# User:Linus44

Things to do:

## Definition:Differential

Let $(E, \| \cdot \|_E)$, $(F, \| \cdot \|_F)$ be normed vector spaces.

Let $U \subseteq E$ be an open set.

Let $f : U \to F$ be a mapping.

Let $a \in U$ be an element of $U$.

Then $f$ is differentiable at $a$ if there exists a continuous and linear map $df_a \in \mathcal L(E,F)$ such that

$\displaystyle \lim_{h \to 0} \| f(a+h) - f(a) - df_a \cdot h \|_F \| h \|_E^{-1} = 0$

Then $df_a$ is called the differental or the tangent map of $f$ at $a$.

We say that $f$ is continuously differentiable if:

$\displaystyle df : (U, \| \cdot \|_E) \to \mathcal (L(E,F),\| \cdot \|_{L(E,F)})$
$\displaystyle \ : a \mapsto df_a$

is continuous.

If $E = \R^n$, this is true iff the first order partial derivatives of $f$ exist and are continuous.

## Induced polynomial homomorphism

Even this needs serious thought if it's to be any good.

## Permutations

Definition:Cyclic Permutation $k$ well defined. Add canonicality.

Incorrect: Definition:Permutation on n Letters/Cycle Notation permutation/cycle confusion? Also $\rho$ should be $\pi$ for consistency.

Then here: Equality of Cycles

## Rings, properties, equivalent definitions

etc...needs organizing into something more standardized