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A Gentle Introduction to Tensors (2014) [pdf]

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Re: A Gentle Introduction to Tensors (2014) [pdf]

#21

Post by harry8 »

paulpauper wrote: post #12
harry8 wrote: post #7

Yet to see a "Gentle Introduction to" that didn't freely and accurately translate to poorly written, impenetrable prose whose purpose is to gratify the ego of the author in demonstrating how they are unbelievably more clever than any reader who could conceivably require any kind of "introduction" Is that bad luck on my part? (I've given up on thinking I'm particularly or especially dense - while making no claims to b…

How should it have been done. Tensors are pretty complicated. It's like advanced calc plus advanced linear algebra. Maybe watch dome youtube videos on general relativity. Those are sometimes better than texts because clarity is essential.

There are great tutorials, textbooks, explanations, sibling post mentions Grant Sanderson's 3B1B youtube channel and I agree with that example whole heartedly.

I haven't seen or read any that have the words in their title "A Gentle introduction..." It's those words I'm talking about as an indicator of being anything but what they claim in the general case. I'm very open to counter examples.

Re: A Gentle Introduction to Tensors (2014) [pdf]

#22

Post by harry8 »

harry8 wrote: post #7

Yet to see a "Gentle Introduction to" that didn't freely and accurately translate to poorly written, impenetrable prose whose purpose is to gratify the ego of the author in demonstrating how they are unbelievably more clever than any reader who could conceivably require any kind of "introduction" Is that bad luck on my part? (I've given up on thinking I'm particularly or especially dense - while making no claims to b…

There's a flagged dead reply to this that attacks me. I think the author has missed the point and the point isn't very important anyway. I agree totally that the author of this or any other freely available material owes me nothing. Nor do I owe them.

I mention this only because I don't think the comment should be flagged or dead. Ymmv.

Re: A Gentle Introduction to Tensors (2014) [pdf]

#23

Post by dreamcompiler »

lisper wrote: post #17

Argh! People make this so much more complicated than it has to be. A tensor of rank N is a vector in a space whose basis is a set of N-tuples of ordinary vectors. That's it. So a rank-1 tensor is just a regular vector. A rank-2 tensor is a vector in a space whose basis is ordered pairs of regular vectors, a rank-3 tensor is a vector in a space whose basis is ordered triples of regular vectors, and so on.

Here's where this is confusing (to me at least).

The dimensionality of a vector space is the number of scalars needed to build a vector in that space. That's easy.

I also think of "rank" as synonymous with "dimensionality" but it's not. Or at least it implies a different connotation of the word "dimensionality." It's the number of dimensions in the notation rather than in the space. Or something. And now I'm off the rails. And this is before I start trying to think about a basis being "ordered pairs of vectors" or tensor fields or about how tensors transform.

Re: A Gentle Introduction to Tensors (2014) [pdf]

#24

Post by ksd482 »

I am a math major and I am learning GR by myself right now. For GR, you need tensors (among many other things), and as a result I have gone through SEVERAL tutorials, books, videos etc.

There is ONE main thing I find lacking in all of these sources: computational examples/exercises.

My idea of a "gentle introduction to tensors" would be: motivation, definition and then immediately followed by computational problems (lots of them). Only then I would be comfortable with abstract definitions and proofs (which is my goal ultimately).

Edit: https://www.youtube.com/watch?v=5oeWX3NUhMA&list=PLFeEvEPtX_... is the most amazing lecture on Tensors I have watched so far, by far!

Re: A Gentle Introduction to Tensors (2014) [pdf]

#25

Post by sam-2727 »

ksd482 wrote: post #24

I am a math major and I am learning GR by myself right now. For GR, you need tensors (among many other things), and as a result I have gone through SEVERAL tutorials, books, videos etc. There is ONE main thing I find lacking in all of these sources: computational examples/exercises. My idea of a "gentle introduction to tensors" would be: motivation, definition and then immediately followed by computational problems (…

best of luck in your studies! Are you learning GR for a certain task or just for your own enjoyment?

Re: A Gentle Introduction to Tensors (2014) [pdf]

#26

Post by lisper »

dreamcompiler wrote: post #23
lisper wrote: post #17

Argh! People make this so much more complicated than it has to be. A tensor of rank N is a vector in a space whose basis is a set of N-tuples of ordinary vectors. That's it. So a rank-1 tensor is just a regular vector. A rank-2 tensor is a vector in a space whose basis is ordered pairs of regular vectors, a rank-3 tensor is a vector in a space whose basis is ordered triples of regular vectors, and so on.

Here's where this is confusing (to me at least). The dimensionality of a vector space is the number of scalars needed to build a vector in that space. That's easy. I also think of "rank" as synonymous with "dimensionality" but it's not. Or at least it implies a different connotation of the word "dimensionality." It's the number of dimensions in the notation rather than in the space. Or something. And now I'm off the…

That's right. The dimensionality of the tensor space depends on both the rank of the tensor and the dimensionality of the underlying vector space. A rank-2 tensor based on 3-D physical space has 9 dimensions because there are 9 distinct pairs of the 3 underlying basis vectors. In general, a rank-N tensor on an M-dimensional space will have M^N dimensions.

Re: A Gentle Introduction to Tensors (2014) [pdf]

#27

Post by dang »

anon_tor_12345 wrote: post #11
harry8 wrote: post #7

Yet to see a "Gentle Introduction to" that didn't freely and accurately translate to poorly written, impenetrable prose whose purpose is to gratify the ego of the author in demonstrating how they are unbelievably more clever than any reader who could conceivably require any kind of "introduction" Is that bad luck on my part? (I've given up on thinking I'm particularly or especially dense - while making no claims to b…

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As you know, you can't attack other users like this on HN, regardless of how knowledgeable you are or feel you are. Therefore I've banned this account. If you don't want to keep getting banned on HN, please follow the site guidelines.

https://news.ycombinator.com/newsguidelines.html

Re: A Gentle Introduction to Tensors (2014) [pdf]

#28

Post by ksd482 »

sam-2727 wrote: post #25
ksd482 wrote: post #24

I am a math major and I am learning GR by myself right now. For GR, you need tensors (among many other things), and as a result I have gone through SEVERAL tutorials, books, videos etc. There is ONE main thing I find lacking in all of these sources: computational examples/exercises. My idea of a "gentle introduction to tensors" would be: motivation, definition and then immediately followed by computational problems (…

best of luck in your studies! Are you learning GR for a certain task or just for your own enjoyment?

Thank you!

For my own enjoyment, for now.

Re: A Gentle Introduction to Tensors (2014) [pdf]

#29

Post by abdullahkhalids »

Koshkin wrote: post #2

I have always hated when tensors are defined as something whose coordinates are transformed in a certain way. I just find it inherently unfriendly and un-geometric. No matter how much talk is given about simpler cases such as scalars, vectors, covectors, etc., the final defining formula would still look to me as daunting as always. (There is nothing “gentle” about the formulas on page 14.) Surprisingly or not, the wh…

I taught a module on tensors in my undergrad Mathematical Methods for Physics course last semester. The students had a weak mathematical background, so I had to explain better than any of the books and videos do.

The main problem with most introductions to the topic are that they deal with coordinate systems where the basis vectors are orthogonal, so the covariant and contravariant components are the same. You need to deal with non-orthogonal basis vectors, because then you realize that naturally there are two ways of defining basis vectors at a given point in space. And naturally these two different sets of basis vectors have different transformation properties under geometric transformations. This book [https://danfleisch.com/sgvt/] does a decent job of getting you started on this approach - of course, it has no rigor.

Re: A Gentle Introduction to Tensors (2014) [pdf]

#30

Post by maze-le »

Koshkin wrote: post #2

I have always hated when tensors are defined as something whose coordinates are transformed in a certain way. I just find it inherently unfriendly and un-geometric. No matter how much talk is given about simpler cases such as scalars, vectors, covectors, etc., the final defining formula would still look to me as daunting as always. (There is nothing “gentle” about the formulas on page 14.) Surprisingly or not, the wh…

I can also recommend this great series on Tensor Calculus: https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6...
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