He's not talking about abstract vector spaces. He's talking about the undergraduate vectors with arrows on them, the ones with the clunky cross product pseudo algebra. These are stupid and useless.

Well you need to understand vectors in order to understand vector spaces, and vector spaces are crucial in physics. In any event only (3) did I discuss vector spaces. I have no idea what you are calling "stupid and useless".

The undergraduate style vectors with little arrows on top, with their calculus of dot and cross products, is useless. It is restricted to 3d, it is no good for calculations. Try to reproduce Thomas's precession using undergraduate vector calculus: physics.stackexchange.com/questions/14751/…

For the vast majority of undergraduate physics, standard vectors/vector notation work extremely well in communicating concepts and preforming calculations. Even if you are breaking them into components, the idea of it being "vector-like" still remains.

@Bejamin--- You are only under that impression because the problems in the undergraduate curriculum have undergone a process of natural selection to reduce to the cases where vectors work. For example, how do you describe the rotation of a rigid ellipsoid? You don't! You just restrict yourself to symmetrical bodies in the undergraduate curriculum, where you don't need a tensor of inertia. How about the divergence of the nonlinearity in Navier Stokes? You can't express that either, too many contractions. Vectors are a 20th century butchering of quaternions. They should be shelved.

I believe tensors have a time and place, after students gain a solid physical intuition. Anyone who is having trouble understanding basic vectors will definitely be completely lost with tensors. Also for those interested in engineering, tensors are usually overkill for most things.

I don't agree about this. Consider Newton's laws in tensor form:

${d\over dt} p^i = - \partial_i \phi$

Undergraduate students immediately appreciate that this is three equations, because they can mentally substitute 1,2,3 for the indices and get three equations. Similarly, if you have angular momentum

$ L_i = \epsilon_{ijk}x^i p^k $

you can see immediately that it's just a set of three equations of the form xp_y -yp_x, etc. The cross product obscures this. In addition, the epsilon identity

$\epsilon_{ijk}\epsilon_{ipq} = \delta{jp}\delta{kq} - \delta_{jq}\delta_{kp}$

is obvious enough to carry in your head, and check metally, while the analogous cross product identities are impossible to verify.

The main point is that tensors allow you to write down covariant equations before you even know what covariant means, because they are already in a simple component form. You don't have to abstract out the notion of a "vector", you can think of three numbers! That means that the conceptual leap to thinking of a vector can be done at your own pace, without inhibiting understanding.

The example of Thomas precession is instructive. In Thomas's paper, he uses vector notation to express the amount of precession. This site got a question regarding the precession because somebody had a hard time reproducing the calculation. I sat down to reproduce it, and got bogged down in two seconds. After choosing components, I solved the problem in two seconds. This is completely typical of vector notation.

It's the shittiest calculus people teach.

Well, for the first example, I see no real difference between the use of subscripts and omitting them. Here is an example I think shows well the power of vectors:

F=q[E+v X b]

instantly you can tell many things about the direction of the force

if E=0, the direction of the force is perpendicular to v and B

even if we write it in the component notation, I find the ways pretty much equivalent

f_i = q [ E_i + e_{ijk} v^i B^k

i just think it is far more notational than anything else

Also I suppose it comes down to this:

I learned tensors and vectors

i believe strongly that it was good i learned vectors before tensors

and pedagogically i will support the standard physics curriculum in that regard.

Also effectively all resources that teach things about tensors assume knowledge of vectors

The example you give is interesting, because this is not how you would write the Lorentz force in the tensor world. You never use the epsilon tensor in the tensor world unless you have to. B is an antisymmetric tensor, not a vector, and this explains the otherwise mysterious property that it is unchanged under reflections in a point, unlike any other vector (it's a pseudovector). The Lorentz force is

$$ F_i = q E_i + B_{ij} v^j$$

This is important, because it is already four-dimensional, it is

Also, it is just as obvious that the motion is in the plane of B (notice that it is natural to think of B as a plane, not arrow), and perpendicular to v (since contracting B_{ij}v^j with v^i gives zero by antisymmetry).

The notion that B lies in planes resolves many intuitive paradoxes. For example, when you have a current in a wire, why does B go around the wire? The reason is that there is only one plane which is picked out by the wire, the plane spanned by the radial outward direction and the direction of the wire, and the B field must lie in this plane.

Similarly, the B plane for a solenoid is just the plane of symmetry of the solenoid.

Everything is obvious once you realize that B is an antisymmetric tensor. But if you represent it as a vector, you get intuitive nightmares.

About your first point--- the difference between keeping the subscripts and omitting them is that if you keep the subscripts, you can easily teach a student what the equation means---- it means plug in 1,2,3 for the subscripts! There is not a single student who would not understand what that involves. There is no need to think of taking an abstract "vector" and projecting out its "components". This type of abstraction is counterproductive.

The difference is immediate: undergraduates who write out all the indices will never, ever, get confused about what a vector equation means.

The book by Schutz teaches tensors without ever dealing with vectors. If you read Einstein's papers, or that era, this was when tensors were at their heyday, and vectors had not yet been developed! So it is completely wrong that there are no books that deal with tensors without vectors--- all the literature before the 1920s or thereabouts does this. Schutz does this in a modern book ( I learned from Schutz).

I believe that the real reasons people use vectors is to scare people away from physics. Physics in the cold war was a high status profession, and many people wanted in. Under such conditions, the curriculum develops barbed wire, which is designed to keep people out, in the form of useless calculi and counterproductive pedagogy. Physics is no longer a high-status profession, and it is time it acknowledged this fact by making its pedagogy clear, so that it can train people properly.