Part of the Poincaré Project.
We've already been introduced to linear spaces and three different types of linear space: n-tuples, arrow-type vectors and stack-type vectors.
In the last post, I promised to motivate stack-type vectors from the perspective of physics.
Both arrow-type vectors and stack-type vectors (in contrast to plain n-tuples) exist in relation to a manifold. From the point of view of (at least pre-relativistic) physics, the manifold introduces a notion of position and distance, but not other quantities such as time or mass or electrical charge, which exist in addition to the structure of the manifold.
We've previously stated that displacement, velocity, acceleration, force, etc can all be modeled as arrow-type vectors. The reason is that they are all fundamentally about a change in position. If you do a dimensional analysis—L, LT-1, LT-2, MLT-2—you see that they all have a single length dimension: no length squared; no per unit length.
Arrow-type vectors can't be used for area, energy or power (all of which have L2). Nor can they be used for quantities that are measured per unit area (L-2) or per unit volume (L-3 e.g. density which is ML-3).
Finally, arrow-type vectors cannot be used for quantities that are measured per unit length (L-1) but this is where stack-type vectors come in.
Borrowing the example from Gabriel Weinreich's wonderful book, Geometrical Vectors (the first book to set me straight on this stuff), consider two plates 1cm apart with a potential difference of 30V. We can model the distance between the plates as an arrow-type vector d. The electric field E between the plates is 30V/cm. E is not an arrow-type vector, but a stack-type vector.
One way to see this is the different behaviour of the numerical values of d and E under a change of scale. If we change our measurements from centimetres to metres, the numerical value of d goes from 1 to 0.01 but the numerical value of E goes from 30 to 3000.
This difference can also been seen in the fact that, when stretched, the magnitude of an arrow-type vector increases whereas the magnitude of a stack-type vector decreases.
Arrow-type vectors are about position and rates of change of position. Stack-type vectors, on the other hand, are about rates of change of some other quantity as position changes.
If you are familiar with gradient vectors from vector calculus, it should start to become clear that gradient vectors are not of the arrow-type but rather the stack-type.
In conclusion, while arrow-type vectors and stack-type vectors both follow the axioms for linear spaces, their behaviour in relation to the manifold they get their geometric interpretation from differs. It could almost be said they are opposites. Notice that while the numerical value of d decreased and that of E increased in our scaling example, if you multiple them together, they stay constant.
Exploring this last observation will be topic of the next post in the series.
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.