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Tensors, the geometric tool that solved Einstein's relativity problem

quantamagazine.org

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Re: Tensors, the geometric tool that solved Einstein's relativity problem

#91

Post by Koshkin »

mitthrowaway2 wrote: post #11
AnotherGoodName wrote: post #9

Earlier quoted context omitted.

For those without a strong math background but more of a programmers background; You know the matrices you work with in 2D or 3D graphics environments that you can apply to vectors or even other matrices to more easily transform (rotate, translate, scale)? Well tensors are the generalisation of this concept. If you’ve noticed 2D games transformation matrices seem similar (although much simpler) to 3D games transforma…

Is there a difference between a 4x4 matrix and a 4x4 tensor?

"4x4 tensor" is not a useful notion. A (4, 4)-rank tensor would make more sense...

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#92

Post by Koshkin »

AnotherGoodName wrote: post #18
mitthrowaway2 wrote: post #11

Earlier quoted context omitted.

Is there a difference between a 4x4 matrix and a 4x4 tensor?

A matrix is a subset of a tensor and the 4x4 matrix is absolutely a tensor. Tensors can be way more complex and do more but the 4x4 matrix you use in 3D operations is a great starting point.

A matrix is an array of numbers. It usually represents a linear map (in some basis). A tensor is more like that linear map.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#93

Post by Koshkin »

MathMonkeyMan wrote: post #14
max_likelihood wrote: post #3

I've always thought the use of "Tensor" in the "TensorFlow" library is a misnomer. I'm not too familiar with ML/theory, is there a deeper geometric meaning to the multi-dimensional array of numbers we are multiplying or is "MatrixFlow" a more appropriate name?

The joke I learned in a Physics course is "a vector is something that transforms like a vector," and "a tensor is something that transforms like a tensor." It's true, though. The physicist's tensor is a matrix of functions of coordinates that transform in a prescribed way when the coordinates are transformed. It's a particular application of the chain rule from calculus. I don't know why the word "tensor" is used in…

> something that transforms

Well, they don't, it is their components that do (under a change of the coordinate system).

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#94

Post by Koshkin »

mvaliente2001 wrote: post #59

The idea of tensors as "a matrix of numbers" or the example of a cube with vectors on every face never clicked for me. It was this (NASA paper)[ https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Ten... ] what finally brought me clarity. The main idea, as others already commented, is that a tensor or rank n is a function that can be applied up to n vector, reducing its rank by one for each vector it consumes.

But, in practice, how often do we ask a tensor to consume a vector?

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#95

Post by Koshkin »

nyrikki wrote: post #5

I would argue that today, geometric algebra/Clifford calculus and space time algebra are more intuitive and useful. Gibbs/Heavysides vectors were more popular at the time. At least for me.

Ah, been waiting for this. Now the discussion is complete. (At least for me.)

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#96

Post by cfgauss2718 »

Koshkin wrote: post #89
cfgauss2718 wrote: post #70

Earlier quoted context omitted.

Here here! Functions do not depend on your choice of coordinates, only the components of tensors do! I think this is why it’s important to keep covariance and contravariance in mind. While tensor(fields) do not depend on coordinates intrinsically, the way we represent them when doing calculations most certainly does, and this is usefully characterized by co/contravariance.

"Here here!" is indeed topological. "Hear, hear!", on the other hand, is conversational.

I love when grammatical mistakes become unintentional puns

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#97

Post by ndriscoll »

g15jv2dp wrote: post #81
xanderlewis wrote: post #79

Earlier quoted context omitted.

It’s not that abstract. You can literally write down the isomorphism: v —> [e_v: V* —> F, e_v(f) = f(v)].

Are you even trying to understand my point? Yes, in that direction, explicitly constructing an element of the bidual of V from an element of V is easy. To explicitly find an element of V from the element of the bidual, you need to choose a basis. Just try it, come on! Write down the inverse isomorphism. Let \alpha be an element of V**. Then v \in V such that f(v) = \alpha (where f is the isomorphism you wrote down) i…

```case class Bidual[V](v: V) { def apply(f: Dual[V]) = f(v) }

def unwrap[V](ff: Bidual[V]): V = ff.v```

There's both directions of the isomorphism explicitly defined in a programming language. No choice of basis needed to define the maps, only to prove that the constructor for Bidual really gives you all linear functionals on the dual.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#98

Post by enugu »

eigenket wrote: post #80
enugu wrote: post #76

Earlier quoted context omitted.

Finite dimension tensors is interesting both in physics(ex: mechanics, electromagnetism, general relativity) and mathematics(ex: representation theory, differential geometry). Infinite dimensions is also used in physics(quantum theory) and in mathematics (Operator Algebras, representation theory again).

I guess? Mostly the cool applications in physics and differential geometry are about tensor fields, which are more complicated than bare tensors. You could argue that they're talking about finite dimensional tensors but tensor fields are kinda a different object (at least subjectively, to me).

A tensor field is a finite dimensional object varying over space. The space of all tensor fields is infinite dimensional. An operator in QM is infinite dimensional at a single point itself, and in QFT we have fields of such operators.
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