# Understanding Tensors for Beginners in Special Relativity

• aeroboyo
In summary, this online book looks pretty good, but you do not need to know about tensors to learn about special relativity. There are other books that cover tensors and manifolds in more detail.
aeroboyo
Hello,

I'd like to learn about tensors so i can start learning about special relativity. I understand nothing right now about tensors, except that they mean different things to mathematicians and to physicists, which is where my confusion begins!

Should i learn about the modern way of doing tensors without coordinates, or should i learn the 'classic method'. I'm very confused. Anyone know any good websites with basic lectures or notes on tensors? Also, does any1 happen to have msn so i might be able to ask a few questions in real time as they prop up? Thank you.

wow that online book is complicated! I guess I've been living on the moon all my life.

one of those online books makes more sense... is an understanding of linear algebra is necessary to begin learning tensor calculus? I'm a 1st year engineering student just now and the linear algebra course is in the 2nd semester so I've never done any linear algebra before. I guess you could say I'm like a baby in a swimming pool wearing inflatable arm bands.

aeroboyo said:
one of those online books makes more sense... is an understanding of linear algebra is necessary to begin learning tensor calculus?

Yes, ideas like linear independence and change of basis are important.

tensors are just a language for describing multilinear functions. e.g. dot products are tensors. because they are bilinear.

aeroboyo said:
Hello,

I'd like to learn about tensors so i can start learning about special relativity.
you can start learning special relativity without tensors. tensors are essential for general relativity

is this tensor 'language' covered in what is called 'tensor analysis' in some maths methods books?

you can start learning by reading what i said.

Many math methods books cover Cartesian tensors, tensors in (flat) Euclidean space. They may also introduce tensors in non-linear coordinate systems (curvilinear coordinates), but still in flat space, in which concepts like the Christoffel symbols arise for the first time. This is the case for Boas (judging from the TOC).

On a curved space or surface, linear coordinate systems cannot be introduced at all, we must use curvilinear coordinates.

The full mathematics of GR in all its technical glory is "tensor analysis on manifolds", where a manifold is the n-dimensional generalization of the concept of a surface.

So i'd assume that Frankels Geometry of Physics is preety much the 'technical glory' of General Relativity. It covers manifolds in the 1st chapter, tensor fields on manifolds in the 2nd chapter... and gets more insanely complicated as it goes on... finally covering einsteins field equations in chapter 11. I think I'm right in saying that another way to describe 'tensor analysis on manifolds' is 'differential geometry'. I think it might take me 10 years to work through Frankels text!

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Do what I say, not what I do...

aeroboyo said:
So i'd assume that Frankels Geometry of Physics is preety much the 'technical glory' of General Relativity. It covers manifolds in the 1st chapter, tensor fields on manifolds in the 2nd chapter... and gets more insanely complicated as it goes on... finally covering einsteins field equations in chapter 11. I think I'm right in saying that another way to describe 'tensor analysis on manifolds' is 'differential geometry'. I think it might take me 10 years to work through Frankels text!

That book is about much more than the background you need for classical gtr. I think you are worrying to much about all the stuff you think you have to read. It might help to be told that bits of knowledge are highly "superadditive" in mathematical subjects: when you have mastered n techniques, and then master another one, your knowledge and ability has increased by some factor much greater than $$1+1/n$$

aeroboy, I actually think you should stop posting for a few weeks to that you can focus on your reading...

The best book on tensors and manifolds for general relativity is "Tensors and Manifolds with Applications to Relativity" by Wasserman, 1992. I am typing out the solutions to all the exercises right now. The download link below are my solutions (with the questions typed out as well) to the exercises in Chapters 1 and 2:

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Wasserman is for students with abstract algebra, advanced linear algebra, and topology under their belt.

which makes it all the more interesting when we tie it with general relativity.

## 1. What are tensors and why are they important in learning?

Tensors are multi-dimensional arrays that are used to represent and manipulate data in machine learning and other scientific fields. They are important in learning because they can handle complex and high-dimensional data, making them useful in tasks such as image and speech recognition, natural language processing, and many others.

## 2. How can I learn tensors from scratch?

To learn tensors from scratch, it is important to have a strong foundation in linear algebra and calculus. This includes understanding vector and matrix operations, derivatives, and partial derivatives. It's also helpful to have knowledge of programming languages such as Python, which is commonly used for machine learning and has libraries for working with tensors.

## 3. What resources are available for learning tensors from scratch?

There are many online resources available for learning tensors from scratch, such as tutorials, videos, and online courses. Some popular resources include the TensorFlow website, which offers tutorials and documentation for their open-source machine learning library, and the Deep Learning Book by Yoshua Bengio, Ian Goodfellow, and Aaron Courville, which covers the basics of tensors in the context of deep learning.

## 4. Can I use tensors in any programming language?

Tensors can be used in many programming languages, including Python, Java, C++, and more. However, some languages may have better support for tensors and machine learning libraries, so it's important to research and choose the best language for your specific needs.

## 5. What are some common applications of tensors in machine learning?

Tensors have many applications in machine learning, including image and speech recognition, natural language processing, and recommendation systems. They are also used in deep learning for tasks such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs).

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