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/sci/ - Science & Math

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4393689 No.4393689 [Reply] [Original]

So I was wondering how what a tensor is?

>> No.4393691

A matrix.

>> No.4393711

rank-0 tensors are scalars
rank-1 tensors are vectors
rank-2 tensors can be represented as d*d matrices where d is your number of dimensions
rank-3 tensors can be represented as a d*d*d object
and so on...

>> No.4393720

>>4393711
How would you go about finding the determinant of a 3d matrix?

>> No.4393727

>>4393720
I don't think that's a sensible operation to perform.

>> No.4393741

>>4393720
Determinants are only defined for square matrices.

>> No.4393854

so what type of tensors are in Einstein's field equations? i tried to study them for my summer school but i'd not covered the necessary maths and it went over my head.

>> No.4393866

Tensors are objects in the tensor product of an (S,R) bimodule and a left R-module

>> No.4393889

>>4393854
There's the

stress-energy tensor: a rank-2 tensor that describes the distribution of energy that causes gravity
Riemann tensor: a rank-4 tensor that describes the curvature of spacetime
Ricci tensor: a rank-2 tensor that can be computed from the Riemann tensor
Ricci scalar: a scalar computed from the Ricci tensor
Einstein tensor: also can be computed from the Riemann tensor, or from the Ricci tensor. This is what is set proportional to the stress-energy tensor in the Einstein field equations.

These are all tensors in 4-dimensional spacetime, so you will need to understand special relativity and 4-vectors first.

>> No.4393903

For >>4393854
Here's some more information:
http://math.ucr.edu/home/baez/gr/outline2.html

>> No.4393909

>>4393889
i wish my education system was better
I've only just started proper vectors, differential equations and groups and I'm 18 in further maths classes
got any useful tips of what areas to look into next for basic understanding of this?

>> No.4393920

>>4393903
cheers thats actually really good. all the help i can get is appreciated

>> No.4393928

Was it tensors that went over your head or Einstein's field equations?

>> No.4393944

Heh. My undergraduate research actually works with Tensors for object recognition (like robotic vision).

We have everything from lower dimensions in some form. Multiplication, determinant, inverse, transpose. Generalized eigenvalue is a bugger.

A lot of this is relatively new. In the past, we would sort of convert what would intuitively be tensors down to matrices. Like row scanning images, and making each image a column, instead of stacking them up into a tensor.

Doing tensor LDA mostly.

>> No.4393962

>>4393741
And everything works nicely for "cubic" tensors.

Nice powerpoint version of linear algebra tensors, as opposed to physics tensors. If you would like a more technical paper, I has links.
http://www.mcs.sdsmt.edu/kbraman/Research/CU-Denver08.pdf

>> No.4394000

>>4393928
both, we'd only ever covered up to 3x3 matrices at the time and i had to learn it by myself for a presentation in 2 days.
needless to say i waffled on about the implications of the theory, rather than the maths

>> No.4394011

a tensor is an object that stays the same under a change of coordinates.

>>4393909
An 18 year old isn't going to be familiar with maths behind general relativity you silly idiot.

>> No.4394029

>>4394000
Think he is just asking about the tensor as a linear algebra object. A 3d matrix.

>> No.4394037

>>4394011
i think you'll find i merely pointed out that my education system was woefully slow at introducing subjects, rather than demanding that i be taught to understand general relativity at this age
also, i couldnt learn all of it, but i don't see why a gifted individual couldn't learn the basics at such an age.
as you can see, i have asked for help understanding the basics required for this in later years - eigenvectors/abelian groups/partial differential equations. stuff that is the next step for me but obviously a long way off from the totality of the subject
is is such a crime to want to know more?

>> No.4394057

>>4393866
Well, at least one person here can actually give a definition.

>> No.4394075

>>4394037
It's a "crime" to criticise your education system for not introducing you to the maths required for general relativity when you're only 18.
>i tried to study them for my summer school but i'd not covered the necessary maths and it went over my head.
>i wish my education system was better

In terms of your scientific career, you're a zygote. I think you'll gain an appreciation of why I'm chiding you some time after you've started university.

>> No.4394078

>>4394037
I still think you are looking for the linear algebra version of tensors, since you keep spouting (fundamentally) math courses.

As for physics tensors, you probably need to actually learn some physics before even trying to understand their basics.

>> No.4394110
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4394110

Here is one intuitive view of what tensors are

1. A vector is a vector. If you don't know what a vector is, look it up.

2. A one-form is a linear function from a vector to a number.

3. A tensor is a linear function from N vectors and M one-forms to a number.

As it turns out that tensors transform according to certain rules when the frame of reference is changed. Some people use this fact as the definition of tensors but it is very unhelpful and unintuitive to beginners.

Pic related. The transformation age(woman)->woman is not linear.