scineram,
May I know your specialty?
I showed that it is not the result of a cross product that is a pseudovector but the cross product itself.
This is a bunch of doublespeak.
Metus, I think part of your confusion comes from focusing too much on mathematical notation. In physics, quantities like angular momentum are considered to be real and not just sets of numbers. A velocioty might be denoted by (1,0,0). An angular momentum might be denoted by (-1,0,0). However, these are fundamentally different objects. You cannot add these to obtain another vector, just as you cannot add the number 3 and an apple to obtain anything. Yes, you can add the numeric components together but the result is meaningless and can’t be transformed. It is revealed to be meaningless as soon as one chooses to assign a different set of numbers to spatial locations (erect a coordinate system).
Is (0,0,1) polar or axial? Is (1,0,0) x (0,1,0) = (0,0,1) polar or axial?
There is no confusion. I am purposefully focusing on the mathematical notation since the mathematical notation stands for certain logical concepts that the physical concepts are believed to have. Funny thing is that you can formalize your argument neatly by introducing different isomorphic vector spaces. Still does not change anything about my argument. Of course it is senseless to add up a velocity and an angular moment.
Is (0,0,1) polar or axial? Is (1,0,0) x (0,1,0) = (0,0,1) polar or axial?
(0,0,1) by itself is a bunch of symbols. It’s meaning is ambiguous, because the notation is deficient.
(1,0,0)x(0,1,0)=(0,0,1) could mean Polar x Polar = Axial. Or PxA = P or AxP = P or AxA = A.
If (1,0,0) stands for 1 apple, 0 bananas, and 0 oranges, then it’s neither a polar nor an axial vector and the equation is meaningless.
Obviously, the notation doesn’t express the full meaning needed in real life problems. It conflates the “isomorphic spaces” you referred to. It lacks units, such as distance or time.
Edit: mathematics is full of sloppy notation. There is sloppiness when you write f(x)=x^2. Is f a function or is it a variable being multiplied by x? Is x a free variable or is this stating something regarding the value of f at a particular constant, x? Does “sin pi x + cos pi x” mean “(sin(pi))x + (cos(pi))x=-x”? The notation is inadequate so common sense (e.g. context and tradition), have to be relied upon.
It is not ambiguous. Somewhere above I said I was talking about real vector spaces. Ultimately, you are avoiding the question. (1,0,0), (0,1,0) and (0,0,1) are by all means polar because they transform like polar vectors but (1,0,0) x (0,1,0) = (0,0,1) is allegedly axial.
In principle there can not be ultimate unambiguity. However we can try not to use confusing terms. That is my whole argument.
"It is not ambiguous. Somewhere above I said I was talking about real vector spaces. Ultimately, you are avoiding the question. (1,0,0), (0,1,0) and (0,0,1) are by all means polar because they transform like polar vectors"
That’s nonsense*.*
There’s no such thing as a transformation rule for “(1,0,0)”.
Just like there’s no transformation rule for “42”. It could be a scalar or a pseudoscalar.
A bunch of numbers cannot be polar or axial. Only things that related to actual physical space can be polar or axial. If I use (1,0,0) to indicate that I have 1 apple, 0 bananas, and 0 oranges, then it is not polar or axial. In fact it does not transform at ALL under physical rotation or reflection.
Do tensors have a transformation law? Is a polar vector a tensor?
Anyway, now it is just silly. I made it clear that I am talking about real vectors. Talking about velocities, oranges or anything is meaningless when we are talking about the mathematical framework.
Do tensors have a transformation law?
Yes http://en.wikipedia.org/wiki/Covariant_transformation
Is a polar vector a tensor?
Yes, it’s a (0,1) tensor.
I am talking about real vectors… about the mathematical framework.
OK, let’s talk about your real, mathematical vectors.
You claim that (1,0,0) is polar.
If I establish a basis in the dual space, I can denote a member of that dual space by (1,0,0). Is that also polar?
I meant real as in body of real numbers. There is nothing to talk about them.
A base to a vector is always in the same space as the vector, obviously, so I am not sure I understand where this is going. Ultimately you will have to claim that the cross product maps two polar vectors (or contravariant vectors or elements of the vector space) to an axial vector (or covariant vector or element of the dual space) since the cross product is apparently axial. I see now that we are fudging the mathematics even a bit more.
Edit: I just want to make clear what we are discussing. My original point was that the wording “pseudovector” for an axial vector is misleading. Further I argue that the notion of polar and axial is meaningful since it represents properties of physical objects. But their use is erroneous. Finally I suspect that the notion of axial and polar is not equivalent to contra- and covariant, that is being an element of vector or dual space. It all breaks down to bad mathematics.
I am applied math undergrad.
Tensors were taught to us badly back then. I never knew they were what or of what use. I only understood them later, after reading about the abstract approach on my own.