Tensor, second order tensor? Wth is this all about?
>>7684487
Why do you need to know? Why is it of interest to you?
>>7684487
here's the NASA intro to tensors for engineeringfags - long version
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050175884.pdf
>>7684487
An array of numbers, pal. Like 1, (1,2,0,1), or
|1, 2, 3|
|2, 0, 1|
|1, 1, 1|
>>7684487
Different ranks are used. Rank 1 is just X axis, Rank 2 is x,y, rank 3 xyz.
Tensors simply measure the tension in these spaces.
Starting with zero because its neat.
rank 0 tensor = scalar tension
Rank 1 tensor = vector tension
Rank 2 tensor = plane tension
Rank 3 tension = cuboid tension
Then you take these tensions and can measre how tense the space is. A really tense plane is like a flat balloon stretching out.
Tensors are the dual of relaxors. With relaxors measuring the relaxation (inverse tension).
For instance rank 0 relaxor is infinite rank.
I never really understood why "tensor analysis" is considered some separate thing from algebra.
The way I understand it is that a tensor is an object similar to a vector, but generalized to be symmetric under some higher dimensional transformation, such as a rotation in the space it's define in.
>>7684496
What is alegbra?
NASA can put a man on the moon but spellcheck is impossible?
TFW when 162000 google-results for alegbra...
>>7684517
So, it's a fucking matrix then.
>>7684586
you make me sad
Ty guys. Cached a little
>>7684583
How many times have you proof read a table of contents?
>>7684586
[spoiler]Yes.[/spoiler]
>>7684567
>Tensors simply measure the tension in these spaces
Poor, poor physics/engineering student. There is so much you don't know and it hurts my feeling to know that out there there is someone like you who believes he/she got the idea behind tensors but is in fact only able to operate with them.
With a single sentence you proved how little you know about what you are trying to explain. I bet you also believe that vectors are arrays of numbers. But that is simply an hypothesis. Well, I pity you.
I'd love it if there were a clean answer to this, but afaik the forget functor concept is nowhere formalized, really.
You can do it in terms of sets via equivalence relations, but this is cumbersome and ugly as night.
>>7684652
>Mathfag trying to talk down to talk down to applied master race.
Do you even know how tensors work in practice mathfag? He was obviously referencing a commonly encountered tensor in realspace so OP can intuitively understand it because some of us have to actually get shit done instead of wanking over the formalities of trivial mathematical constructions.
>Alright let's get this spaceship built already we should be halfway to the moon right now.
>No! We have to rigorously define what we mean by "spaceship" first. In fact we still need to obsess over the tools we'll use to construct it - (You)
>>7684652
Hey, physicists aren't as dumb as him
>>7684691
Not the guy you're replying to, but it's so obvious you are still in high school and/or trying to bait. If it's the latter, congratulations, you got me.
>>7684691
>I bet you also believe that vectors are arrays of numbers. But that is simply an hypothesis.
>hypothesis
I don't think the guy you're replying to is a mathfag. He's probably a bad physicist who supplements his poor understanding of material by harping on a poor understanding of things other people frequently oversimplify.
>>7684652
o wise man, can you impart any knowledge i have not yet acquired? because i already know how to shitpost so you really havent tought me anything.
>>7684852
Firstly, the first letter of the first word of a sentences is to be capitalized.
Second, "I" is a proper noun, capitalize it. What you're doing right now looks foolish.
Third, taught*.
It's not about perfect grammar or conforming to certain syntactical rules, it's about basic little obvious things. Like capitalizing your shit. At the very least capitalize "I", it looks fucking stupid otherwise.
>>7684487
Watch this video OP. It makes stuff super clear.
https://www.youtube.com/watch?v=f5liqUk0ZTw
>>7684868
that's just your opinion fuccboi
i like the way this kind of text looks, to me yours just looks like a tryhard redditor's
>>>/b/654173024
>>7684868
I never got this Anglo fad of capitalizind the pronoun "I". It just doesn't make sense. It isn't a noun, it's a pronoun.
>>7684487
an indexed object: the (natural!) number of indexes is the rank.
I just skimmed through more shit so this is a retard's thoughts.
Tensors seem to be something that looks for the metric-independent properties of an object and create some sort of function/pattern to match them to any arbitrary metric/basis/graph etc
So for instance a vector defined by it's slope and magnitude but in terms of how to translate the slope to any weirdly chosen coordinate system.
It can be higher rank than the space is its in. So the 2d vector can have a really complex tensor that helps it translate to something like 6d.
?
Mfw random retard skimming Google knows more than I do after taking tensor analysis in uni
>>7685130
>metric-independent
It has nothing to do with metrics in the first place.
A tensor is an element of a tensor space, which can be seen as being formed from basis vector
Tensor product is a purely algebraic construction. For any two R-modules (R being a ring), you can construct their tensor product which satisfies the universal property.
Thanks for all replies anons. U helped me a lot. Nothin like 4chan :)
>>7684487
Tension, apprehension and dissension have begun
>>7684652
What are vectors, senpai? I thought they were arrays of numbers. :^)
>>7684487
For everyone in this thread, I believe that this talks some about relations between tensors and physics and the typical tensor product of modules in commutative algebra. Please give this a read.
http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf
>>7685390
Yeah that 7th chapter was nice.
I learned about differential forms last spring, from a geometric perspective, and this fall I learned ring theory and module theory and tensor products of modules, and have been trying to connect the two ideas. Thanks.
>>7685324
HOW CAN YOU HAVE A QUARTIC ARRAY OF NUMBERS?
>>7686361
it's just an array of cube arrays :)
>>7684652
>gets trolled
>jumps at the opportunity to try assetting the superiority of a shitty major
You can't fake being this dumb
A tensor is a multilinear map of vectors and covectors into the reals (or complex numbers).
Tensors have a rank (k,l) if they map k covectors and l vectors. A matrix is a representation of a (0,2) tensor in a certain base, since it takes two vectors and maps them into the reals.
Since covectors are linear maps of vectors to the reals, and co-covectors (linear maps of covectors to the reals) are isomorphic to vectors, you can create a tensor of rank (k,l) by stitching together k vectors and l covectors.
And since vectors (covectors) are ultimately defined as directional derivatives on a manifold, by stiching them together you get the characteristic tensor equation for a change of coordinate bases, which I'm too lazy to write down here.
>>7686371
This guy gets it
>>7685328
They are elements of a vector space, vector space is a structure satisfying several axioms. Euclidian spaces satisfy these axioms, and so points in them are vectors (and arrays of numbers). The space does not have to have anything to do with numbers, though.
>>7684852
Topkek
>>7686371
thanks for explaining, so the two vectors need to share the same base?
what do you think of lambdoma?
>>7685324
You can represent that as a matrix moron.
>>7686371
At least one person on this site knows what they're talking about.
>>7684586
not JUST a matrix
a matrix of numbers with specific properties/constraints that make a few kinds of transformations trivial
>>7684583
What is alegbra?
Well it's a leg
[spoiler]bra[/spoiler]
>>7688066
>thanks for explaining, so the two vectors need to share the same base?
Unless you do higher level calculations that don't deal with the explicit representation of the tensor, yes.
