The Unapologetic Mathematician

Mathematics for the interested outsider

Derivatives in Coordinates

Let’s take the derivative and see what it looks like in terms of coordinates. Say we have a smooth manifold M and a smooth map f:U\to N from an open subset of M to another smooth manifold N. If p\in U is any point, we define the derivative f_{*p}:\mathcal{T}_pM\to\mathcal{T}_{f(p)}N as before.

Now, if (U,x) is a coordinate patch — even if there isn’t a single coordinate patch on the whole domain of f we can restrict f down to a coordinate patch containing p — we get a basis of coordinate vectors at p. Similarly, if (V,y) is a coordinate patch around f(p) we get a basis of coordinate vectors at f(p). We want to write down the matrix of f_{*p} in terms of these two bases.

So, the obvious path is to take one of the coordinate vectors at p, hit it with f_{*p}, and write the result out in terms of the coordinate vectors at f(p). The generic problem, then, is to calculate the jth component — the one corresponding to \frac{\partial}{\partial y^j}(f(p)) — of f_{*p}\left(\frac{\partial}{\partial x^i}(p)\right). But we know that this coefficient comes from sticking y^j into this vector and seeing what pops out!

\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 ith partial derivative of the jth component of the function y\circ f\circ x^{-1}, which goes from the open set x(U)\in\mathbb{R}^m into \mathbb{R}^n, where m and n are the dimensions of M and N, respectively. Like we saw for coordinate transforms in place, this is just the Jacobian again.

So if we want to write out the derivative f_{*p} in terms of local coordinates, we first write out our local coordinate version of f as a function from one Euclidean space to another, and then we take the Jacobian of that function at the appropriate point.

April 6, 2011 - Posted by | Differential Topology, Topology


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