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.

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

4 Comments »

  1. [...] have the identity map on this vector space. And indeed, if we let be any coordinate patch around we know that the matrix of with respect to these local coordinates is the Jacobian of the coordinate [...]

    Pingback by Functoriality of the Derivative « The Unapologetic Mathematician | April 7, 2011 | Reply

  2. [...] Notice now that this does not guarantee that itself is injective. For instance, if and , then we can form the mapping . Using the coordinates on and on , we can calculate the derivative in coordinates: [...]

    Pingback by Immersions and Embeddings « The Unapologetic Mathematician | April 18, 2011 | Reply

  3. [...] In components, this is , where the are the canonical coordinates on . We can easily calculate the derivative in these coordinates: . This is the zero function if and only if , and so has rank at any nonzero [...]

    Pingback by Spheres as Submanifolds « The Unapologetic Mathematician | April 25, 2011 | Reply

  4. [...] us the basic coordinate vector fields in the patch. If is a coordinate patch around , then we know how to calculate the derivative applied to these [...]

    Pingback by Isometries « The Unapologetic Mathematician | September 27, 2011 | Reply


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