Exploring Tensor Calculus: A Brief Introduction

In summary, the conversation discusses the use of tensor operations such as addition, contraction, tensor product, lowering and raising indices, and their role in differential geometry. It also explores the concept of tensors as multilinear mappings and generalizations of vectors and matrices. The conversation mentions the use of tensors in various theorems and calculations, such as the Riemann curvature tensor, the Ricci curvature tensor, and the scalar curvature. The need for understanding these operations and their applications is emphasized, as well as the importance of studying and practicing to use them correctly.
  • #1
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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
martinbn said:
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.
 
  • #5
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.
 
  • #6
I get stuck after i memorise them on if i use them correctly.
 
  • #7
infinitely small said:
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.
 
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  • #8
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?
 
  • #9
infinitely small said:
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.
 

1. What is tensor calculus?

Tensor calculus is a branch of mathematics that deals with the study of tensors, which are mathematical objects that describe linear relations between different sets of data. It combines concepts from linear algebra, calculus, and geometry to solve problems related to physical systems.

2. Why is tensor calculus important in science?

Tensor calculus is important in science because it provides a powerful mathematical framework for describing and analyzing physical systems. It is used in various fields such as physics, engineering, and computer science to model and understand complex systems and phenomena.

3. What are some real-world applications of tensor calculus?

Tensor calculus has many real-world applications, including but not limited to:

  • General relativity: Tensors are used to describe the curvature of spacetime in Einstein's theory of general relativity.
  • Fluid mechanics: Tensors are used to describe the velocity, pressure, and stress of fluids in motion.
  • Image and signal processing: Tensors are used to analyze and manipulate digital images and signals.
  • Machine learning: Tensors are used in deep learning algorithms to process and analyze large datasets.

4. What are the basic operations in tensor calculus?

The basic operations in tensor calculus include:

  • Addition and subtraction: Tensors can be added or subtracted if they have the same dimensions and order.
  • Multiplication: Tensors can be multiplied by scalars or other tensors according to certain rules.
  • Contractions: This operation involves summing over indices of a tensor to produce a new tensor with fewer indices.
  • Covariant and contravariant derivatives: These operations are used to differentiate tensors with respect to a particular coordinate system.

5. Are there any resources available to learn tensor calculus?

Yes, there are many resources available to learn tensor calculus, including textbooks, online courses, and video tutorials. Some popular books on the subject include "Tensor Calculus for Physics" by Dwight E. Neuenschwander and "A Student's Guide to Vectors and Tensors" by Daniel Fleisch. Additionally, there are many online resources and forums where one can find helpful explanations and practice problems for tensor calculus.

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