The Derivative
It turns out that the tangent bundle construction is actually a functor. Given a smooth map 
between smooth manifolds, we will get a smooth map 
. Yes, we’d usually write 
for a functor’s action on a map, but the 
notation is pretty classical.
So if we’re given a tangent vector 
we want to get a tangent vector 
. And since we already have 
sending points of 
to points of 
, it only makes sense to ask that 
. That is, in terms of the tangent bundle projection functions, we can write 
. In other words, the projection 
will be a natural transformation from the tangent bundle functor to the identity functor.
Anyway, this means that for each 
we’ll get a map 
. Since these are both vector spaces, it only stands to reason that we’d have a linear map. We haven’t yet established the connection between our “tangent vectors” and the geometric notion, but we do have a notion from multivariable calculus of a linear map that takes tangent vectors to tangent vectors: the Jacobian, which we saw as a certain extension of the notion of the derivative. We will find that our map is the analogue of the same concept on manifolds, and so we will call it the derivative of .
So here’s our definition: if 
is a differentiable map in some open set 
and if 
, then we define our map by
=v(\phi\circ f)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5Bf_%7B%2Ap%7D%28v%29%5Cright%5D%28%5Cphi%29%3Dv%28%5Cphi%5Ccirc+f%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[f_{*p}(v)\right](\phi)=v(\phi\circ f)")
where 
is any smooth function on a neighborhood of 
. That is, 
is a linear functional on 
; if 
represents a germ at 
we can compose it with to represent a germ at 
, and then we can apply 
itself to this germ. It should be immediately clear that this construction is linear in .
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April 6, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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