Chronology | Current Month | Current Thread | Current Date |
[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
That's not how it works. Each location in a vector field has
its own proprietary vector space. Different point, different
space. Within each vector /space/ the vector has magnitude and
direction but not location. The space as a whole has a location
within the field, but that's the answer to a different question.
The idea of a separate vector /space/ at each point in the
vector /field/ is particularly important and obvious when
you consider 2D vectors on the surface of a sphere. Each
vector space is /tangent/ to the sphere. A tangent vector
such as the velocity vector does not live in or on the
surface of the sphere; it lives in the tangent space. See
the second diagram (tangent space and tangent vector) here:
https://en.wikipedia.org/wiki/Tangent_space
I don't see any issues worth debating. The exterior derivativeTheplain old cross product AXB did not require any relocations. If A is position vector of a point in a solidbody with fixed center of mass, and I apply force B at A, then anyrelocation of B to another point A' is not only unnecessary but most ofthem are forbidden if we want to find the torque tau = AXB.The latest modifications towards outer products, bivectors etc., whilebeing useful, do not warrant universal cancellation of locality.
∇∧B would require a bit of work, but that's the answer to a
different question. The plain old wedge product between two
plain old vectors A∧B is child's play. Indeed the fact that
A and B can be freely relocated makes it /easier/ to construct
the parallelogram representing A∧B.