To define tensors as they are used in physics with the clarity and rigour that is used in mathematics, we would have to develop the notions of:

• duals of vector spaces
• manifolds
• vector bundles
• the tangent bundle
• tensors
• and finally, tensor bundles

This is quite a bit of work and it's not surprising then that physicists eschew all this and choose to stick to their traditional definition and which most likely helped motivate the discovery of the above notions.

Still, given the clarity it brings to the subject I think that this is a pity. Even more so when one realises how omnipresent the bundle picture is in gauge theories like Yang-Mills. However, to bring all this to the attention of physicists would mean reforming the physics curricula and this itself is no small bit of work.

Personally, I can't have been the first person to have noticed that a contravariant and covariant vector is simply not a vector, either in its axiomatic or geometric definition. Even without chasing the full curriculum reform that the subject deserves, it might be worth pointing out where these new concepts fit in with the traditional definition.

I also think it's worth pointing out that the categorical definition of a tensor makes it quite clear we don't need to think of tensors as multilinear functions - we can define them directly. This merely underlines that there are a number of mathematical technologies that have been teased out of the traditional physical picture.

edit

I'm not able to enter comments with the phone that I have at the moment. So I'm answering comments here.

@Bachtold: The geometric picture of vectors that is usually found in most classical mechanics texts, and in the description of a vector that Einstein described in his semi-popular book, The Evolution of Physics. And the axiomatic definition that is found in most introductory books in linear algebra. And if my high-school curricula in a comprehensive school is any guide, in most schools.

To define tensors as they are used in physics with the clarity and rigour that is used in mathematics, we would have to develop the notions of:

• the tensor product
• duals of vector spaces
• manifolds
• vector bundles
• the tangent bundle
• and finally, tensor bundles

This is quite a bit of work and it's not surprising then that physicists eschew all this and choose to stick to their traditional definition and which most likely helped motivate the discovery of the above notions.

Still, given the clarity it brings to the subject I think that this is a pity. Even more so when one realises how omnipresent the bundle picture is in gauge theories like Yang-Mills. However, to bring all this to the attention of physicists would mean reforming the physics curricula and this itself is no small bit of work.

Personally, I can't have been the first person to have noticed that a contravariant and covariant vector is simply not a vector, either in its axiomatic or geometric definition. Even without chasing the full curriculum reform that the subject deserves, it might be worth pointing out where these new concepts fit in with the traditional definition.

I also think it's worth pointing out that the categorical definition of a tensor makes it quite clear we don't need to think of tensors as multilinear functions - we can define them directly. This merely underlines that there are a number of mathematical technologies that have been teased out of the traditional physical picture.

edit

I'm not able to enter comments with the phone that I have at the moment. So I'm answering comments here.

@Bachtold: The geometric picture of vectors that is usually found in most classical mechanics texts, and in the description of a vector that Einstein described in his semi-popular book, The Evolution of Physics. And the axiomatic definition that is found in most introductory books in linear algebra. And if my high-school curricula in a comprehensive school is any guide, in most schools.

To define tensors as they are used in physics with the clarity and rigour that is used in mathematics, we would have to develop the notions of:

• duals of vector spaces
• manifolds
• vector bundles
• the tangent bundle
• tensors
• and finally, tensor bundles

This is quite a bit of work and it's not surprising then that physicists eschew all this and choose to stick to their traditional definition and which most likely helped motivate the discovery of the above notions.

Still, given the clarity it brings to the subject I think that this is a pity. Even more so when one realises how omnipresent the bundle picture is in gauge theories like Yang-Mills. However, to bring all this to the attention of physicists would mean reforming the physics curricula and this itself is no small bit of work.

Personally, I can't have been the first person to have noticed that a contravariant and covariant vector is simply not a vector, either in its axiomatic or geometric definition. Even without chasing the full curriculum reform that the subject deserves, it might be worth pointing out where these new concepts fit in with the traditional definition.

I also think it's worth pointing out that the categorical definition of a tensor makes it quite clear we don't need to think of tensors as multilinear functions - we can define them directly. This merely underlines that there are a number of mathematical technologies that have been teased out of the traditional physical picture.

To define tensors as they are used in physics with the clarity and rigour that is used in mathematics, we would have to develop the notions of:

• duals of vector spaces
• manifolds
• vector bundles
• the tangent bundle
• tensors
• and finally, tensor bundles

This is quite a bit of work and it's not surprising then that physicists eschew all this and choose to stick to their traditional definition and which most likely helped motivate the discovery of the above notions.

Still, given the clarity it brings to the subject I think that this is a pity. Even more so when one realises how omnipresent the bundle picture is in gauge theories like Yang-Mills. However, to bring all this to the attention of physicists would mean reforming the physics curricula and this itself is no small bit of work.

Personally, I can't have been the first person to have noticed that a contravariant and covariant vector is simply not a vector, either in its axiomatic or geometric definition. Even without chasing the full curriculum reform that the subject deserves, it might be worth pointing out where these new concepts fit in with the traditional definition.

I also think it's worth pointing out that the categorical definition of a tensor makes it quite clear we don't need to think of tensors as multilinear functions - we can define them directly. This merely underlines that there are a number of mathematical technologies that have been teased out of the traditional physical picture.

edit

I'm not able to enter comments with the phone that I have at the moment. So I'm answering comments here.

@Bachtold: The geometric picture of vectors that is usually found in most classical mechanics texts, and in the description of a vector that Einstein described in his semi-popular book, The Evolution of Physics. And the axiomatic definition that is found in most introductory books in linear algebra. And if my high-school curricula in a comprehensive school is any guide, in most schools.