# The Unapologetic Mathematician

## 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 $j$th 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 $i$th partial derivative of the $j$th 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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