Lethe, To complement your notes, I have a few textbooks in front of me: Shutz 'Geometrical methods...' and Frankels 'Geometry of physics'. I've been through these before but they're making more sense now. One point, Frankel says: 'tangent vector, or contravariant vector, or simply vector (section 1.3a)' and: 'covariant vector or covector or 1-form' (sect 2.1b) That's the opposite of what you said above. Can you clarify? I understand the need to have a coordinate-free description but I guess that, to spit out numbers for design purposes, you have to eventually define a coordinate system. That may well be why engineering mechanics courses tend to be lagging behind here. Maybe not for long! Cheers, haven't enjoyed this stuff so much before! Ron.
hmm... it s true, most books have it exactly opposite from what i said. i actually only have one book where the terms are switched, and the reason i chose to use this terminology is that this one book that uses it this way also explains why the terms are used the way they are. since it is the only such explanation, i was inclined to believe it over the multitude of other books claiming the opposite.
Fukin typical, isn't it! Just when you think you know the difference between black and white. I'd better put a footnote in my own lecture notes to warn students of the possible confusion. Maybe best to just stick with 'vector' and '1-form' here? Cheers, Ron. Just returned d'Inverno to the library - might dig it out again.
More on 'co' and 'contra' This from: http://www.mathpages.com/rr/s5-02/5-02.htm 'By the way, it may seem that the naming convention is backwards, because the "contra" components go with the axes, whereas the "co" components go against the axes. Historically these names were given on the basis on the transformation laws that apply to these two different interpretations. In any case, we're stuck with the names.' Typical!
yeah, so apparently this situation is a big mess in terms of terminology. if i have to choose sides, i will go with the modern mathematicians, but i think your suggestion is best, ron. we will just avoid using the terms altogether. tangent vector (or sometimes simply vector), and 1-form. that s what i ll call them. i think quite a few books follow this rule too. for example, if you look for the terms contravariant and covariant in schutz, you will not find them at all.
Don't get me started on terminology and representations. Take the dirac euation. You have the Wyle representation, the pauli representation as well as 2 different representations of the lorentz 4-vector, one where the time component is written as ct and the other as ict. They make it difficult for us to learn this stuff.
Agreed - and I remember your previous advice - even if all my tensors are symmetric - I might benefit from the formulism of diff forms. don't stop now! cheers, ron.
Don t worry, i haven t forgotten about my thread. it s just that this is a very busy time for me (finals and what not), and i haven t had time to really flesh out some new posts. i can always find time to pop off a few replies to the crackpots, when i m procrastinating from my work, just not to write something more serious. however, the semester will be over for me about a week and a half, hopefully i ll get something new up there sooner than that, but after that, i can spend a lot more time on it. i had a lot of fun writing that thread, thinking about the material, and i would like to see more stuff like that on this forum. i was thinking about other threads i might like to write after that one is done. some things i would be qualified to write about that i thought might be nice: abstract algebra, galois theory, representation theory, lie groups/lie algebras. we will see what time allows for. as for making the thread sticky, well, while my opinion is that it s a damn fine thread, i m not sure that there is anyone reading it besides you, ron. maybe 2 or 3 others, but probably it is not of much interest to 99% of the visiters of this board. certainly you re the only one who has asked any questions pertaining to the material, i think james made a few comments. i guess it s hard to gauge.
OK, i m totally swamped. i ve been up for probably 48 hrs now, and spent too much time looking at this damn message board when i should have been working. and no end in sight. but i promise, when things lighten up, we ll go back to the differential forms thread. not too much more on it, and we ll have everything we need to reformulate maxwell s equations with differential forms, do stoke s theorem, and maybe we will look at yang-mills fields.... well... that might be a long shot.... we ll see.
I was basically told the same thing by my major professor. I think this is quoting him, so I will use quotation marks: "The partial differential operator is the paradigm for the covariant first rank tensor." This was in the context of the 4-D nabla operator, basically the grad operator. He was contrasting it to the differential displacement 4-vector, which he referred to as the "paradigm for the contravariant first rank tensor." BTW, lethe, your thread kicks ass! That post where you laid out the different "whatever-they-are" in order of, I'm assuming, increasing complexity has finally started to turn the light bulb on in my brain about this stuff. THANKS!
well, that depends on which convention you are using. in my thread, and in certain books, forms are contravariant, vectors are covariant. the gradient is naturally a form, although given a metric, you can construct a dual vector to it, but it is naturally a contravariant object. however, quite a few books use the opposite convention, in which case the gradient is a covariant object. to alleviate confusion, i adopt the standpoint that the words covariant and contravariant should simply be avoided altogether. say instead vectors and covectors. or tangent vectors and forms. or dual vectors.
Lethe, would you be willing to briefly explain what you mean by "vector" vs. what you mean by "form?"
hey man!! i wrote a whole thread about it! why don t you read that! it s not so long. i worked hard on it! you want the short version?? well here it is: a vector is something that points, and a form is a dual to that vector: a function(al) that eats a vectors and spits out a number. that s it in a nutshell.
I didn't know you had another threads. Apparently this one is not yours. I just peaked at the other thread and will have to go read that one now.
What is the maximum degree that can have a differential k-form, that is, what is the maximum value permitted for k?
