Total derivative

TODO: I have decided to move strong derivative here and go with directional/total/partial terminology, as total/partial are common. As I need to rewrite the current strong page anyway, I've started here Alec (talk) 01:27, 15 November 2016 (UTC)

Definition

Let $(U,\Vert\cdot\Vert_U)$ and $(V,\Vert\cdot\Vert_V)$ be normed spaces over the same field (either both real or complex), let $A\in\mathcal{P}(U)$ be an arbitrary subset of $U$ and let $a\in A$ be a point such that $A$ is a neighbourhood of $a$ in $U$, then a map, $f:A\rightarrow V$ is "differentiable at $a$" if::

• there exists a linear map, $L:U\rightarrow V$ such that
  • $$\lim_{h\rightarrow 0}\left(\frac{\Vert f(a+h)-f(a)-L(v)\Vert_V}{\Vert h\Vert_U}\right)\eq 0$$ (i.e. the limit exists for some $L:U\rightarrow V$)

The linear map, $L$, is called the derivative of $f$ at $a$ and $f$ is said to be differentiable at $a$.

• Claim 1: the derivative of $f$ at $a$ (if it exists) is unique.

We denote the derivative of $f$ at $a$ by: $df\vert_a$.

If the vector spaces $U$ and $V$ are finite dimensional then recall all norms on finite dimensional vector spaces are equivalent and thus the choice of norm doesn't matter.

• See: For finite dimensional vector spaces the derivative at a point is independent of the norm

Alternative definition

There is a "remainder" such that $\lim_{h\rightarrow 0}\left(\frac{\Vert R(h)\Vert_V}{\Vert h\Vert_U}\right)\eq 0$