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Questions & Discussions on Vectors & relationships to differentia l geometry



Vectors live in Tangent Spaces to "space";

i.e. A vector is an element of the tangent space to your manifold in
question.

a) E.g. A velocity vector of a particle moving through manifold S, lies in
the tangent space to S at the location of the particle.

b)I take this to mean that the ordinary (and useful) pictures we draw for
students of velocity vectors, angular momentum vectors etc; are in some
sense incorrect since they are drawn in the picture of the manifold? They
are really pictures of two or more spaces superposed on top of each other.

a) and b) are rather easily visualized when you think of the motion of an
ant on the surface of the sphere. Plot its trajectory on the surface of the
sphere and then draw the velocity vector as we are all taught to do, tangent
to the trajectory and you see the arrow representing the velocity isn't in
the surface of the sphere but is rather tangent to it.

Where does a position vector reside?

Joel Rauber
Joel_Rauber@sdstate.edu

More questions to follow (perhaps)