From: mikko.levanto@iki.fi   
      
   On 2024-08-08 07:02:29 +0000, Stefan Ram said:   
      
   > moderator jt wrote or quoted:   
   >> calculus.  In this usage, these phrases describe how a vector (a.k.a   
   >> a rank-1 tensor) transforms under a change of coordintes: a tangent   
   >> vector (a.k.a a "contravariant vector") is a vector which transforms   
   >> the same way a coordinate position $x^i$ does, while a cotangent vector   
   >> (a.k.a a "covariant vector") is a vector which transforms the same way   
   >> a partial derivative operator $\partial / \partial x^i$ does.   
   >   
   >   Yeah, that explanation is on the right track, but I got to add   
   >   a couple of things.   
   >   
   >   Explaining objects by their transformation behavior is   
   >   classic physicist stuff. A mathematician, on the other hand,   
   >   defines what an object /is/ first, and then the transformation   
   >   behavior follows from that definition.   
   >   
   >   You got to give it to the physicists---they often spot weird   
   >   structures in the world before mathematicians do. They measure   
   >   coordinates and see transformation behaviors, so it makes sense   
   >   they use those terms. Mathematicians then come along later, trying   
   >   to define mathematical objects that fit those transformation   
   >   behaviors. But in some areas of quantum field theory, they still   
   >   haven't nailed down a mathematical description. Using mathematical   
   >   objects in physics is super elegant, but if mathematicians can't   
   >   find those objects, physicists just keep doing their thing anyway!   
   >   
   >   A differentiable manifold looks locally like R^n, and a tangent   
   >   vector at a point x on the manifold is an equivalence class v of   
   >   curves (in R^3, these are all worldlines passing through a point   
   >   at the same speed). So, the tangent vector v transforms like   
   >   a velocity at a location, not like the location x itself. (When   
   >   one rotates the world around the location x, x is not changed,   
   >   but tangent vectors at x change their direction.)   
   >   
   >   A /cotangent vector/ at x is a linear function that assigns a   
   >   real number to a tangent vector v at the same point x. The total   
   >   differential of a function f at x is actually a covector that   
   >   linearly approximates f at that point by telling us how much the   
   >   function value changes with the change represented by vector v.   
      
   For physicists' purposes this definition looks more asymmetric that   
   necessary. It is simple to postulate that there are vectros and   
   covectors and there is a multiplication of a covector and a vector,   
   and there is a relation that connects directions of vectors to   
   directions of curves at a point, and likewise drections of covectors   
   and directions of gradients of scalar fields. Of course the exact   
   details depend on what kind of space one wants to have and that   
   may depend on why one wants to have it.   
      
   >   When one defines the "canonical" (or "generalized") momentum as   
   >   the derivative of a Lagrange function, it points toward being a   
   >   covector.   
      
   Derivative of scalar with respect to position is a covector.   
   Canonical momentum is the derivative of Lagrangian with   
   respect to one of its arguments and that argument is not a position   
   but a velocity, which is a vector.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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