# User:Linus44

Things to do:

## Definition:Differential

Let $\struct {E, \norm {\, \cdot \,}_E}$, $\struct {F, \norm {\, \cdot \,}_F}$ 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 $\d f_a \in \map \LL {E, F}$ such that

$\ds \lim_{h \mathop \to 0} \norm {\map f {a + h} - \map f a - \d f_a \cdot h}_F \norm h_E^{-1} = 0$

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

We say that $f$ is continuously differentiable if and only if:

$\d f : \struct {U, \norm {\, \cdot \,}_E} \to \struct {\map \LL {E, F}, \norm {\, \cdot \,}_{\map \LL {E, F} } }$
$: a \mapsto \d f_a$

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