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 | 4 Comments

The Derivative

It turns out that the tangent bundle construction is actually a functor. Given a smooth map f:M\to N between smooth manifolds, we will get a smooth map f_*:\mathcal{T}M\to\mathcal{T}N. Yes, we’d usually write \mathcal{T}f for a functor’s action on a map, but the f_* notation is pretty classical.

So if we’re given a tangent vector v\in\mathcal{T}_pM we want to get a tangent vector f_*(v)\in\mathcal{T}_qN. And since we already have f sending points of M to points of N, it only makes sense to ask that q=f(p). That is, in terms of the tangent bundle projection functions, we can write f(\pi(v))=\pi(f_*(v)). In other words, the projection \pi:\mathcal{T}M\to M will be a natural transformation from the tangent bundle functor to the identity functor.

Anyway, this means that for each p\in M we’ll get a map f_{*p}:\mathcal{T}_pM\to\mathcal{T}_{f(p)}N. 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 f_* is the analogue of the same concept on manifolds, and so we will call it the derivative of f.

So here’s our definition: if f:U\to N is a differentiable map in some open set U\subseteq M and if p\in U, then we define our map f_{*p}:\mathcal{T}_pM\to\mathcal{T}_{f(p)}N by

\displaystyle\left[f_{*p}(v)\right](\phi)=v(\phi\circ f)

where \phi\in\mathcal{O}(V) is any smooth function on a neighborhood of f(p)\in N. That is, f_{*p}(v) is a linear functional on \mathcal{O}_{f(p)}; if \phi represents a germ at f(p) we can compose it with f to represent a germ at p, and then we can apply v itself to this germ. It should be immediately clear that this construction is linear in v.

April 6, 2011 Posted by | Differential Topology, Topology | 23 Comments

   

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