## Franklin's Notes

### Derivatives in affine spaces

Given a map $f:X\to X'$ between affine spaces , a derivative of $f$ at $x\in X$ is a linear map between the tangent spaces of $x\in X$ and $f(x)\in X'$, with the property that for any neighborhood $N$ of $0\in L(T_x X; T_{f(x)}X')$, there is a neighborhood $N'$ of $0\in T_x X$ such that for some $A\in N$, the following equality holds for all $t\in N'$: In other words, if the derivative exists, then $f(x+t)$ can be approximated by $f(x)+D_x f(t)$ with arbitrarily high accuracy (i.e. with an arbitrarily small error term $A$), so long as $t$ is sufficiently small. If it exists, the derivative is given by the formula

If $f$ is differentiable, or has a derivative everywhere in $X$, then we have a map $\hat D f:X\to L(T;T')$ mapping each $x\in X$ to the transformation $D_x f:T\to T'$. If $\hat D f$ is continuous, then $f$ is continuously differentiable or $C^1$, and if $\hat D f$ is differentiable then we say that $f$ is $C^2$, and so on. If $f$ is $C^k$ for all $k\in\mathbb N$, then we say that $f$ is $C^\infty$, or smooth. Note that the higher derivatives can be reformulated as functions $\hat D^k f:X\to L^k(T;T')$ by uncurrying.