 Solution

The Fréchet derivative at a point $a$ is the unique linear map $D{f}_a:{\mathbb{R}}^n\to {\mathbb{R}}^m$ such thatThe error divided by $\Vert h\Vert$ tends to zero as $h\to 0$. In standard coordinates this linear map is represented by the Jacobian matrix, but the derivative itself is the map acting on increments, not merely a list of directional rates.