Derivatives in Coordinates
Let’s take the derivative and see what it looks like in terms of coordinates. Say we have a smooth manifold 
and a smooth map 
from an open subset of to another smooth manifold 
. If 
is any point, we define the derivative 
as before.
Now, if 
is a coordinate patch — even if there isn’t a single coordinate patch on the whole domain of 
we can restrict down to a coordinate patch containing 
— we get a basis of coordinate vectors at . Similarly, if 
is a coordinate patch around 
we get a basis of coordinate vectors at . We want to write down the matrix of 
in terms of these two bases.
So, the obvious path is to take one of the coordinate vectors at , hit it with , and write the result out in terms of the coordinate vectors at . The generic problem, then, is to calculate the 
th component — the one corresponding to 
— of 
. But we know that this coefficient comes from sticking 
into this vector and seeing what pops out!
&=\left[\frac{\partial}{\partial x^i}(p)\right](y^j\circ f)\\&=D_i\left(y^j\circ f\circ x^{-1}\right)\\&=D_i\left(u^j\circ(y\circ f\circ x^{-1})\right)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Bf_%7B%2Ap%7D%5Cleft%28%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ei%7D%28p%29%5Cright%29%5Cright%5D%28y%5Ej%29%26%3D%5Cleft%5B%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ei%7D%28p%29%5Cright%5D%28y%5Ej%5Ccirc+f%29%5C%5C%26%3DD_i%5Cleft%28y%5Ej%5Ccirc+f%5Ccirc+x%5E%7B-1%7D%5Cright%29%5C%5C%26%3DD_i%5Cleft%28u%5Ej%5Ccirc%28y%5Ccirc+f%5Ccirc+x%5E%7B-1%7D%29%5Cright%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[f_{*p}\left(\frac{\partial}{\partial x^i}(p)\right)\right](y^j)&=\left[\frac{\partial}{\partial x^i}(p)\right](y^j\circ f)\\&=D_i\left(y^j\circ f\circ x^{-1}\right)\\&=D_i\left(u^j\circ(y\circ f\circ x^{-1})\right)\end{aligned}")
We’re taking the 
th partial derivative of the th component of the function 
, which goes from the open set 
into 
, where 
and 
are the dimensions of and , respectively. Like we saw for coordinate transforms in place, this is just the Jacobian again.
So if we want to write out the derivative in terms of local coordinates, we first write out our local coordinate version of as a function from one Euclidean space to another, and then we take the Jacobian of that function at the appropriate point.
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April 6, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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