1. Covariant vs contravariant

Thanks gvk, I'm alerted. I'll need some time to think this over, before I post back. It's a very important issue.
2. Covariant vs contravariant

Hiya StatusX et all The two expressions for [pq,r] are equivalent, and this is independent of the metric signature. I had a little trouble interpreting the y-coordinate notation, but I think it makes sense. Assuming so, the formal proof goes as follows. We start with a set of basis...
3. Covariant vs contravariant

Let me tell a story. Its relevance will become apparent. When I had my first taste of analytic geometry, I got the idea that if (x,y) were the coordinates of a point, and a x + b y = 0 was the equation of a line, then (a,b) should be "the coordinates of a line". But...
4. Math Newb Wants to know what a Tensor is

Hiya mathwonk I was away for a few days, came back and settled down to sort out the relations between curvature and the several kinds of derivatives on amanifold; and I'm still going. Anyway, my apology for imprecision above. I think the following is correct, though I can't prove it...
5. Math Newb Wants to know what a Tensor is

Hiya, Peterdevis and mathwonk In the spirit of chroot's beautifully economical characterization of tensors, the Riemann curvature tensor is a machine which takes a bivector area and returns a rotation matrix. In other words, it's a (1,1)-tensor-valued 2-form. The motivation is "parallel...
6. Cross product question

matt grime writes: You're right, exterior products don't require a metric. However, inner products do, even if the metric is just the boring old sum-of-squares norm. Here's my reasoning. (1) How do you measure a vector (in a vector space V), i.e. resolve it to its components v^i? Answer...
7. Cross product question

lethe writes: You're right, I'm wrong: see http://courses.cs.vt.edu/~cs5485/notes/ch1-2/linalg.pdf [Broken] Sure. But I can quote MTW: "Contraction seals off two of the tensor's slots, reducing the rank by two." That has to include the contraction of the tensor product of a contravariant...
8. Cross product question

lethe writes: I'm afraid the terminology depends on context. The only ambiguity here is that sometimes only certain kinds of vector should have inner products formed with other kinds. Really, vectors should only have inner products formed with vectors from the dual space. When you have a...
9. Cross product question

Let me throw some more geometric intution into this. I'm going to argue that (1) the cross product defines an area in space, (2) you get the perpendicular vector only after you've defined what "perpendicular" means, (3) you get perpendiculars when you define an inner or dot product. (1) We...
10. 4-volume is a rank (0,0) tensor?

Hey, consistency in formalism should never go unrewarded. Turin you're right, it's a (0,4) tensor. If you built it up as dx^0 \wedge dx^1 \wedge dx^2 \wedge dx^3 , then it alternates, i.e. dx^1 \wedge dx^0 \wedge dx^2 \wedge dx^3 , or any other exchange of the 1-forms, multiplies the volume...
11. Differential forms. Why?

I've had my difficulties with this, but fundamentally it's not too hard. In the small limit, there are only derivatives and the geometry is necessarily linear: a vector space with its dual. Any point of a manifold, by definition must have a neighbourhood which can be mapped to an n-dimensional...