At 11:10 AM 6/12/2006, you wrote:
the same units. Since the coordinate system is another thing that should be problem dependent, even simple vectors like the position of a particle may have mixed units.
John Phillips
(and other comments on vectors, matrices, tensors etc.). I have no idea how practical this would be but I think a different take on this might resolve these problems, at least in principal. I would say that it is a mistake to view the elements of a vector (matrix, whatever -- for simplicity, I'll just refer to matrices henceforth) as having "units" per se. Rather a matrix is itself a quantity, just as a scalar is, but one with, for lack of a better term, structure. Just as a scalar can have associated units (type), so can a matrix. The "unit" in this case is structured in the same way as the quantity. One consequence, perhaps the most direct, for this "structure" in the quantity, is that there are accessor functions on that matrix that return scalar quantities with scalar units. To multiply two matrices they must be compatible, not only in shape but in other aspects of their structure. Given that the units (complete structure) of the two matrices are compatible, they can be multiplied together and the "unit" for the result is well defined. Once compatibility has been checked the unitless "values" for the quantities (e.g., old-fashioned matrices of floats) are numerically multiplied and the result is cast to the proper structured unit. This is just a matter of separating the abstract interface from the particular implementation. It is a conceptual error to automatically equate a position in 3-space as being interchangeable with an array of floats or doubles with one dimension and three elements representing orthogonal distances from an origin. The latter is how the concept might (or might not) be implemented. Topher Cooper