Here is a really good resource for a beginner: https://grinfeld.org/books/An-Introduction-To-Tensor-Calculu...
Is that you Pavel?
Why tensors? A beginner's perspective
41–50 of 109 posts
Re: Why tensors? A beginner's perspective
#42That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…
It has nothing to do with tensor fields, uniform/constant tensors still obey the proper coordinate transformations, that's the defining property of any tensor. (With non-uniform tensor fields, covariant derivatives also pick up a correction, but that's a separate thing.)
TensorFlow "tensor"(and most other use of "tensor" in programmer jargon) is not a tensor at all, it's just a multidimensional array.
Re: Why tensors? A beginner's perspective
#43That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…
Thanks for this summary.
Even as a physicist I found it highly confusing when I got told in physics classes that a tensor is "just a thing (or object) that behaves like so under coordinate transformation". Like, what do you mean by "thing"? I have no intuition to this yet, I need it concise definitions! Fortunately I took a differential geometry class at the same time, which was really helpful.
Re: Why tensors? A beginner's perspective
#44My version is just: Tensors allow us to write data and operations on data in a way which does not depend on how we chose to represent them. For example, if I have a vector x in V and a map T from V to W, then I would like the truth of T(x)=y to be independent of how I represent T and x.
I like the concrete example from when I first used tensors in school. Stress in a block of concrete. You can choose any basis you like to represent the stresses and transform between them.
Whether or not the concrete block breaks under that stress obviously does not depend on your choice of basis or units, so your transformation rules had better reflect that reality.
Re: Why tensors? A beginner's perspective
#45That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…
It has nothing to do with tensor fields, uniform/constant tensors still obey the proper coordinate transformations, that's the defining property of any tensor. (With non-uniform tensor fields, covariant derivatives also pick up a correction, but that's a separate thing.) TensorFlow "tensor"(and most other use of "tensor" in programmer jargon) is not a tensor at all, it's just a multidimensional array.
What do you think the tensor product of finite dimensional vector spaces looks like?
Re: Why tensors? A beginner's perspective
#46That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…
It has nothing to do with tensor fields, uniform/constant tensors still obey the proper coordinate transformations, that's the defining property of any tensor. (With non-uniform tensor fields, covariant derivatives also pick up a correction, but that's a separate thing.) TensorFlow "tensor"(and most other use of "tensor" in programmer jargon) is not a tensor at all, it's just a multidimensional array.
Mathematicians would disagree with you there. There are no coordinates to transform in an ordinary tensor space and therefore no way for a tensor to be affected by such a transformation.
Matrices (or linear transformations in general) are important examples of tensors. There's a nice adjunction between tensor spaces A(x)B and the space of linear transformations B=>C given by:
Hom(A(x)B, C) = Hom(A, B=>C)
In the case of Tensorflow I think they do actually still talk about linear transformations of some kind so it's perfectly fine to call them tensors.
Re: Why tensors? A beginner's perspective
#47That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…
The post was also a poor explanation for someone doing modern physics. [edit: not true actually I should have read the rest of the post - it’s a good post]
Wald's approach in General Relativity is much better - he treats Tensors as a multilinear map from vectors and dual vectors to scalars.
He then derives the underlying coordinate transformaton rules, for the vector spaces used in differential geometry. But
Re: Why tensors? A beginner's perspective
#48I was happy to see that this article is actually talking about tensors, not just multidimensional arrays (which for some reasons are often called tensors by machine learning folks).
This is mostly a semantic argument, but I find this to be a very annoying perspective. Given a basis, there is a natural isomorphism between tensors of a certain type and multidimensional arrays of certain dimensions.
Of course there is, but if you perform an operation on a multidimensional array, there is no guarantee it corresponds to an operation on tensors, ie. the resulting tensor may depend on the basis.
Re: Why tensors? A beginner's perspective
#49That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…
The post was also a poor explanation for someone doing modern physics. [edit: not true actually I should have read the rest of the post - it’s a good post] Wald's approach in General Relativity is much better - he treats Tensors as a multilinear map from vectors and dual vectors to scalars. He then derives the underlying coordinate transformaton rules, for the vector spaces used in differential geometry. But
That’s the approach I used as well in the second half of the article - I just mentioned the transformation law in the beginning since that’s what most physics students encounter first.
Most of the article tries to provide some intuition behind why multilinear maps, which sound like a fairly abstract concept, might be relevant in physics. The key link being the importance of coordinate invariance.
I didn’t go into deriving the coordinate transforms from the multilinear map definition as I didn’t feel that it’d provide much better intuition, but I did mention the equivalence near the end.
Re: Why tensors? A beginner's perspective
#50Earlier quoted context omitted.
> author describes tensors as things behaving according to tensor transformation formula In grade school it drove me nuts when the homework required us to describe a word without using the word (or it’s Latinate siblings). And yet as an adult there are few enough weeks that go by where some grownup doesn’t try to pull that same trick. If you think developers are guilty of circular logic, check out some of the math pa…
I think a lot of circularity occurs in mathematics because we don't typically qualify our utterances when it can be implicitly understood. Eg "Numbers (formal) are those objects which behave like numbers (informal)."
Bertrand Russell turns over in grave.