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How can we prove that the tensor product between two tensors of lower rank forms the basis for ANY tensor of higher order? also WHY is it it true?
ANY TENSOR of higher order.
ANY TENSOR of higher order.
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I find that an odd saying. Could you please reference several physics texts which say this so that the likelyhood that I'll have one of them will be good? Thanksmost physics books say that the tensor product between two tensors is the most general higher order tensor. how can we prove this?
I find that an odd saying. Could you please reference several physics texts which say this so that the likelyhood that I'll have one of them will be good? Thanks
Best wishes
Pete
That one I have. What page should I turn to?a first course in general relativity.
most physics books say that the tensor product between two tensors is the most general higher order tensor. how can we prove this?
The most general [itex](0,2)[/itex] tensor is not a simple outer product, but it can always be represented as a sum of such tensors.
I wouldn't worry if you're in high school and finding tensor analysis quite difficult!Ah that was just a simple misunderstanding. however there is another section i have a hard time with. It's on page 70. It's the part where they talk about the absis of the gradient one form. i don't quite understand what's being done. could you guide me through it step by step?
I'm finding tensor analysis to be quite difficult actually. Is this normal for high schoolers studying the subject?
He introduces [tex]x^{\alpha}_{, \beta}\equiv \delta^{\alpha}_{\beta}[/tex] [sorry, I don't know how to offset the indices like in the text]. Note he is using , to denote partial derivative, so the LHS is [tex]\frac{\partial x^{\alpha}}{\partial x^{\beta}}[/tex] Do you know what the kronecker delta is?to be more precise:
I'm having a hard time with the section on page 70 where he talks about basis one forms for the gradient vector.
To be more precise here's a quote" note that the index aappears as a supercript in the denominator and as a subscript on the right hand side. As we have seen this is consistent eith the transformation properties of the expression.
In particular we have:
Then he introduces a symbol that I don't understand at all.
The conclusion comes from looking at (3.12). Do you understand this equation?I don't understand the conclusion after that.
My other problem on page 71 deals with the fact that he says that "since each index has four values there are 16 components". could you explain that in more detail?
Its probably one of the most difficult branches of math that there are. Graduate students have trouble with that math too. So if you're having trouble then you're normal.I'm finding tensor analysis to be quite difficult actually. Is this normal for high schoolers studying the subject?
high schoolers? uh, no it is not normal for high schoolers to find it difficult, because most high schoolers are prudent enough to leave the topic for 4-5 years later. do you understand trig well? plane geometry? solid geometry? logic? algebra of polynomials? probability? matrices? calculus of one and several variables? topology?
if not, my suggestion is leave the tensors alone.
Ok so right now my main problems are equation 3.24, which i'd like explained in more detail
If that's the case then I wouldn't worry about it. I myself don't know topology and my probability is only enough to work in quantum mechanics.Everything there but topology.
If that's the case then I wouldn't worry about it. I myself don't know topology and my probability is only enough to work in quantum mechanics.
It appears that all your questions were answered before I was able to get to them. Is there anything else I can help with? How much do you know about special relativity and physics?
Pete
Right now I'm refreshing myself in EM. I'm using Ohanian's book "Classical Electrodynamics - 2nd Ed.". I highly recommend this book.I know enough physics to manage with some things things, though it would be nice to know more electromagnetism.
As I said, it was simply the introduction of new notation. Instead of writing out the entire expression of the partial of [itex]\phi[/itex] with respect to x^{[itex]\alpha[/itex]} you can just express that as [itex]\phi[/itex],_{[itex]\alpha[/itex]}. Its merely a simpler way of expressing it for the ease of reading. Without a lot of the notational shortcuts that we use in this area of math the equations would look more hellish than they already are.Right now i just don't understand equation 3.19. everything else is fine.
Everything there but topology.
forgive my scepticism, but if you understahnd all that stuff, why is gradf perpendicular to a level set of f?
and what does greens theorem say and why is it true? (in a rectangle, say.)
and how do you prove a continuous function has a maximum on a closed bounded interval?
green's theorem says that the circulation is around an interval is equal to the sum of iinfintessimal circulations within the area. Say that you have a square, you can find the cxirculation around it through very simple methods. then put another square beside it and calculate it's circulation. For obivous reasons(related to orientation of the sides), the sum of the circulations will be the circulation around both squares. since any area can be apporximated by squares we can approximate the circulation around the boundary of the region. If we take the limit as the number of squares approach infinity we get the total circulation around the boundary. if you want me to give a better explanation just ask.
I'm kind of offended, though I undersrtand your skepticism.
As for the second thing, I've nver seen such a proof but it seems very easy to prove in an informal way.
now if your wondering why I'm having difficulty with tensors, i guess I'm stupid.
Terilien said:what must I explain now? stokes? gauss's? change of variables? this is exactly why I would never ask questions in the past.
I'm not asking you to prove yourself and I'm certainly not saying that you're stupid. Regardless of how old you are and regardless of your background, the fact that you're asking questions proves that you're not thick.
Seriously, if you're in high school and you've been able to understand and absorb this much physics and mathematics then you should do very well if you choose to study physics at university. I would however suggest that you look at a book such as Chern, Chen, and Lam to learn differential geometry before trying to learn too much about general relativity.
Oh, and complex analysis is an absolute must. I'm still amazed at how many undergrads I see that don't understand it at all.
I've never found a resource on complex analysis, I am VERY interested in ANY introduction to rigorous mathematics.chern chem and lam? what's the book like? does it start with gaussian geometry or full blown generality?
Terilien said:I'd actually prefer to study mathematics outside of physics but i can't find any non physics books that treat advanced topics with a conversational tone.
Terilien said:"A "insert fancy term here" is a bijective mapping between two surjective Y spaces".
I know that's not anything specific, but I think you get the point. I'd love to major in physics, but trouble with things like school are getting in the way. sometimes it takes a while for me to get over my fear of studying a certain topic. I'm generally very afraid of not understanding something.
Oh well I'm getting a tad sentimental. My problem is that i generally keep my thoughts on topics to myself because I'm very sensitive to criticism.
I've always wanted to meet a math professor of some sort. Unfortunately I have no idea how to.