Curves
Now we can start coming back down to geometric earth. A smooth curve in a smooth manifold 
is nothing but a smooth map 
, where 
is some interval in the real line with its standard differentiable structure. The interval can, in principle, be half-infinite or infinite, but commonly we just consider finite open intervals like 
.
At any point 
of an open interval, the tangent space 
is one-dimensional. And, in fact, it comes equipped with a canonical vector to use as a basis: 
, the derivative operator at the point itself! Any other linear functions on germs at that satisfies a product rule must be a scalar multiple of this one.
Since we have a canonical tangent vector in , we can hit it with the derivative 
and see what happens. We get a tangent vector
which we call the tangent vector of 
at , and we write it as 
.
Let’s say that 
and let 
be a germ at 
. What does do to ? We can calculate:
&=\left[c_{*t_0}\left(\frac{d}{dt}(t_0)\right)\right](f)\\&=\left[\left(\frac{d}{dt}(t_0)\right)\right](f\circ c)\\&=\frac{d}{dt}\left(f\circ c\right)\bigg\vert_{t_0}\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Bc%27%28t_0%29%5Cright%5D%28f%29%26%3D%5Cleft%5Bc_%7B%2At_0%7D%5Cleft%28%5Cfrac%7Bd%7D%7Bdt%7D%28t_0%29%5Cright%29%5Cright%5D%28f%29%5C%5C%26%3D%5Cleft%5B%5Cleft%28%5Cfrac%7Bd%7D%7Bdt%7D%28t_0%29%5Cright%29%5Cright%5D%28f%5Ccirc+c%29%5C%5C%26%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Cleft%28f%5Ccirc+c%5Cright%29%5Cbigg%5Cvert_%7Bt_0%7D%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
That is, we pull the function back along to define a smooth real-valued function on the interval itself, then we hit it with the derivative operator and evaluate at .
If our curve lies with a coordinate patch 
— or if we cut out a segment of the curve that does — then we have a curve 
. We can also use 
to define a coordinate basis on 
, and thus get components of in those coordinates. As usual, we calculate the 
th component by
&=\frac{d}{dt}\left(x^i\circ c\right)\bigg\vert_{t_0}\\&=\frac{d}{dt}\left(u^i\circ(x\circ c)\right)\bigg\vert_{t_0}\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Bc%27%28t_0%29%5Cright%5D%28x%5Ei%29%26%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Cleft%28x%5Ei%5Ccirc+c%5Cright%29%5Cbigg%5Cvert_%7Bt_0%7D%5C%5C%26%3D%5Cfrac%7Bd%7D%7Bdt%7D%5Cleft%28u%5Ei%5Ccirc%28x%5Ccirc+c%29%5Cright%29%5Cbigg%5Cvert_%7Bt_0%7D%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
But this is just the derivative of the th component of the function 
. That is, when we’re working in local coordinates we get th coefficient of the tangent vector by taking the derivative of the th component function of the curve.
If you remember calculations like this in multivariable calculus, this is almost exactly why it works. There’s one other little caveat, though, that we’ll get to next time.
April 8, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
The Unapologetic Mathematician 
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