Recommendation of a thorough book on suffix/index notation?

[Bobman]

I believe it is also called "einstein notation"?
The "notation-thingy" using kronecker delta, levi-civita and etc to simplify expressions with div, grad, curl (i took the course in my native language so i am not entirely sure what the notation or technique is called in english).

Looking to get a good and thorough book on suffix notation as it is something i would like to revisit to learn properly. Sadly, the book we used for vector calculus only have 1 page on it and emphasis were on "using" the nabla operator when simplifying expressions, which did the trick, but i found suffix notation more compact and elegant as well as i feel like it will be more useful later on when you get to tensors and so on.

Any recommendations?

Oh, and my "math level" is basically up to vector calculus.

 

[fresh_42]

I'm not sure you will find what you are looking for, except perhaps on Wikipedia, which explains e.g. the Einstein notation, Kronecker delta, structure constants, or the Levi-Civita symbol.

The basic difference here, and this is typical for the difference between math and physics, is that physics often writes its equations in coordinate form. So instead of $W= F \cdot x$ or $W=\vec{F}\,\vec{x}$ we find $W=\sum_i F_i \,x_i$ and with Einstein's sum convention $W=F^i\, x_i\,.$

This means it gets messy, especially if many dimensions are involved, but one can do coordinate transformations, i.e. the change of reference frames more explicitly. It also requires, that the corresponding basis are known, and they are usually not mentioned. It's not a big deal for ordinary Cartesian coordinates, but sometimes leads to confusion if the basis vectors are, e.g. differential forms $dx_i$: where is the tangent space in $\sum_{j=1}^n\frac{\partial}{\partial x_j}\,dx_j = \partial x^j \,dx_j\,?$ The distinction between scalars, components, and coordinates $F_i=ma_i$ can also become more difficult compared to a notation $\vec{F}=m\cdot \vec{a}\,$

Physics often looks like index acrobatic at first sight. Personally I think it is. And with any acrobatic skill, only practice, practice and practice will help. I would start to do this with the following question: "What is what and what are the bases?" whenever you see an equation with indices. Especially the difference between component and coordinate can be tricky!

 

[Bobman]

I think we might be talking about different things, I probably made a weak point.
I meant a book on the actual skill on manipulating expressions "using" the notation.

And for two examples on what i am actually talking about i added two pictures below from my professors slide.
https://imgur.com/Yo6Ok6j
https://imgur.com/ZFvoZ78

 

[fresh_42]

That was exactly what I was talking about.

Ex.1: How to write (compute) $\vec{a}\times (\vec{b}\times \vec{c})$ in coordinates?
Ex.2: How to write (compute) $\operatorname{div}(\vec{A}\times\vec{B})$ in coordinates?

"What is what and what are the bases?" are the questions to be asked, and the usage of the notation can be found here:

• Kronecker delta
• Levi-Civita symbol
• Einstein notation
• cross product
• triple product
• tensor product
• co- and contravariance
• Laplace operator
• divergence
• curl resp. rotation
• tensor field, e.g. vector or scalar field
• metric tensor

That's basically it. Sometimes it makes sense to switch to other languages of those Wikipedia pages as they use different presentations and some are better there, but this is only an additional option on Wikipedia compared to other sites.
There will be no shortcut to practice, however.

 

[Bobman]

Ah, sorry for misunderstanding you.
I did give wiki a good look when i actually took the course, but i remember not thinking it being very helpful for me. But perhaps its worth revisiting then, still hoping for a book that would cover it though :)

 

[fresh_42]

I know what you mean and I can understand this, as me, too, has his trouble with those indices. The difficulty here is that these are just notations. They are introduced - if at all - in one or two lines. A matrix is $A=(A)_{ij}=(a_{ij})_{i,j}$ and a vector $v=(v_i)_i$ and nobody makes a bit deal of it. That was why I pointed toward Wikipedia, although I have to admit that the English pages here aren't really good in this respect, others are better. Nevertheless, as it is a notation, the main difficulties are actually:

• What is it (matrix, vector, covector, tensor, metric, connection etc.)?
• What does it (function, static vector, factor, product, transformation etc.)?
• How is it written (coordinates, how many, in which dimensions)?
• What are the basis vectors (e.g. the standard Euclidean vectors $\vec{e}_i$, one-forms $dx_i$ etc.)?
• Is there an Einstein summation involved (upper indices, or powers?)?
• Which index written where belongs to which quantity written according to which basis?

This is confusing, and all those questions are usually unanswered. Authors assume the readers would know, and often it is answered somewhere when the quantity has been introduced. However, if they say e.g. $X \in \bigwedge^d V$, it is already assumed, that the reader knows there are $d$ copies of an $n-$dimensional vector space which results in $m\cdot n\cdot d$ scalar factors resp. coordinates, because we have linear combinations of some length $m$ here. This doesn't mean that you can describe $X_{ijk}$ in a meaningful way, but if multiplied with another object, which itself again has coordinates, those $X_{ijk}$ appear in the calculation. Just as in your examples.

 

[Bobman]

Indeed, and the swedish ones (it's my native language) are utterly rubbish and almost as a rule contains only 10% of the information. That's why i feel it's a shame if there aren't any textbooks on the matter. It is probably possible to just put yourself through it all by trial and error, but sadly i don't have the time for that.

 

[fresh_42]

Indeed, and the swedish ones (it's my native language) are utterly rubbish and almost as a rule contains only 10% of the information. That's why i feel it's a shame if there aren't any textbooks on the matter. It is probably possible to just put yourself through it all by trial and error, but sadly i don't have the time for that.

If you're Swedish, there is a chance you understand a bit of the German Wikipedia pages. They are better than the English ones when it comes to formulas and indices as they are in general less general and more specific. I know, that's not the same as a book, and I agree, someone should finally write this da.. book, because it is a part of physics, which is quasi another language and a dictionary would be fine.

 

[Orodruin]

the swedish ones (it's my native language) are utterly rubbish and almost as a rule contains only 10% of the information.

Now this makes me curious. Which ones have you read?

 

[Bobman]

If you're Swedish, there is a chance you understand a bit of the German Wikipedia pages. They are better than the English ones when it comes to formulas and indices as they are in general less general and more specific. I know, that's not the same as a book, and I agree, someone should finally write this da.. book, because it is a part of physics, which is quasi another language and a dictionary would be fine.

Good idea, i'll check it out.

Now this makes me curious. Which ones have you read?

So i may have been a bit harsh in the heat of the moment, but i would stand by that they contain about 10% of the information.

I want to try to avoid to cherrypick, but in addition to most of the pages that fresh_42 linked, there are a lot of examples that i visited a few times while taking vector calculus:
https://en.wikipedia.org/wiki/Scalar_potential
https://en.wikipedia.org/wiki/Stokes'_theorem
https://en.wikipedia.org/wiki/Divergence_theorem
https://en.wikipedia.org/wiki/Gradient
https://en.wikipedia.org/wiki/Line_integral

Our books are usually much shorter than international literature as well (i could take some photos of my books if u want to), but sometimes that's a advantage as well. A lot of american books, i find, tend to be filled with "fluff" covering the actual useful bits.

 

[Orodruin]

Our books are usually much shorter than international literature as well (i could take some photos of my books if u want to), but sometimes that's a advantage as well.

I would prefer if you just told us which books you have.

In general, the notation itself is not much more than some conventions. After that it is a matter of understanding how different things look in this notation. The very basics are introducing the Kronecker delta and the Levi-Civita symbol and apart from that everything is typically rather straight-forward. It is therefore a bit unclear to me what you mean should go into the literature that you are after. Manipulating index expressions is no different from manipulating normal expressions with just numbers.

Some time ago I wrote an Insight on common things that you should keep in mind when dealing with index expressions. You might find it helpful.