# The Unapologetic Mathematician

## Functoriality of the Derivative

We’ve said that the tangent bundle construction is a functor with the derivative as the action on morphisms. But we haven’t actually verified that it obeys the conditions of functoriality.

First off, if $f=1_M:M\to M$ is the identity map on a smooth manifold $M$, then $f_*:\mathcal{T}M\to\mathcal{T}M$ should be the identity map between the tangent bundles. That is, at each point $p$ we should have $f_{*p}:\mathcal{T}_pM\to\mathcal{T}_pM$ the identity map on this vector space. And indeed, if we let $(U,x)$ be any coordinate patch around $p$ we know that the matrix of $f_{*p}$ with respect to these local coordinates is the Jacobian of the coordinate function $x\circ1_M\circ x^{-1}=1_{x(U)}$. But this Jacobian is clearly the identity matrix, proving our claim.

More importantly, if $f:L\to M$ and $g:M\to N$ are two smooth maps, then their composition $g\circ f$ is also smooth. Given a point $p\in L$ we can define the derivatives $f_{*p}:\mathcal{T}_pL\to\mathcal{T}_{f(p)}M$, $g_{*f(p)}:\mathcal{T}_{f(p)}M\to\mathcal{T}_{g(f(p))}N$, and $(g\circ f)_{*p}\mathcal{T}_pL\to\mathcal{T}_{g(f(p))}N$. I say that $g_{*f(p)}\circ f_{*p}=(g\circ f)_{*p}$. And since this holds at every point we can write $(g\circ f)_*=g_*\circ f_*$, proving functoriality.

So, let’s take a vector $v\in\mathcal{T}_pL$ and see what happens. Taking a test function $\phi:N\to\mathbb{R}$ we calculate

\displaystyle\begin{aligned}\left[(g\circ f)_{*p}(v)\right](\phi)&=v\left(\phi\circ(g\circ f)\right)\\&=v\left((\phi\circ g)\circ f\right)\\&=\left[f_{*p}(v)\right](\phi\circ g)\\&=\left[g_{*f(p)}\left(f_{*p}(v)\right)\right](\phi)\\&=\left[\left[g_{*f(p)}\circ f_{*p}\right](v)\right](\phi)\end{aligned}

And so $(g\circ f)_{*p}=g_{*f(p)}\circ f_{*p}$, just as we claimed.

We should note, here, how this recalls the Newtonian notation for the chain rule, where we wrote $\left[g\circ f\right]'(p)=g'(f(p))f'(p)$. Of course, multiplication is changed into composition of linear maps, but that little detail will be cleared up soon (if you don’t already see it).