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Why tensors? A beginner's perspective

mfaizan.github.io

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Re: Why tensors? A beginner's perspective

#51

Post by lalaithion »

edflsafoiewq wrote: post #48
lalaithion wrote: post #33

Earlier quoted context omitted.

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.

Sure, if you perform an arbitrary operation on a multidimensional array. But the same is true of any representation of any mathematical object. It makes no physical sense to take the sine of a mass, or two to the power of a length. But that doesn't mean that whenever someone says "oh, the mass of an object is a real number" I need to nitpick them.

Re: Why tensors? A beginner's perspective

#52

Post by cygx »

contravariant wrote: post #46
tagrun wrote: post #42

Earlier quoted context omitted.

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…

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.

Sure there are: Any basis of the underlying vector space(s) induces a basis of the tensor space. Components respective to some basis are coordinates. You can then investigate what happens to the induced basis (or rather, the respective components) under a basis transformation of the underlying vector space(s), which is where the "physicist's" definition of tensors originates.

Re: Why tensors? A beginner's perspective

#53

Post by joppy »

jhokanson wrote: post #31
hinkley wrote: post #29

Earlier 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…

Speaking of math pages on Wikipedia ... and math text more generally Is it just me or are we horrible at teaching advanced math? Where are the examples (with actual numbers)? Where is the motivation? Where are the pictures?

Wikipedia is a terrible place to learn advanced mathematics, for the reasons you raise (and more). There are lots of terrific short books, and many terrific lectures online.

Re: Why tensors? A beginner's perspective

#54

Post by contravariant »

cygx wrote: post #52
contravariant wrote: post #46

Earlier quoted context omitted.

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…

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. Sure there are: Any basis of the underlying vector space(s) induces a basis of the tensor space. Components respective to some basis are coordinates. You can then investigate what happens to the induced basis (or rather, the respective components) under a basis transformation of…

The components of a vector aren't the same as the coordinates physicists talk about when they're dealing with tensors. The components would be something like the value of the magnetic potential, or the local wind speed. The coordinates would be the location where that particular vector is 'anchored'.

A change of coordinates does indeed induce a change of basis, but a change of basis isn't really a change of coordinates. And strictly speaking some vector spaces don't really have an obvious basis (without invoking choice), so having a basis be a prerequisite for the definition is not ideal.

The whole requirement that a tensor is 'something that transforms like [...] under a coordinate transformation' is just how physicists have chosen to phrase that a vector bundle is only well defined if it's definition isn't dependent on some arbitrary choice of coordinates. In my opinion this requirement is more easily apparent in the mathematical definition where there is no choice of coordinates in the first place, rather than the physicists way of working with some choice of coordinates and checking how things transform.

Re: Why tensors? A beginner's perspective

#55

Post by Beldin »

The way I think of it: you have 0-dimensional arrays of numbers (plain numbers or scalars). You have 1-dimensional arrays of numbers (a list of N numbers or an N-vector). You have 2-dimensional arrays of numbers (an NxM matrix). We can extend this concept to 3- and 4-dimensional arrays and even further.

The kicker? All of them are tensors. Tensor is just a generalisation of the concept.

I am no licensed mathematician, so this could be off. However, every time I dive into this topic, I have to wade through way too complex mathnobabble to arrive at that notion. So let's keep it simple: tensors are a mathematician's template for arrays of any dimension.

Re: Why tensors? A beginner's perspective

#56

Post by bmitc »

billfruit wrote: post #10
bmitc wrote: post #7

Anyone interested in a visual exploration should checkout Geometrical Vectors by Gabriel Weinreich. https://www.maa.org/press/maa-reviews/geometrical-vectors

Is there any book that treats whole off geometry using vectors?

I’m not sure I understand the question enough to answer. Do you mean something like differential geometry? There, the theory is built upon vectors and covectors (i.e., differential forms) that are associated with tangent spaces and cotangent spaces, respectively. But that is modern differential geometry and not classical geometry.

Re: Why tensors? A beginner's perspective

#57

Post by cygx »

contravariant wrote: post #54
cygx wrote: post #52

Earlier quoted context omitted.

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. Sure there are: Any basis of the underlying vector space(s) induces a basis of the tensor space. Components respective to some basis are coordinates. You can then investigate what happens to the induced basis (or rather, the respective components) under a basis transformation of…

The components of a vector aren't the same as the coordinates physicists talk about when they're dealing with tensors. The components would be something like the value of the magnetic potential, or the local wind speed. The coordinates would be the location where that particular vector is 'anchored'. A change of coordinates does indeed induce a change of basis, but a change of basis isn't really a change of coordinat…

I'm aware. Though if we want to be more precise, that's about tensor fields, where the basis transformations of the underlying vector bundles (the tangent and cotangent bundle) are in turn induced by coordinate transformations of the base manifold.

However, physicists get introduced to tensors far earlier than any excursions into differential geometry when discussing rigid bodies.

Re: Why tensors? A beginner's perspective

#58

Post by contravariant »

cygx wrote: post #57
contravariant wrote: post #54

Earlier quoted context omitted.

The components of a vector aren't the same as the coordinates physicists talk about when they're dealing with tensors. The components would be something like the value of the magnetic potential, or the local wind speed. The coordinates would be the location where that particular vector is 'anchored'. A change of coordinates does indeed induce a change of basis, but a change of basis isn't really a change of coordinat…

I'm aware. Though if we want to be more precise, that's about tensor fields, where the basis transformations of the underlying vector bundles (the tangent and cotangent bundle) are in turn induced by coordinate transformations of the base manifold. However, physicists get introduced to tensors far earlier than any excursions into differential geometry when discussing rigid bodies.

Yes I'd also call those tensor fields. The main point I'm trying to make is that the tensor transformation law only makes sense for such fields.

Re: Why tensors? A beginner's perspective

#59

Post by kaashif »

ericphanson wrote: post #3

I 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).

It's not clear to me what you're annoyed about exactly. The way I see it, there are a few options:

You're getting annoyed that people are confusing the map with the territory [1]. Multidimensional arrays with certain properties can be used to represent tensors, but aren't tensors. In the same way a diagram of torus isn't a topological space, or a multiplication table isn't a group, or a matrix is not a linear map. Isomorphic but not literally the thing.

Or you're annoyed that people forget an array representing a tensor needs to satisfy some transformation law and can't just be any big array with some numbers in it.

Or maybe you're a fan of basis-free linear algebra!

Which one is it?

1: https://en.wikipedia.org/wiki/Map%E2%80%93territory_relation

Re: Why tensors? A beginner's perspective

#60

Post by cygx »

contravariant wrote: post #58
cygx wrote: post #57

Earlier quoted context omitted.

I'm aware. Though if we want to be more precise, that's about tensor fields, where the basis transformations of the underlying vector bundles (the tangent and cotangent bundle) are in turn induced by coordinate transformations of the base manifold. However, physicists get introduced to tensors far earlier than any excursions into differential geometry when discussing rigid bodies.

Yes I'd also call those tensor fields. The main point I'm trying to make is that the tensor transformation law only makes sense for such fields.

The terms co- and contravariant make sense on a purely algebraic basis, with components of tensors transforming 'the same as' or 'opposite to' the basis vectors. That the basis transformation is induced by transformations of some base manifold is incidental.
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