On Jun 19, 2006, at 7:23 PM, Oleg Abrosimov wrote:
Gerhard Wesp wrote:
On Mon, Jun 19, 2006 at 10:04:21AM +0200, Matthias Troyer wrote:
As a physicist I am completely baffled and confused. What do you mean by rank of a quantity? Do you mean the size of a vector/matrix? If
I understood Olegs post about rank such that it was something that would allow to distinguish energy from torque, e.g. Is there such a thing? Off the top of my head, I cannot think of a situation where you might want to add energy to torque, even if in the SI system they have the same dimension (Nm).
Same thing for angular velocity [rad/s] and frequency (1/s). You probably don't want to add both, even if in SI they're both in s^-1. Is this maybe a deficiency of SI? Would it make sense to add the unit "radians" that is used colloquially anyway, to the system?
You are almost right, but in the end you've chosen a wrong direction.
There is a very good article in wikipedia about tensors in which tensor rank is also described: http://en.wikipedia.org/wiki/Tensor
The wrong direction is about deficiency in SI. The SI is all about units, rank is absolutely orthogonal concept to units (of course it deals not only with units, but also with dimensions, but it is natural because units are just scales in multidimensional space defined by dimensions). The truth is that angular velocity is (surprise!) a pseudovector. Its direction is the same as a direction of rotation axis. it is known to be involved in the following equation: velocity_vector = crossproduct(angular_velocity_pseudovector, radius_vector)
On the other hand, frequency is simply a scalar. So, here is the same situation as we have in energy vs torque "paradox" ;-)
Roots of such a problems lies in existence of two operations that produce the same dimension from given ones: 1) dot-product (results in a scalar, like energy) 2) cross-product (results in a vector or a pseudovector, like torque)
This problem can not be solved only inside dimensional analysis. I hope that it is clear from this post. The solution is to take into account the rank of quantity (rank is 0 -> scalar; rank is 1 -> vector, ...)
It can be implemented absolutely in the same manner as dimensional analysis: rank would be a compile time constant that is bound to quantity and all operations would be defined with respect to quantities' rank. It means, in particular, that dimensional vector _must_ be defined as: 1) length< vector<double> > l;
and not as: 2) vector< length<double> > l;
reason is simple - we should define operations for vectors (rank is one) that respect rank of it's results. For example: dot(v1, v2) -> scalar (rank is 0) of dim = dim1 * dim2 cross(v1, v2) -> vector (rank is 1) of dim = dim1 * dim2
I feel that this discussion about "rank" is trying to achieve too much. If I understand it correctly, one wants to prevent users from mixing quantities with the same units but different semantics, and to use the "rank" or some other identifier to disambiguate them. I would propose to completely drop this idea, since there are *many* more examples where semantics play a role. Just consider - Total energy of a system or energy per particle have the same units but care must be taken in adding them - Free energy, energy, and enthalpy have the same units but care must be taken in adding them - Price of a produce including or excluding tax has the same units, but but care must be taken in adding them - ... There are many cases where it does not make any sense to ad two quantities even if they have the same units. No matter how one extends the unit library, one will never be able to catch all of them. I thus think that the unit library should focus on unit checking and conversions and not attempt to prevent any possible user error, especially if this will make the library more complex and will require a completely new linear algebra library to be written Matthias