biVector forum

Dimensional Analysis and GA

In physics and engineering, one of our most reliable analytical tools is dimensional analysis. Briefly, by labeling numbers by their physical use (length, time, force, etc.) we can troubleshoot calculations by checking for consistent dimensions (i.e. physics is a “strongly typed” language). Also, we can often gain insight into relationships between quantities by looking for dimensionless combinations. Particularly, when teaching undergraduate students we’re always nagging them about properly labeling units and dimensions.

One thing that’s a little confusing in this regard about GA is that each grade of a multivector has a different dimensionality. This is particularly vexing when trying to parse geometric products for dimensional content.

Does anyone have suggestions on easy ways to track dimensionality in GA? For my own use I keep notes about the grades present in any expressions, but that seems to clunky and tedious to teach to students.

I would think that every dimension drop in a term of a geometric product is because a scalar square has cancelled out. Keeping the dimensionality of those cancellations (like meter squared or so) should still make things homogeneous from a dimensional analysis point of view.

In CGA, Anthony Lasenby makes sure explicitly that each of those vectors have a sensible unit anyway, are dimensionally uniform. That is a prerequisite for the above.

Yes, of course. Thank you for pointing that out. I was doing a mental shortcut by equating grades with powers of units but that can’t be correct. I think that if all quantities can be traced back to products of vectors then things will remain dimensionally consistent.