integrals of tensors? Or taking limits of tensors?

We know there are some tensors used in differential geometry, like the Riemann curvature tensor, the Weyl tensor, The Ricci curvature, the metric tensor,or the stress energy tensor, the electromagnetic tensor, the Einstein tensor, the Einstein metric, what other tensors do you know? We know in vector analysis theorems for mappings of several variables like in vector integral calculus the Gauss theorem or the Stokes theorem, what about generalisations of these in tensor calculus?Can they be generalised?

Thank you.

 

[trees and plants]

I have read somewhere i think about the first order derivative of the Riemann curvature tensor, about the contraction of the Riemann curvature tensor with the metric tensor which gives the Ricci curvature tensor.

 

[martinbn]

So, you want someone to magically answer all that in a post, when it usually takes at least one textbook. Quoting Euclid "There is no royal way to geometry". You need to roll up your sleeves and do the work.

 

[trees and plants]

So, you want someone to magically answer all that in a post, when it usually takes at least one textbook. Quating Euclid "There is no royal way to geometry". You need to roll up your sleeves and study.

I have read some of it, but after i memorise them, i am not sure if i use them correctly or if i make some mistakes.

 

[trees and plants]

I have read about the Ricci flow introduced by Hamilton, about the proof of the Poincare conjecture and its proof, which uses other theorems as well if i know correctly. There are also some publications on arxiv about the applications in physics of the Ricci flow and a generalised Ricci flow if i remember correctly.

 

[trees and plants]

I get stuck after i memorise them on if i use them correctly.

 

[Infrared]

I have read some of it, but after i memorise them, i am not sure if i use them correctly or if i make some mistakes.

Perhaps it would be better to show us one of your attempts to use tensor calculus that you're unsure of.

 

[trees and plants]

Two examples: $R_{jl}=g^{ik}R_{ijkl}$ , $R=g^{ij}R_{ij}$ . Could someone explain these two equations, how the operations were done to have the Ricci curvature tensor and the scalar curvature?

 

[LCSphysicist]

Two examples: $R_{jl}=g^{ik}R_{ijkl}$ , $R=g^{ij}R_{ij}$ . Could someone explain these two equations, how the operations were done to have the Ricci curvature tensor and the scalar curvature?

I am not sure if i understand your question, maybe not, but it is worthy to mention that this operations has a name, https://en.wikipedia.org/wiki/Raising_and_lowering_indices , once you understand the concept it becomes almost immediate.

 

[guru_4]

You can search summation convention. It means sum all possible combinations of indices. https://rpubs.com/lizhi1800/Riemann_curvature_tensor_2nd_kind