A Gentle Introduction to Tensors (2014) [pdf]

https://www.ese.wustl.edu/~nehorai/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf

134 51 5 years ago

ese.wustl.edu

Koshkin4• 5 years ago

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 whole thing clicked for me when I eventually learned about tensors in a more abstract algebraic setting, where they are defined as multilinear forms. The coordinate transformation laws were very easy to understand and remember after that.

But if you want to learn about tensors from how their coordinates transform, here’s a treat:

https://www.youtube.com/watch?v=CliW7kSxxWU

tmearnest• 5 years ago

Could you recommend a text/paper/blog article which develops tensor theory from an algebraic setting?

hrwvjigegby• 5 years ago

Chapter 12 of this book https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth...

Koshkin4• 5 years ago

Hm, that’s a good question! Pretty much any modern book on, or with a chapter on, multilinear algebra, or even a text with a modern treatment of differential geometry (smooth manifolds) would, I imagine, do a decent job. Also, searching the web for “tensors as multilinear forms” turns up quite a few promising links. For a video lecture that could give you a flavor of what tensors are from the linear-algebraic perspective, try

https://www.youtube.com/watch?v=4l-qzZOZt50

(You might get more mileage if you also watch the preceding lecture(s) of this excellent series.)

abdullahkhalids2• 5 years ago

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.

thraxil• 5 years ago

Yes! As a former Physics major who was puzzled by tensors for years (despite being relatively comfortable using and manipulating them), I actually had a similar breakthrough reading Fleisch's book and finally "getting" covariant and contravariant components via thinking about non-orthogonal basis vectors. Definitely one of those "why couldn't they have just explained it this way back when I was an undergrad?" moments. I got so excited when I was reading it that I literally got up, ran to the living room and subjected my partner (not a math or physics person at all) to an impromptu fifteen minute lecture with diagrams in multiple colors of ink. She wasn't as impressed.

(Fleisch's "Student's Guide to Waves" is also highly recommended as a book I wish I'd had when I was a student).

JadeNB4• 5 years ago

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.

Or, preferably, not with basis vectors at all, but that notion seems to do violence to a physicist's way of thinking about linear algebra—I've never quite understood why the physical approach to the subject is so bound up in co-ordinates, when it seems like physicists would be one of the groups most likely to benefit from fully grokking a co-ordinate-free approach.

maze-le• 5 years ago

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

andi999• 5 years ago

What is also confusing that for physics to be a tensor/vector only the transformation behavior under the orthogonal group is considered, so acceleration is a vector. Math and GR (ok, this is also physics) then uses the coordinate transforms on a manifold, so these are different things.

gerdesj• 5 years ago

The author is clearly more comfortable with deploying beautifully laid out mathematics than prose. The introduction or "Opening Remarks" is lovely and well written but is nearly a wall of text. There is one long first paragraph followed by some staccato afterthoughts.

The author wavers between I and we. Am I your friend, guiding you through the maze or are we lecturing you on something? The author needs to settle on either one identity or spell out when they become one or another.

Sometimes, you might wish to appear as a friend hovering over the shoulder and provide hints as to the right direction to follow and at other times you might deploy something that will give LaTeX a headache and pull out the Vox Dei stop on the 250 ton Organ and destroy nearby eardrums.

I love the paper and it is saved locally.

kowlo• 5 years ago

I've always thought of it as a journey that we're going on together, where we may do something, expect something, find something, or conclude something - together!

Perhaps I'm finding camaraderie where there is none. How disappointing.

725686• 5 years ago

I never studied tensors, but for a very first ultra basic understanding of tensors I really liked Dan Fleisch's video:

https://www.youtube.com/watch?v=f5liqUk0ZTw&t=3s

He also has some great "Student's Guide" books about tensors and other subjects.

harry84• 5 years ago

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 be some species of super-genius) Or is that the generally accepted genre?

Koshkin4• 5 years ago

Many of the scientific treatises of today are formulated in a half-mystical language, as though to impress the reader with the uncomfortable feeling that he is in the permanent presence of a superman.

Cornelius Lanczos

SantalBlush• 5 years ago

After having had difficult topics explained remarkably well by a few math teachers, I'm convinced that the "gentle introductions" you're talking about are meant to be confusing. Or perhaps, in some cases, they don't grasp the material as well as they let on.

edit: For anyone who wants examples of good introductions, check out 3Blue1Brown's YouTube channel.

Koshkin4• 5 years ago

Or, the infamous Curse of Knowledge?

https://en.wikipedia.org/wiki/Curse_of_knowledge

anon_tor_12345• 5 years ago

[flagged]

dang2• 5 years ago

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

paulpauper• 5 years ago

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.

harry84• 5 years ago

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.

harry84• 5 years ago

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.

lisper2• 5 years ago

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.

dreamcompiler• 5 years ago

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.

lisper2• 5 years ago

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.

soVeryTired8• 5 years ago

Take a look at my definition of the "mathematician's tensor" in the top-level comments. The rank of a tensor is the number of row- and column-vectors you need to feed the tensor to get it to return a real number. AFAIK it's not related to 'rank' in the sense of the rank-nullity theorem.

To move from the mathematician's definition to the ML definition, pick a basis for your row and colum vectors. Now if you want the (i, j, ...) element of the multidimensional array, gather the vectors with one in the i-th position and zero elsewhere, the j-th position and zero elsewhere, etc. Then feed them to the tensor in that order.

It's easiest to see how it works with row vectors (rank (0,1)), column vectors (rank (1, 0)) and matrices (rank (1, 1)) and work from there.

effthisis• 5 years ago

A Gentle Introduction to Tensors: just learn Clifford/Geometric Algebra

miles7• 5 years ago

An easy to read (minimal theorems) but authoritative and modern introduction to tensors can be found in the book “An Introduction to Tensors and Group Theory for Physicists” by Jevanjee (https://www.springer.com/gp/book/9783319147932).

He takes the more geometrical perspective as a tensor as a multi-linear function of vectors, from which all other statements about tensors (eg how the components transform) follows straightforwardly. Lots of other great material in this book and best of all there are loads of examples.

glaucon• 5 years ago

"A Gentle Introduction to Tensors" which starts with not one word about what a tensor is, or why you might wish to be introduced to one. Good stuff.

ksd4822• 5 years ago

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!

sam-2727• 5 years ago

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

ksd4822• 5 years ago

Thank you!

For my own enjoyment, for now.

soVeryTired8• 5 years ago

I think one of the issues that makes tensors difficult to understand is that different disciplines use the word in different ways. I'm going to generalize here a bit about how each group uses the term, but I suspect not everyone will agree with me. Apologies in advance!

For machine-learning types, a tensor is a multidimensaional array of numbers that admits certain operations (addition, subtraction, hadamard product, etc) [0]

For mathematicians, a tensor is a function F. We pass the function an ordered tuple of N vectors and M covectors (or, to keep things simple, N column vectors and M row vectors), and the function returns a scalar. The function F needs to be linear in each of the N vectors and M covectors. In this view, a matrix is a tensor with N=1, M=1. The operations used by machine learning types arise naturally once you crank the mathematical handle a little bit.

For physicists, a tensor is what mathematicians would refer to as a tensor field. So they take a space X (think R^N), and for each point in the space they associate a mathematician's tensor F as defined above. The properties of the function F are permitted to change from point to point. The Riemannian metric tensor is a classic example of 'tensor' according to this usage.

[0] https://machinelearningmastery.com/introduction-to-tensors-f...

ktpsns• 5 years ago

This is so true!

For machine-learning and data science, tensors are largely what is called n-dimensional array. Numpy is a popular python library built basically around this data type.

Mathematical tensors in a particular base (sorry for the sloppy language, theoretical physicist here) can be displayed with such a data type, depsite the n-dimensional array data type (as available in numpy) lacks the (co)vector properties.

Funny enough, in theoretical physics there are two groups of people: The one prefering to write out complicated objects (i.e. tensors) with indices and the ones who do not (typically called abstract tensor notation in general relativity). The latter are much more close to mathematical physics and prefer to understand tensors as mappings from (co)vector spaces to scalars, similar to as you already wrote.

Most general relativity computer codes are written by the index people, not by the mathematical physicists ;-)

JadeNB4• 5 years ago

For mathematicians, a tensor is a function F. We pass the function an ordered tuple of N vectors and M covectors (or, to keep things simple, N column vectors and M row vectors), and the function returns a scalar. The function F needs to be linear in each of the N vectors and M covectors. In this view, a matrix is a tensor with N=1, M=1. The operations used by machine learning types arise naturally once you crank the mathematical handle a little bit.

As with almost all non-technical statements one can make about math (including my amendment—recursive nerd-sniping away), this is neither quite true nor quite false.

A tensor is an element of a tensor product—that's it. (But what tensor product? There are a lot of notions. I'll use the bare tensor product of non-topologised vector spaces.)

One kind of thing you can tensor is copies of a single vector space, and/or its dual. This is the sort of tensor product you likely have in mind. Tensoring copies of the dual allows you to feed in elements of the original vector space. Tensoring copies of the vector space allows you to feed in elements of the dual vector space (there is mathematically no intrinsic difference between a vector and a co-vector; having a favourite vector space in mind, you can talk about the elements of that vector space or of its dual, and—which is where the vector vs. co-vector terminology in physics comes from—about how coefficient vectors change when you change the basis of V), although, importantly, if your original vector space is infinite dimensional then the natural embedding into its double dual is not an isomorphism.

Which is to say, yes, the objects you describe all arise as mathematical tensors; but mathematical tensors can also describe much more.

soVeryTired8• 5 years ago

if your original vector space is infinite dimensional then the natural embedding into its double dual is not an isomorphism

Ok, so this feels like the crux of the matter. So how does the fact that V** is not isomorphic to V make the tensor product construction a more general concept than the linear function construction?

joppy5• 5 years ago

I think that tensors have a much more broad meaning to mathematicians. At least in pure mathematics and algebra, there is much more of a focus on the tensor product operation, a way of taking two (or more) vector spaces and producing a new vector space: their tensor product. Tensors are elements of this new space. In particular they don’t need to be multilinear functions on a product of vector spaces (for finite dimensional spaces there is not much difference, but for infinite-dimensional spaces there is a real difference between these two things).

Tensor products in this generality don’t need to have an (n, m) rank in the sense you’re describing, since they might be put together from completely different spaces. For example it’s perfectly fine to form the tensor product of a 2-dimensional space with a 5-dimensional space, yielding a 10-dimensional space.

soVeryTired8• 5 years ago

That's a fair point about the tensor product being more central in algebra. But I think the 'friendliest' introduction to tensors is the multilinear function viewpoint. IMO it motivates the tensor product.

I'm not familiar with the infinite-dimensional case. Do you have an example of a tensor that isn't a multilinear function on a product of vector spaces? I'd be interested to refine my understanding here.

sgregnt• 5 years ago

What is a tensor part 1: https://youtu.be/MkYEh0UJKcE What is a tensor Part 2: https://youtu.be/Lkpmd5-mpHY Paper: https://arxiv.org/abs/2106.08090

A Gentle Introduction to Tensors (2014) [pdf]

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