From: ram@zedat.fu-berlin.de   
      
   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.   
      
     When one defines the "canonical" (or "generalized") momentum as   
     the derivative of a Lagrange function, it points toward being a   
     covector. But I was confused because I saw a partial derivative   
     instead of a total differential. But possibly this is just a   
     coordinate representation of a total differential. So, broadly,   
     it's plausible that momentum is a covector, but I struggle   
     with the technical details and physical interpretation. What   
     physical sense does it make for momentum to take a velocity   
     and return a number? (Maybe that number is energy or action).   
      
     (In the world of Falk/Ruppel ["Energie und Entropie", Springer,   
     Berlin] it's just the other way round. There, they write   
     "dE = v dp". So, here, the speed v is something that maps   
     changes of momentum dp to changes of the energy dE. This   
     immediately makes sense because when the speed is higher   
     a force field is traveled through more quickly, so the same   
     difference in energy results in a reduced transfer of momentum.   
     So, transferring the same momentum takes more energy when the   
     speed is higher. Which, after all, explains while the energy   
     grows quadratic with the speed and the momentum only linearly.)   
      
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